morfismos, vol 19, no 1, 2015

VOLUMEN 19NÚMERO 1

ENERO A JUNIO DE 2015 ISSN: 1870-6525

Chief Editors - Editores Generales

• Isidoro Gitler • zelaznoGsuseJ

Associate Editors - Editores Asociados

• Ruy Fabila • zednanreHleamsI• amreL-zednanreHomisenO • Hector Jasso Fuentes

• Sadok Kallel • Miguel Maldonado• Carlos Pacheco • Enrique Ramırez de Arellano

• Enrique Reyes • Dai Tamaki• Enrique Torres Giese

Apoyo Tecnico

• zehcnaSadnarAanairdA • Irving Josue Flores Romero• zelaznoGsetneuFoinotnAocraM • oczorOzednanreHramO

• Roxana Martınez • Laura Valencia

Morfismos noicceridalneelbinopsidatse http://www.morfismos.cinvestav.mx.Para mayores informes dirigirse al telefono +52 (55) 5747-3871. Toda corres-

-ametaMedotnematrapeD,aicnelaVaruaL.arSalaadigiridriebedaicnednopticas del Cinvestav, Apartado Postal 14-740, Mexico, D.F. 07000, o por correo

:noicceridalaocinortcele [email protected].

VOLUMEN 19NÚMERO 1

ENERO A JUNIO DE 2015ISSN: 1870-6525

noicacilbupanuse,5102oinujaorene,1oremuN,91nemuloV,somsfiroMsodaznavAsoidutsEedynoicagitsevnIedortneCleropadatidelartsemes

del Instituto Politecnico Nacional (Cinvestav), a traves del DepartamentoordePnaS.loC,8052.oNlanoicaNocincetiloPotutitsnI.vA.sacitametaMed,00837475-55.leT,.F.D,06370.P.C,oredaM.AovatsuGnoicageleD,ocnetacaZ

www.cinvestav.mx, [email protected], Editores Generales: Drs.sohcereDedavreseR.sorraBonipsEzelaznoGsuseJyreltiGorodisI

No. 04-2012-011011542900-102, ISSN: 1870-6525, ambos otorgados por elInstituto Nacional del Derecho de Autor. Certificado de Licitud de TıtuloNo. 14729, Certificado de Licitud de Contenido No. 12302, ambos otorga-

aledsadartsulIsatsiveRysenoicacilbuPedarodacfiilaCnoisimoCalropsodledsacitametaMedotnematrapeDleroposerpmI.noicanreboGedaıraterceS

Cinvestav, Avenida Instituto Politecnico Nacional 2508, Colonia San PedronerimirpmiedonimretesoremunetsE.F.D,ocixeM,06370.P.C,ocnetacaZ

julio de 2015 con un tiraje de 50 ejemplares.

Las opiniones expresadas por los autores no necesariamente reflejan la.noicacilbupaledserotidesoledarutsop

-nocsoledlaicrapolatotnoiccudorperaladibihorpetnematcirtseadeuQlednoicazirotuaaiverpnis,noicacilbupaledsenegamiesodinet Cinvestav.

Information for Authors

The Editorial Board of Morfismos calls for papers on mathematics and related areas tobe submitted for publication in this journal under the following guidelines:

• Manuscripts should fit in one of the following three categories: (a) papers covering thegraduate work of a student, (b) contributed papers, and (c) invited papers by leadingscientists. Each paper published in Morfismos will be posted with an indication ofwhich of these three categories the paper belongs to.

• Papers in category (a) might be written in Spanish; all other papers proposed forpublication in Morfismos shall be written in English, except those for which theEditoral Board decides to publish in another language.

• All received manuscripts will be refereed by specialists.

• In the case of papers covering the graduate work of a student, the author shouldprovide the supervisor’s name and affiliation, date of completion of the degree, andinstitution granting it.

• Authors may retrieve the LATEX macros used for Morfismos through the web sitehttp://www.math.cinvestav.mx, at “Revista Morfismos”. The use by authors of thesemacros helps for an expeditious production process of accepted papers.

• All illustrations must be of professional quality.

• Authors will receive the pdf file of their published paper.

• Manuscripts submitted for publication in Morfismos should be sent to the email ad-dress [email protected].

Informacion para Autores

El Consejo Editorial de Morfismos convoca a proponer artıculos en matematicas y areasrelacionadas para ser publicados en esta revista bajo los siguientes lineamientos:

• Se consideraran tres tipos de trabajos: (a) artıculos derivados de tesis de grado dealta calidad, (b) artıculos por contribucion y (c) artıculos por invitacion escritos porlıderes en sus respectivas areas. En todo artıculo publicado en Morfismos se indicarael tipo de trabajo del que se trate de acuerdo a esta clasificacion.

• Los artıculos del tipo (a) podran estar escritos en espanol. Los demas trabajos deberanestar redactados en ingles, salvo aquellos que el Comite Editorial decida publicar enotro idioma.

• Cada artıculo propuesto para publicacion en Morfismos sera enviado a especialistaspara su arbitraje.

• En el caso de artıculos derivados de tesis de grado se debe indicar el nombre delsupervisor de tesis, su adscripcion, la fecha de obtencion del grado y la institucionque lo otorga.

• Los autores interesados pueden obtener el formato LATEX utilizado por Morfismos enel enlace “Revista Morfismos” de la direccion http://www.math.cinvestav.mx. La uti-lizacion de dicho formato ayudara en la pronta publicacion de los artıculos aceptados.

• Si el artıculo contiene ilustraciones o figuras, estas deberan ser presentadas de formaque se ajusten a la calidad de reproduccion de Morfismos.

• Los autores recibiran el archivo pdf de su artıculo publicado.

• Los artıculos propuestos para publicacion en Morfismos deben ser dirigidos a la di-reccion [email protected].

Editorial Guidelines

Morfismos is the journal of the Mathematics Department of Cinvestav. Oneof its main objectives is to give advanced students a forum to publish their earlymathematical writings and to build skills in communicating mathematics.

Publication of papers is not restricted to students of Cinvestav; we want to en-courage students in Mexico and abroad to submit papers. Mathematics researchreports or summaries of bachelor, master and Ph.D. theses of high quality will beconsidered for publication, as well as contributed and invited papers by researchers.All submitted papers should be original, either in the results or in the methods.The Editors will assign as referees well-established mathematicians, and the accep-tance/rejection decision will be taken by the Editorial Board on the basis of thereferee reports.

Authors of Morfismos will be able to choose to transfer copy rights of theirworks to Morfismos. In that case, the corresponding papers cannot be consideredor sent for publication in any other printed or electronic media. Only those papersfor which Morfismos is granted copyright will be subject to revision in internationaldata bases such as the American Mathematical Society’s Mathematical Reviews, andthe European Mathematical Society’s Zentralblatt MATH.

Morfismos

Lineamientos Editoriales

Morfismos, revista semestral del Departamento de Matematicas del Cinvestav,tiene entre sus principales objetivos el ofrecer a los estudiantes mas adelantadosun foro para publicar sus primeros trabajos matematicos, a fin de que desarrollenhabilidades adecuadas para la comunicacion y escritura de resultados matematicos.

La publicacion de trabajos no esta restringida a estudiantes del Cinvestav; de-seamos fomentar la participacion de estudiantes en Mexico y en el extranjero, asıcomo de investigadores mediante artıculos por contribucion y por invitacion. Losreportes de investigacion matematica o resumenes de tesis de licenciatura, maestrıao doctorado de alta calidad pueden ser publicados en Morfismos. Los artıculos apublicarse seran originales, ya sea en los resultados o en los metodos. Para juzgaresto, el Consejo Editorial designara revisores de reconocido prestigio en el orbe in-ternacional. La aceptacion de los artıculos propuestos sera decidida por el ConsejoEditorial con base a los reportes recibidos.

Los autores que ası lo deseen podran optar por ceder a Morfismos los derechos depublicacion y distribucion de sus trabajos. En tal caso, dichos artıculos no podranser publicados en ninguna otra revista ni medio impreso o electronico. Morfismossolicitara que tales artıculos sean revisados en bases de datos internacionales como loson el Mathematical Reviews, de la American Mathematical Society, y el ZentralblattMATH, de la European Mathematical Society.

Morfismos

Contents - Contenido

Real projective space as a space of planar polygons

Donald M. Davis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Motion planning in tori revisited

Jesus Gonzalez, Barbara Gutierrez, Aldo Guzman, Cristhian Hidber, MarıaMendoza, and Christopher Roque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Toeplitz operators with piecewise quasicontinuous symbols

Breitner Ocampo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Morfismos, Vol. 19, No. 1, 2015, pp. 1–6

Real projective space as a space of planar

polygons ∗

Donald M. Davis

Abstract

We describe an explicit homeomorphism between real projectivespace RPn−3 and the space Mn,n−2 of all isometry classes of n-gons in the plane with one side of length n−2 and all other sides oflength 1. This makes the topological complexity of real projectivespace more relevant to robotics.

2010 Mathematics Subject Classification: 58D29, 55R80, 70G40, 51N20.Keywords and phrases: Topological complexity, robotics, planar polygonspaces.

1 Introduction

The topological complexity, TC(X), of a topological spaceX is, roughly,the number of rules required to specify how to move between any twopoints of X ([4]). This is relevant to robotics if X is the space of allconfigurations of a robot.

A celebrated theorem in the subject states that, for real projectivespace RPn with n = 1, 3, or 7, TC(RPn) is 1 greater than the dimensionof the smallest Euclidean space in which RPn can be immersed ([5]).This is of interest to algebraic topologists because of the huge amountof work that has been invested during the past 60 years in studyingthis immersion question. See, e.g., [6], [9], [1], and [2]. In the populararticle [3], this theorem was highlighted as an unexpected applicationof algebraic topology.

∗Invited paper.

1


Real projective space as a space of planar

polygons ∗

Donald M. Davis

Abstract

We describe an explicit homeomorphism between real projectivespace RPn−3 and the space Mn,n−2 of all isometry classes of n-gons in the plane with one side of length n−2 and all other sides oflength 1. This makes the topological complexity of real projectivespace more relevant to robotics.

2010 Mathematics Subject Classification: 58D29, 55R80, 70G40, 51N20.Keywords and phrases: Topological complexity, robotics, planar polygonspaces.

1 Introduction

The topological complexity, TC(X), of a topological spaceX is, roughly,the number of rules required to specify how to move between any twopoints of X ([4]). This is relevant to robotics if X is the space of allconfigurations of a robot.

A celebrated theorem in the subject states that, for real projectivespace RPn with n = 1, 3, or 7, TC(RPn) is 1 greater than the dimensionof the smallest Euclidean space in which RPn can be immersed ([5]).This is of interest to algebraic topologists because of the huge amountof work that has been invested during the past 60 years in studyingthis immersion question. See, e.g., [6], [9], [1], and [2]. In the populararticle [3], this theorem was highlighted as an unexpected applicationof algebraic topology.

∗Invited paper.

1

2 Donald M. Davis

But, from the definition of RPn, all that TC(RPn) really tells ishow hard it is to move efficiently between lines through the origin inRn+1, which is probably not very useful for robotics. Here we showexplicitly how RPn may be interpreted to be the space of all polygonsof a certain type in the plane. The edges of polygons can be thought ofas linked arms of a robot, and so TC(RPn) can be interpreted as tellinghow many rules are required to tell such a robot how to move from anyconfiguration to any other.

Let Mn,r denote the moduli space of all oriented n-gons in the planewith one side of length r and the rest of length 1, where two suchpolygons are identified if one can be obtained from the other by anorientation-preserving isometry of the plane. These n-gons allow sidesto intersect. Since any such n-gon can be uniquely rotated so thatits r-edge is oriented in the negative x-direction, we can fix verticesx0 = (0, 0) and xn−1 = (r, 0) and define

(1) Mn,r = (x1, . . . ,xn−2) : d(xi−1,xi) = 1, 1 ≤ i ≤ n− 1.

Here d denotes distance between points in the plane.

Most of our work is devoted to proving the following theorem.

Theorem 1.1. If n − 2 ≤ r < n − 1, then there is a Z/2-equivarianthomeomorphism Φ : Mn,r → Sn−3, where the involutions are reflectionacross the x-axis in Mn,r, and the antipodal action in the sphere.

Taking the quotient of our homeomorphism by the Z/2-action yieldsour main result. It deals with the space Mn,r of isometry classes ofplanar (1n−1, r)-polygons. This could be defined as the quotient of (1)modulo reflection across the x-axis.

Corollary 1.2. If n − 2 ≤ r < n − 1, then Mn,r is homeomorphic toRPn−3.

These results are not new. It was pointed out to the author afterpreparation of this manuscript that the result is explicitly stated in [8,Example 6.5], and proved there, adapting an argument given much ear-lier in [7]. The result of our Corollary 1.2 was also stated as “well known”in [10]. Nevertheless, we feel that our explicit, elementary homeomor-phism may be of some interest.

Real projective space 3

2 Proof of Theorem 1.1

In this section we prove Theorem 1.1. Let Jm denote the m-fold Carte-sian product of the interval [−1, 1], and S0 = ±1. Our model forSn−3 is the quotient of Jn−3×S0 by the relation that if any componentof Jn−3 is ±1, then all subsequent coordinates are irrelevant. That is,if ti = ±1, then

(2) (t1, . . . , ti, ti+1, . . . , tn−2) ∼ (t1, . . . , ti, t′i+1, . . . , t

′n−2)

for any t′i+1, . . . , t′n−2. This is just the iterated unreduced suspension of

S0, and the antipodal map is negation in all coordinates. An explicithomeomorphism of this model with the standard Sn−3 is given by

(t1, . . . , tn−2) ↔ (x1, . . . , xn−2),

with

xi = ti

i−1∏j=1

√1− t2j , ti =

xi√1− x21 − · · · − x2i−1

ifi−1∑j=1

x2i < 1.

Then ti = ±1 for the smallest i for which x21 + · · ·+ x2i = 1.Let P ∈ Mn,r be a polygon with vertices xi as in (1). We will define

the coordinates ti = φi(P) of Φ(P) under the homeomorphism Φ ofTheorem 1.1.

For 0 ≤ i ≤ n− 2, we have

(3) n− 2− i ≤ d(xi,xn−1) ≤ n− 1− i.

The first inequality follows by induction on i from the triangle inequalityand its validity when i = 0. The second inequality also uses the triangleinequality together with the fact that you can get from xi to xn−1 byn − 1 − i unit segments. The second inequality is strict if i = 0 andis equality if i = n − 2. Let i0 be the minimum value of i such thatequality holds in this second inequality. Then the vertices xi0 . . . ,xn−1

must lie evenly spaced along a straight line segment.Let C(x, t) denote the circle of radius t centered at x. The in-

equalities (3) imply that, for 1 ≤ i ≤ i0, C(xn−1, n − 1 − i) cuts offan arc of C(xi−1, 1), consisting of points x on C(xi−1, 1) for whichd(x,xn−1) ≤ n − 1 − i. Parametrize this arc linearly, using parame-ter values −1 to 1 moving counterclockwise. The vertex xi lies on this

4 Donald M. Davis

arc. Set φi(P) equal to the parameter value of xi. If i = i0, thenφi(P) = ±1, and conversely.

The following diagram illustrates a polygon with n = 7, r = 5.2, andi0 = 5. We have denoted the vertices by their subscripts. The circlesfrom left to right are C(xi, 1) for i from 0 to 4. The arcs centered atx6 have radius 1 to 5 from right to left. We have, roughly, Φ(P) =(.7, .6, .5,−.05, 1).

.......

.......

..............................................................

........................

.......................................................................................................................................................................................................................................................................................................................................................................................

.............................................................................................

..............................................................................................................................................

............................................................................................................................................................................................................................................................................................

..........................................................................................................................................................................................................................................................................................................................................................................................................................................

...........................................................................................................................................................................................................................................................................................................................................................................................

...................................................................................................................................................................................................................................................................................................................................................................

.......

.......

..............................................................

........................

.......................................................................................................................................................................................................................................................................................................................................................................................

.............................................................................................

.......

.......

..............................................................

........................

.......................................................................................................................................................................................................................................................................................................................................................................................

............................................................................................. .......

.......

..............................................................

........................

.......................................................................................................................................................................................................................................................................................................................................................................................

.............................................................................................

...................................................................................................................

................................................................

....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.......

.......

..............................................................

........................

.......................................................................................................................................................................................................................................................................................................................................................................................

.............................................................................................

•6•0

•1 •2 •3

•4

•5

Here is another example, illustrating how the edges of the polygoncan intersect one another, and a case with i0 < n − 2. Again we haven = 7 and r = 5.2. This time, roughly, Φ(P) = (.2,−.4, .4, 1, t5), witht5 irrelevant. Because i0 = 4, we did not draw the circle C(x4, 1).

.......

.......

..............................................................

........................

.......................................................................................................................................................................................................................................................................................................................................................................................

.............................................................................................

..............................................................................................................................................

.............................................................................................................................................................................................................................................

.....................................................................................................................................................................................................................................................................

............................................................................................................................................................................................................................................................................................

....................................................................................................................................................................................................................................................................................

.......

.......

..............................................................

........................

.......................................................................................................................................................................................................................................................................................................................................................................................

.............................................................................................

.......

.......

..............................................................

........................

.......................................................................................................................................................................................................................................................................................................................................................................................

.............................................................................................

.......

.......

..............................................................

........................

.......................................................................................................................................................................................................................................................................................................................................................................................

.............................................................................................

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..........................................

.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................•6•0

•1

•2

•3

•4•5

That Φ is well defined follows from (2); once we have ti = ±1, whichhappens first when i = i0, subsequent vertices are determined and thevalues of subsequent tj are irrelevant. Continuity follows from the factthat the unit circles vary continuously with the various xi, hence so dothe parameter values along the arcs cut off. Bijectivity follows from the

Real projective space 5

construction; every set of ti’s up to the first ±1 corresponds to a uniquepolygon, and ±1 will always occur. Since it maps from a compact spaceto a Hausdorff space, Φ is then a homeomorphism. Equivariance withrespect to the involution is also clear. If you flip the polygon, you flipthe whole picture, including the unit circles, and this just negates allthe ti’s.

We elaborate slightly on the surjectivity of Φ. The arc on C(x0, 1)cut off by C(xn−1, n− 2) is determined by n and r. Given a value of t1in [−1, 1], the vertex x1 is now determined on this arc. Now the arc onC(x1, 1) cut off by C(xn−1, n − 3) is determined, and a specified valueof t2 determines the vertex x2. All subsequent vertices of an n-gon aredetermined in this manner.

Donald M. DavisDepartment of Mathematics,Lehigh University Bethlehem,PA 18015, USA,[email protected]

References

[1] Davis D. M., Immersions of projective spaces: a historical survey,Contemp Math Amer Math Soc 146 (1993) 31–38.

[2] , Immersions of real projective spaces,www.lehigh.edu/∼dmd1/imms.html.

[3] , Algebraic topology: there’s an app for that, Math Hori-zons 19 (2011), 23–25.

[4] Farber M., Invitation to topological robotics, European Math So-ciety (2008).

[5] Farber M,; Tabachnikov S.; Yuzvinsky S., Topological robotics:motion planning in projective spaces, Intl Math Research Notices34 (2003) 1853–1870.

[6] Gitler S., Immersions and embeddings of manifolds, Proc Sympin Pure Math Amer Math Soc 22 (1971) 87–96.

[7] Hausmann J. C., Sur la topologie des bras articuls, Lecture Notesin Math, Springer, 1474 (1991) 146–159.

6 Donald M. Davis

[8] Hausmann J. C., Rodriguez E., The space of clouds in Euclideanspace, Experiment. Math. 13 (2004) 31–47.

[9] James I. M, Euclidean models of projective spaces, Bull LondonMath Soc 3 (1971) 257–276.

[10] Kamiyama Y., Kimoto K., The height of a class in the cohomologyring of polygon spaces, Int Jour of Math and Math Sci (2013) 7pages.


Motion planning in tori revisited∗

Jesus Gonzalez, Barbara Gutierrez, Aldo Guzman,Cristhian Hidber, Marıa Mendoza, Christopher Roque

AbstractThe topological complexity (TC) of the complement of a com-plex hyperplane arrangement, which is either linear generic orelse affine in general position, has been computed by Yuzvinsky.This is accomplished by noticing that efficient homotopy mod-els for such spaces are given by skeletons of Cartesian powers ofcircles. Soon after, Cohen and Pruidze noticed that the topologi-cal complexity of the complement of the corresponding redundantsubspace arrangement, as well as of right-angled Artin groups, canbe obtained by considering general subcomplexes of cartesian pow-ers of higher dimensional spheres. Unfortunately Cohen-Pruidze’sTC-calculations are flawed, and our work describes and mends theproblems in order to validate the extended applications. In addi-tion, we generalize Farber-Cohen’s computation of the topologicalcomplexity of oriented surfaces, now to the realm of Rudyak’shigher topological complexity.

2010 Mathematics Subject Classification: 20F36, 52C35, 55M30.Keywords and phrases: Topological complexity, motion planner, zero-divisior cup-length.

1 Introduccion

Michael Farber proposed in [5, 6] a topological model to study the con-tinuity instabilities of the motion planning problem in robotics. Fol-lowing Farber, a motion planning algorithm (or motion planner) P =

∗This work is the result of the activities of the authors in the student workshop en-titled “Applied Topology at ABACUS: Motion Planning in Robotics” held in August2013. The authors thank all participants of the workshop for useful discussions, andkindly acknowledge the financial support received from ABACUS through CONA-CyT grant EDOMEX-2011-C01-165873.

7


Motion planning in tori revisited∗

Jesus Gonzalez, Barbara Gutierrez, Aldo Guzman,Cristhian Hidber, Marıa Mendoza, Christopher Roque

AbstractThe topological complexity (TC) of the complement of a com-plex hyperplane arrangement, which is either linear generic orelse affine in general position, has been computed by Yuzvinsky.This is accomplished by noticing that efficient homotopy mod-els for such spaces are given by skeletons of Cartesian powers ofcircles. Soon after, Cohen and Pruidze noticed that the topologi-cal complexity of the complement of the corresponding redundantsubspace arrangement, as well as of right-angled Artin groups, canbe obtained by considering general subcomplexes of cartesian pow-ers of higher dimensional spheres. Unfortunately Cohen-Pruidze’sTC-calculations are flawed, and our work describes and mends theproblems in order to validate the extended applications. In addi-tion, we generalize Farber-Cohen’s computation of the topologicalcomplexity of oriented surfaces, now to the realm of Rudyak’shigher topological complexity.

2010 Mathematics Subject Classification: 20F36, 52C35, 55M30.Keywords and phrases: Topological complexity, motion planner, zero-divisior cup-length.

1 Introduccion

Michael Farber proposed in [5, 6] a topological model to study the con-tinuity instabilities of the motion planning problem in robotics. Fol-lowing Farber, a motion planning algorithm (or motion planner) P =

∗This work is the result of the activities of the authors in the student workshop en-titled “Applied Topology at ABACUS: Motion Planning in Robotics” held in August2013. The authors thank all participants of the workshop for useful discussions, andkindly acknowledge the financial support received from ABACUS through CONA-CyT grant EDOMEX-2011-C01-165873.

7

8 Gonzalez et al

Fi, sii=1,...,k for a space X consists of a collection of k pairwise dis-joint subsets Fi of X × X, each admitting a continuous section si :Fi → X [0,1] for the end-points evaluation map π : X [0,1] → X × X,π(γ) = (γ(0), γ(1)), such that Fii is a covering of X × X by ENR’s.The sets Fi and the maps si are respectively called the local domainsand the local rules of P. The motion planner is said to be optimal whenthe number of local domains is minimal possible. The topological com-plexity of X, TC(X), is one less than the number of local domains inany optimal motion planner for X.

A lower bound for TC(X) is described in Proposition 1.1 belowthrough the concept of the zero-divisor cup-length of X with respect toa cohomology theory with products h∗. An h∗-zero-divisor of X is anelement in the kernel of the induced map

(1) π∗ : h∗(X × X) → h∗(X [0,1]).

The h∗-zero-divisor cup-length of X, denoted by zclh∗(X), is the maxi-mal number of h∗-zero-divisors whose product in h∗(X ×X) is non-zero.The “zero-divisor” adjective comes from the fact that π : X [0,1] → X×Xis a fibrational substitute for the diagonal map X → X×X. Thus, if thestrong form of the Kunneth formula holds for h∗, the kernel of (1) canbe identified with the kernel of the cup-product map h∗(X) ⊗ h∗(X) →h∗(X).

Proposition 1.1 ([5, Theorem 7]). The topological complexity of X isbounded from below by the h∗-zero-divisor cup-length of X, i.e.

zclh∗(X)) ≤ TC(X).

Instead of the usual upper bound for TC(X) given by homotopytheory (see [7, 8, 11]), the novel ingredient in Yuzvinsky’s [16] andCohen-Pruidze’s [4] works relies on the explicit construction of motionplanners whose optimality is then guaranteed by Proposition 1.1. Therelevant spaces arise as follows. Fix a positive integer k and considerthe standard (minimal) cellular structure in the k-dimensional sphereS = Sk = e0 ∪ ek. Here e0 is the base point, which we denote by e.Then take the product cell decomposition in

(2) Sn = S × · · · × S︸︷︷︸n times

=⊔J

eJ ,

where the cells eJ , indexed by subsets J ⊆ [n] = 1, . . . , n, are definedby eJ =

∏ni=1 edi with di = 0 if and only if i ∈ J . Explicitly, eJ =

Motion planning in tori revisited 9

(x1, . . . , xn) ∈ Sn | xi = e0 if and only if i /∈ J

. Cohen and Pruidze’s

main result is stated next.

Theorem 1.2. For a subcomplex X of the cell decomposition (2),

1. TC(X) = 2 dim(X)k for even k;

2. TC(X) = zclH∗(X)

= max |J | + |K| : J ∩ K = ∅, eJ and eK cells of X

for odd k.

It is illustrative to compare Theorem 1.2 to its Lusternik-Schnirel-mann category counterpart (in terms of the polyhedral power notation).Felix and Tanre prove in [10] the equality

cat((Sk, )L) = cat(Sk)(dim(L) + 1).

Here L is an abstract simplicial complex with vertices in [n], and catdenotes the (reduced) Lusternik-Schnirelmann category of X. For evenk, this corresponds to the equality TC((Sk, )L) = TC(Sk)(1 + dim(L))in item 1 of Theorem 1.2. However, for an odd k, the answer

TC((Sk, )L) = zclH∗((Sk, )L)

in item 2 of Theorem 1.2 has a value which is arbitrarily lower than thatin item 1, as we explain next.

Item 2 in Theorem 1.2 yields the calculations in [4, 16] of the topo-logical complexity of complements of complex hyperplane arrangements(either linear generic, or affine in general position), and of Eilenberg-MacLane spaces K(π, 1) for π a right-angled Artin group. It is alsointeresting to notice that, while the value of TC(X) in item 1 of Theo-rem 1.2 is maximal possible (see [6, Theorem 5.2]), item 2 in Theorem 1.2gives instances where the actual value of TC(X) can be arbitrarily lowerthan the dimension-vs-connectivity bound. In fact item 2 in Theo-rem 1.2 implies that the general estimate “cat ≤ TC ≤ 2 cat” in [5,Theorem 5] can reach any possible combination1—besides the stan-dard facts that TC(X) = cat(X) for H-spaces ([12, Theorem 1]), and

1The authors learned of this fact at Dan Cohen’s lecture during the student work-shop Applied Topology at ABACUS: Motion Planning in Robotics, that took place aweek after the 2013 Mathematical Congress of the Americas, in Mexico.

10 Gonzalez et al

TC(X) = 2 cat(X) for closed simply connected symplectic manifolds ([9,Corollary 3.2]). Indeed, for any positive integers c and t with c ≤ t ≤ 2c,there is a space X with cat(X) = c and TC(X) = t. In detail, for afixed non-negative odd integer k, let X = Sc ∨St−c. Since cat(Sc) = c ≥t−c = cat(St−c), we see cat(X) = max

cat(Sc), cat(St−c)

= c. On the

other hand, item 2 in Theorem 1.2 yields TC(X) = t. (The case k = 1is treated in [14] with different techniques.)

We noticed that the inequality TC(X) ≤ 2 dim(X)k in item 1 of The-

orem 1.2 is standard. Cohen and Pruidze assert to have constructedan explicit motion planning algorithm realizing this upper bound, butas described in the next section, their construction is flawed on severalfronts. Similar problems hold in item 2 of Theorem 1.2, but in this casethe situation is critical because the needed upper bound is not availableby other means.

The first main goal of this paper, addressed in the next section, isto fix the problems in [4]. Then, in Sections 3, we compute the highertopological complexity of oriented surfaces.

2 Correction of gaps in Cohen-Pruidze’s work

The simplest motion planner on Sn holds for k odd, assumption whichwill be in force in this section until further notice. When n = 1, themotion planner has two local domains described as follows: Let F1 ⊂S × S and s1 : F1 → S[0,1] be given by

F1 = (x, −x)|x ∈ S

and, for a fixed nowhere zero tangent vector field ν on S, s1(x, −x) is thepath from x to −x at constant speed along the semicircle determinedby the tangent vector ν(x). The second local domain is given by thecomplement of F1,

F0 = S × S − F1,

with local rule s0 : F0 → S[0,1] where s0(x, y) is the path from x to yat constant speed along the shortest geodesic arc. It is elementary tosee that zclH∗(S) = 1, so TC(S) = 1 and the above motion planner isoptimal.

The corresponding product motion planner in Sn (described in [5,Theorems 11 and 13], and simplified in [6, p. 24]) is recalled in Propo-sition 2.1 below. The needed preliminaries go as follows: For a subset


I ⊂ [n], let

FI = (x, y) ∈ Sn × Sn| xi = −yi iff i ∈ I

and define sI : FI → (Sn)[0,1] using the maps s1 and s0 defined above,namely

(3) sI(x, y) = (t1(x1, y1), . . . , tn(xn, yn))

where ti = s1 if i ∈ I, and ti = s0 if i /∈ I. (Here and below we usethe shorthand x = (x1, . . . xn), y = (y1, . . . , yn), etc.) The sets FI ’sare conveniently separated as FI ∩ FJ = ∅ for I J . In particular,FI ∩ FJ = ∅ = FI ∩ FJ when |I| = |J | with I = J . This allows us to set

(4) Wj =⋃

|I|=n−j

FI∼=

⊔

|I|=n−j

FI

for j = 0, 1, . . . , n, and define a local rule σj : Wj → (Sn)[0,1] by σj |FI=

sI .

Proposition 2.1 ([6, p. 24]). For k odd, the subsets Wj ⊂ Sn × Sn andmaps σj : Wj → (Sn)[0,1], j = 0, 1, . . . , n, determine an optimal motionplanner for Sn, thus TC(Sn) = n.

Still assuming k is odd, let X be a subcomplex of the cell decomposi-tion (2) and, for J ⊂ [n], let TJ denote the subcomplex of Sn generatedby eJ , i.e. TJ = eJ = x ∈ Sn | xi = e0 if i /∈ J. If J ∩ K = ∅, TJ ∪TK

sits inside Sn as the wedge union TJ ∨ TK . Therefore the term on theright of the second item in Theorem 1.2 takes the form

z(X) := max |J | + |K| | J ∩ K = ∅ and TJ ∨ TK ⊆ X .

Cohen-Pruidze’s critical assertion TC(X) = z(X) in [4, Theorem 3.4] isargued by (i) constructing a motion planner for X with z(X) + 1 localdomains, and then (ii) showing that the H∗-zero-divisor cup-length ofX is at least z(X). Their proof of (ii) is correct and straightforward,but their construction in (i) is flawed. Explicitely, the authors assertthat the local rules in the product motion planner for Sn constructedin Proposition 2.1 restrict to give local rules for any subcomplex X ofSn. But such an assertion is false in most of the cases. We exhibit anexplicit (but typical) counterexample (Example 2.2), and then show howthe combinatorics of the cell decomposition of X need to be taken intoconsideration to fix the construction—and, therefore, Cohen-Pruidze’sproof of Theorem 1.2.

12 Gonzalez et al

Example 2.2. Take n = 2 and k = 1, so Sn = T 2, the 2-torus. LetX = S1 ∨S1 be the 1-dimensional skeleton in the minimal cell structureof T 2, so X has the cell decomposition X = e∅ ∪e1 ∪e2, and z(X) =2. The local domain decomposition proposed in [4] is

X × X =2⋃

j=0(X × X) ∩ Wj

with local rules given by the restricted sections σj(X) = σj |Wj∩(X×X).Now let us focus attention on the local domain

(5) (X × X) ∩ W0 = (−1, 1, 1, −1) ∪ (1, −1, −1, 1)

with corresponding local rule given by

σ0(X)(x, −x, −x, x) = (s1(x, −x), s1(−x, x)).

The authors of [4] claim that the local rule σ0(X) lands in X [0,1] —rather than in (T 2)[0,1]. But such an assertion is clearly false. In fact,for any p ∈ (X × X) ∩ W0, the path σ0(X)(p) takes values in X only att = 0, 1.

The assertion in [4] that (X × X) ∩ Wj = ∅ for j < n − z(X) istrue (and easy to verify). As illustrated above, the problem comes withthe claim that the restrictions σj(X) of σj to (X × X) ∩ Wj , where(n − z(X) ≤ j ≤ n), give local motion planners in X. The gap notedin Example 2.2 is typical and can be corrected by taking into accountthe combinatorial properties of the cell structure in X. For instance, inthe explicit situation considered in Example 2.2, rather than insistingon performing the motion in both coordinates in a “parallel” way, oneshould move in two halves; the first part of the motion should be onthe coordinate different to 1, keeping the other coordinate fixed. Onlywhen this part of the motion is complete, and we have arrived to thebase point (1, 1), it will be safe to move the missing coordinate. Forinstance, the motion planning algorithm from (−1, 1) to (1, −1) —thefirst of the two “tasks” represented in (5)— is depicted by the thickcurve in

• •

(−1, 1) (1, −1).

The correct general motion planner is described next.


Fixed motion planner for item 2 in Theorem 1.2 (k odd ). The descrip-tion simplifies by normalizing S so to have great semicircles of length1/2. For x, y ∈ S, we let d(x, y) stand for the length of the shortestgeodesic between x and y over S, so d(x, −x) = 1/2. Likewise, the localrules s0 and s1 for S defined at the beginning of the section need to beadjusted. For i = 0, 1 and (x, y) ∈ Fi with x = y, set

Si(x, y)(t) =

si(x, y)(

1d(x,y) t

), 0 ≤ t ≤ d(x, y);

y, d(x, y) ≤ t ≤ 1

(if x = y, Si(x, y)(t) = x = y for all t ∈ [0, 1]). Thus, Si reparametrizessi so to perform the motion at speed 1, keeping still at the final positiononce it is reached—which happens at most at time 1/2. In view ofthe homeomorphism in (4), it suffices to define a local rule on each(X × X) ∩ FI taking values in X [0,1]. Replace (3) by the map SI : FI ∩(X × X) → (Sn)[0,1] defined by SI(x, y) = (T1(x1, y1), . . . , Tn(xn, yn))where Ti(xi, yi) : [0, 1] → S is the path

Ti(xi, yi)(t) =

xi, 0 ≤ t ≤ txi ,

Σi(xi, yi)(t − txi), txi ≤ t ≤ 1.

Here txi = 12 − d(xi, 1), and Σi = S1 if i ∈ I while Σi = S0 if i /∈ I.

It is clear that SI is a continuous section on FI ∩ (X × X) for theend-points evaluation map π : (Sn)[0,1] → Sn × Sn. We only need tocheck that SI takes values in X [0,1]. With that in mind, note thatthe motion described by the local rule SI , from an “initial coordinate”xi to the corresponding “final coordinate” yi, is executed according tothe relevant instruction Sj (j ∈ 0, 1), except that the movement isdelayed a time txi ≤ 1/2. The closer xi gets to 1, the closer thedelaying time txi gets to 1/2. It is then convenient to think of thepath SI(x, y) as happening in two stages. In the first stage (t ≤ 1/2)all initial coordinates xi = 1 keep still, while the rest of the coordi-nates (eventually) start traveling to their corresponding final positionyi. Further, at the time the second stage starts (t = 1/2), any fi-nal coordinate yi = 1 will already have been reached. As a result, SI

never leaves X. In more detail: Let eJ , eK ⊂ X be cells of X. For(x, y) = ((x1, . . . , xn), (y1, . . . , yn)) ∈ FI ∩ (eJ × eK), coordinates corre-sponding to indexes i not in J , keep their initial position xi = 1 throughtime t ≤ 1/2. Therefore SI(x, y)[0, 1/2] stays within TJ ⊆ X. On the

14 Gonzalez et al

other hand, by construction, Ti(xi, yi)(t) = yi = 1 whenever t ≥ 1/2and i /∈ K. Thus, SI(x, y)[1/2, 1] stays within TK ⊆ X.

The motion planning algorithm constructed above is a variation ofthe one carefully described in [16] for skeletons of the minimal cell de-composition in the n-th Cartesian power of the circle. In the currentcase, we are considering all possible subcomplexes of the minimal CW-complex (Sk)n for any odd k.

Proof of Theorem 1.2 (an additional fixing). It remains to check thatzclH∗(X) is bounded from below by 2 dim(X)/k when k is even, and byz(X) when k is odd. The case for odd k is addressed correctly in [4,Proposition 3.7], however the case for even k requires some tune-up. Wethus assume in this proof that k is even.

Say dim(X) = k. Cohen and Pruidze’s argument starts by noticingthat X must contain a copy of S as a subcomplex, and that the inclusionι : S → X induces an epimorphism ι∗ : H∗(X) → H∗(S). From this,they infer

(6) zclH∗(X) ≥ zclH∗(S),

and obtain the desired conclusion from the well-known equality

zcl(H∗(S)) = 2.

The subtlety here is that the surjectivity of the ring morphism ι∗ isnot enough to deduce (6). One actually needs to know that each factorin a non-zero product of zero-divisors realizing zcl(H∗(S))—as the onedescribed in the proof of Proposition 6.2 in [4]—is the image of a zero-divisor in zcl(H∗(X)). But such a property does hold in the currentsituation, as Proposition 3.6 in [4] holds true also for k even (cf. [1,Theorem 2.35]). Alternatively, the composition of the inclusion X → Sn

with the obvious projection Sn → S gives a retraction ρ for the inclusionι : S → X and, evidently, both ι and ρ are compatible with diagonalinclusions.

Remark 2.3. The problem noted in Example 2.2 (for k odd) also holdsin [4] for k even. The new issue is more subtle, and this is reflected inpart by noticing an additional gap in the proof of [4, Theorem 6.3]. Herewe illustrate the new error (and some of the subtleties needed to sortit out), so we assume in this remark that the reader is familiar withthe notation set in the final section of [4] (where k is even). For X =


S2 ∨ S2 ⊂ S2 × S2, the first paragraph in the proof of [4, Theorem 6.3]asserts that (X ×X)∩Wj = ∅ for j = 0, 1. In particular (AJ ×AK)∩Fα

would have to be empty for J = 1, K = 2, and α = (1, 0). However((−e, e), (e, −e)) clearly lies in the latter intersection. Of course, Cohenand Pruidze’s gap in the argument of their Theorem 6.3 comes from theirassertion (in the second paragraph of their proof) that there should besome index in 1, 2 missing J ∪ K.

As part of her Ph.D. studies, the second author of this paper hasmanaged to construct an optimal motion planner for any subcomplexof Sn when k is even. The construction, carried over in more generalterms (for Rudyak’s higher TC and any parity of k), will be discussedelsewhere.

3 Higher TC of oriented surfaces

In [13] Yuli B. Rudyak introduced the concept of the higher topologicalcomplexity of a path connected space X, denoted by TCs(X). In thissection we extend Farber and Cohen’s calculation of TC2(Σg) in [3]to the realm of higher topological complexity. Here Σg stands for anoriented surface of genus g.

Let Js, s ∈ N, denote the wedge of n closed intervals [0, 1]i, i =1, . . . , s, where the zero points 0i ∈ [0, 1]i are identified. Consider Xs

(the s-th cartesian product of X) and XJs , where X is a path connectedspace. There is a fibration

(7) es : XJs −→ Xs, es(f) = (f(11), . . . , f(1s))

where 1i ∈ [0, 1]i. Recall that the s-th topological complexity of X,denoted by TCs(X), is defined as the reduced Schwarz genus of es.Note that (7) is a fibrational substitute of the iterated diagonal mapdX

s : X −→ Xs. Hence TCs(X) coincides with the Schwarz genus ofdX

s : X −→ Xs. Using the iterated diagonal map dXs and allowing coho-

mology with local coefficients we have the following standard definition:

Definition 3.1. Given a space X and a positive integer n, we denoteby zcls (H∗(X)) the maximal length of non-zero products of elements inthe kernel of the map induced in cohomology by dX

s . Thus, zcls (H∗(X))is the largest integer m for which there exist cohomology classes ui ∈H∗(Xs, Ai) with dX

s (ui) = 0, i = 1, . . . , m, and

0 = u1 ⊗ · · · ⊗ um ∈ H∗(Xs, A1 ⊗ · · · ⊗ Am).

16 Gonzalez et al

Note that zcl2 (H∗(X)) recovers zclH∗(X). The following resultbounds TCs(X) from below by zcls(H∗(X)) and from above by a num-ber which involves the homotopy dimension of X, hdim(X) (the smallestdimension of CW complexes having the homotopy type of X), and theconnectivity of X, conn(X).

Theorem 3.1. For any path-connected space X we have:

zcls (H∗(X)) ≤ TCs(X) ≤ s hdim(X)conn(X) + 1

For a proof of Theorem 3.1 see [15, Theorems 4 and 5].

Proposition 3.2. For g, s ≥ 2, the s-th higher topological complexityof Σg is TCs(Σg) = 2s.

This should be compared to the facts that TCs(Σ0) = s and

TCs(Σ1) = 2(s − 1)

proved in [2, Corollary 3.12] (see also [13, Section 4]). In addition, itshould be noted that Proposition 3.2 was also mentioned by Ibai Basabeduring his talk at the conference “Applied Algebraic Topology” held atthe Centro Internacional de Encuentros Matematicos on July 2014.

Proof of Proposition 3.2. We use cohomology with rational coefficients.Let ai, bi, i = 1, . . . , g, be the generators of H1(Σg) which satisfy aibj =aiaj = bibj = 0 for i = j, a2

i = b2i = 0 and aibi = ω for any i, where

ω generates H2(Σg). Let HΣg = H∗(Σ×sg ) = [H∗(Σg)]⊗s. For each

i = 2, . . . , s, consider the elements

αi = a1 ⊗ 1 ⊗ · · · ⊗ 1 − 1 ⊗ · · · ⊗ a1 ⊗ · · · ⊗ 1,

βi = b1 ⊗ 1 ⊗ · · · ⊗ 1 − 1 ⊗ · · · ⊗ b1 ⊗ · · · ⊗ 1,

where the factor a1 (resp. b1) in 1 ⊗ · · · ⊗ a1 ⊗ · · · ⊗ 1 (resp. 1 ⊗ · · · ⊗b1 ⊗ · · · ⊗ 1) appears in the i-th tensor coordinate, and

γ1 = a2 ⊗ 1 ⊗ · · · ⊗ 1 − 1 ⊗ a2 ⊗ 1 ⊗ · · · ⊗ 1,

γ2 = b2 ⊗ 1 ⊗ · · · ⊗ 1 − 1 ⊗ b2 ⊗ 1 ⊗ · · · ⊗ 1

of HΣg . These elements lie in the kernel of the cup-product map

[H∗(Σg)]⊗s → H∗(Σg),

and satisfy γ1 · γ2 · α2 · β2 · · · αs · βs = 2ω ⊗ · · · ⊗ ω = 0. Therefore,2s ≤ zcls(H∗(Σg)) ≤ TCs(Σg). On the other hand, using the upperbound in Theorem 3.1, we get TCs(Σg) ≤ 2s. Thus, TCs(Σg) = 2s.


Jesus GonzalezDepartamento de Matematicas,CINVESTAV del I.P.N.,Apartado Postal 14-740,Mexico D.F., C.P. 07360,[email protected]

Barbara GutierrezDepartamento de Matematicas,CINVESTAV del I.P.N.,Apartado Postal 14-740,Mexico D.F., C.P. 07360,[email protected]

Aldo GuzmanDepartamento de Matematicas,CINVESTAV del I.P.N.,Apartado Postal 14-740,Mexico D.F., C.P. 07360,[email protected]

Cristhian HidberDepartamento de Matematicas,CINVESTAV del I.P.N.,Apartado Postal 14-740,Mexico D.F., C.P. 07360,[email protected]

Marıa MendozaDepartamento de Matematicas,CINVESTAV del I.P.N.,Apartado Postal 14-740,Mexico D.F., C.P. 07360,[email protected]

Christopher RoqueDepartamento de Matematicas,CINVESTAV del I.P.N.,Apartado Postal 14-740,Mexico D.F., C.P. 07360,[email protected]

References

[1] Bahri A.; Bendersky M.; Cohen F.; Gitler S., The polyhedralproduct functor: a method of decomposition for moment-anglecomplexes, arrangements and related spaces, Adv. Math. 225:3(2010), 1634–1668.

[2] Basabe I.; Gonzalez J.; Rudyak J.; Tamaki D., Higher topologicalcomplexity and its symmetrization, Algebr. Geom. Topol. 14: 4(2014), 2103–2124.

[3] Cohen D.; Farber M., Topological complexity of collision-free mo-tion planning on surfaces, Compos. Math. 147:2 (2011), 649–660.

[4] Cohen D.; Pruidze G., Motion planning in tori, Bull. Lond. Math.Soc. 40:2 (2008), 249–262.

[5] Farber M., Topological complexity of motion planning, DiscreteComput. Geom. 29:2 (2003), 211–221.

[6] Farber M., Instabilities of robot motion, Topology Appl. 140:2-3(2004), 245–266.

[7] Farber M.; Grant M., Topological complexity of configurationspaces, Proc. Amer. Math. Soc. 137:5 (2009), 1841–1847.

18 Gonzalez et al

[8] Farber M.; Grant M.; Yuzvinsky S., Topological complexity of col-lision free motion planning algorithms in the presence of multiplemoving obstacles, Topology and robotics, Contemp. Math. 438(2007), 75–83.

[9] Farber M.; Tabachnikov S.; Yuzvinsky S., Topological robotics:motion planning in projective spaces, Int. Math. Res. Not. 34(2003), 1853–1870.

[10] Felix Y.; Tanre D., Rational homotopy of the polyhedral productfunctor, Proc. Amer. Math. Soc. 137:3 (2009), 891–898.

[11] Gonzalez J.; Grant M., Sequential motion planning of non-colliding particles in Euclidean spaces, Accepted for publicationin Proceedings of the American Mathematical Society.

[12] Lupton G.; Scherer J., Topological complexity of H-spaces, Proc.Amer. Math. Soc. 141:5 (2013), 1827–1838.

[13] Rudyak Y., On higher analogs of topological complexity, Topologyand its Applications 57 (2010), 916–920 (erratum in Topology andits Applications 57 (2010), 1118.

[14] Rudyak Y., On topological complexity of Eilenberg-MacLanespaces, http://arxiv.org/pdf/1302.1238v2.pdf.

[15] Schwarz A., The genus of a fiber space, Amer. Math. Soc. Transl.Series 2 55 (1966), 49–140.

[16] Yuzvinsky S., Topological complexity of generic hyperplane com-plements, Contemporary Math. 438 (2007), 115–119.

Morfismos, Vol. 19, No.1, 2015, pp. 19–38

Toeplitz operators with piecewise

quasicontinuous symbols ∗

Breitner Ocampo

Abstract

For a fixed subset of the unit circle ∂D, Λ := λ1, λ2, . . . , λn,we define the algebra PC of piecewise continuous functions in∂D \ Λ with one sided limits at each point λk ∈ Λ. Besides, welet QC stands for the C∗-algebra of quasicontinuous functions on∂D defined by D. Sarason in [5]. We define then PQC as theC∗-algebra generated by PC and QC.

A2(D) stands for the Bergman space of the unit disk D, that is,the space of square integrable and analytic functions defined on D.Our goal is to describe TPQC , the algebra generated by Toeplitzoperators whose symbols are certain extensions of functions inPQC acting on A2(D). Of course, a function defined on ∂D can beextended to the disk in many ways. The more natural extensionsare the harmonic and the radial ones. In the paper we describe thealgebra TPQC and we prove that this description does not dependon the extension chosen.

2010 Mathematics Subject Classification: 32A36, 32A40, 32C15, 47B38,47L80.Keywords and phrases: Bergman spaces, C∗-algebras, Toeplitz operator,quasicontinuous symbols, piecewise continuous symbols.

1 Introduction

We consider the C∗-algebra of quasicontinuous functions QC, whichconsists of all functions f : ∂D → C such that both f and its complex

∗This paper is part of the doctoral thesis of Breitner Ocampo under the supervisionof Dr. Nikolai Vasilevski at the Mathematics Department of Cinvestav-IPN.

19

Morfismos, Vol. 19, No.1, 2015, pp. 19–38

Toeplitz operators with piecewise

quasicontinuous symbols ∗

Breitner Ocampo

Abstract

For a fixed subset of the unit circle ∂D, Λ := λ1, λ2, . . . , λn,we define the algebra PC of piecewise continuous functions in∂D \ Λ with one sided limits at each point λk ∈ Λ. Besides, welet QC stands for the C∗-algebra of quasicontinuous functions on∂D defined by D. Sarason in [5]. We define then PQC as theC∗-algebra generated by PC and QC.

A2(D) stands for the Bergman space of the unit disk D, that is,the space of square integrable and analytic functions defined on D.Our goal is to describe TPQC , the algebra generated by Toeplitzoperators whose symbols are certain extensions of functions inPQC acting on A2(D). Of course, a function defined on ∂D can beextended to the disk in many ways. The more natural extensionsare the harmonic and the radial ones. In the paper we describe thealgebra TPQC and we prove that this description does not dependon the extension chosen.

2010 Mathematics Subject Classification: 32A36, 32A40, 32C15, 47B38,47L80.Keywords and phrases: Bergman spaces, C∗-algebras, Toeplitz operator,quasicontinuous symbols, piecewise continuous symbols.

1 Introduction

We consider the C∗-algebra of quasicontinuous functions QC, whichconsists of all functions f : ∂D → C such that both f and its complex

∗This paper is part of the doctoral thesis of Breitner Ocampo under the supervisionof Dr. Nikolai Vasilevski at the Mathematics Department of Cinvestav-IPN.

19

20 Breitner Ocampo

conjugate f belong to H∞ + C. Here H∞ denotes the set (algebra)of boundary functions for bounded analytic functions on the unit diskD, and C stands for the algebra of continuous functions on ∂D. Thespace QC has two natural extensions to the disk, namely, the radial andthe harmonic extension, we denote these extensions by QCR and QCH ,respectively.

We use A2(D) to denote the Bergman space of L2(D) which consistsin all analytic functions. For A2(D) ⊂ L2(D), we denote by BD theBergman projection BD : L2(D) → A2(D). Let K denote the ideal ofcompact operators acting on A2(D).

Recall that, for a bounded function f on D, the Toeplitz operator Tf

acting on A2(D) is defined by the formula Tf (g) = BD(fg). For a linearsubespace A ⊂ L∞(D) we denote by TA the (closed) operator algebragenerated by Toeplitz operators with defining simbols in A.

In this paper we describe the Calkin algebras TQCR/K and TQCH

/K.We use the characterization of QC as the set of bounded functions withvanishing mean oscillation to prove that the Calkin algebras TQCR

/Kand TQCH

/K are commutative, moreover TQCR∼= TQCH

.

For a finite set of points Λ := λ1, . . . , λn of ∂D, we define thespace of piecewise continuous functions PC := PCΛ as the algebra ofcontinuous functions on ∂D \ Λ with one sided limits at each pointλk ∈ Λ. We denote by PQC the C∗-algebra generated by both PCand QC. We use an extension of PQC to the disk and thus define theToeplitz operator algebra TPQC ⊂ B(A2(D)). There are several ways toextend the functions in PQC to D; two of them are: the radial extension,PQCR, and the harmonic extension, PQCH . The main goal of thispaper is the description of the Calkin algebra TPQC/K, which is statedin Theorem 3.15. Finally, in Section 4, we prove that the result doesnot depend on the extension chosen for PQC, that is, TPQCR

= TPQCH.

2 Preliminaries

First of all, we set some notation that will be used throughout the paper.Any mathematical symbol not described here will be used in its morecommon sense, ‖ · ‖A stands for the norm in the space A. We denoteby D the unit disk and by ∂D its boundary, the unit circle. The sets Dand ∂D are endowed with the standard topology and with the Lebesguemeasures dz = dxdy and dθ , where the point z = x + iy belongs to Dand eiθ belongs to ∂D. All the functions in the paper are considered as

Toeplitz operators with piecewise quasicontinuous symbols 21

complex-valued.This section includes some basic facts about the space of Vanishing

Mean Oscillation functions on ∂D, denoted here by VMO. The impor-tance of this space lies in the fact that QC = VMO ∩ L∞ ( see [4]).For the convenience of the reader we recall the relevant material from[5] omitting proofs, thus making the exposition self contained.

We define the following spaces of functions on ∂D:

• L∞ := L∞(∂D) = the algebra of bounded measurable functionsf : ∂D → C,

• H∞ := H∞(∂D) = the algebra of radial limits of bounded analyticfunctions defined on D,

• C := C(∂D) = the algebra continuous functions on ∂D.

Definition 2.1. [5, page 818] We define the C∗-algebra of quasicon-tinuous functions QC as the algebra of all bounded functions f on ∂D,such that, both f and its complex conjugate f belong to H∞ +C, thatis;

QC := (H∞ + C) ∩(H

∞+ C

).

Some of the statements below are formulated for segments in thereal line, but they can also be formulated for arcs in ∂D.

By an interval on R we always mean a finite interval. The length ofthe interval I will be denoted by |I|.

For f ∈ L1(I), the average of f over I is given by

(1) I(f) := |I|−1

∫

If(t)dt.

For a > 0, let

Ma(f, I) := supJ⊂I,|J |<a

1

|J |

∫

J|f(t)− J(f)|dt.

Note that 0 ≤ Ma(f, I) ≤ Mb(f, I) if a ≤ b, then let M0(f, I) :=lima→0

Ma(f, I).

Definition 2.2. [5, page 81] A function f ∈ L1(I) is of vanishing meanoscillation in the interval I ( or the arc I), if M0(f, I) = 0. The set ofall vanishing mean oscillation functions on I is denoted by VMO(I).

In particular, if we replace I by ∂D in definitions above we getVMO := VMO(∂D).

22 Breitner Ocampo

A useful characterization of the space VMO is as follows: a functionf belongs to VMO if and only if for any ε > 0 there exists δ > 0,depending on ε, such that

|J |−2

∫

J

∫

J|f(t)− f(s)|dsdt < ε,

for every interval J ⊂ I with |J | < δ.

Definition 2.3. [5, Page 818] Let f be an integrable function definedin an open interval containing the point λ. We define the integral gapof f at λ by

γλ(f) := lim supδ→0

∣∣∣∣∣∣δ−1

λ+δ∫

λ

f(t)dt− δ−1

λ∫

λ−δ

f(t)dt

∣∣∣∣∣∣.

Obviously, if f belongs to VMO(I), then γλ(f) = 0 for each interiorpoint λ of I. The most important use of Definition 2.3 is stablished inthe following lemma:

Lemma 2.4. [5, Lemma 2] Let I = (a, b) be an open interval, λ a pointof I, and f a function on I which belongs to both VMO((a, λ)) andVMO((λ, b)). If γλ(f) = 0, then f belongs to VMO(I).

We denote by M(QC) the space of all non-trivial multiplicative lin-ear functionals on QC, endowed with the Gelfand topology. In the sameway define M(C) and identify it with ∂D via the evaluation function-als. Since C is a subset of QC, every functional in M(QC) induces, byrestriction, a functional in C.

Here and subsequently, f0 denotes the function f0(λ) = λ. TheStone-Weierstrass theorem implies that f0 and the function f(λ) = 1generate the C∗-algebra of all continuous functions on ∂D.

Definition 2.5. [5, Page 822] For every λ ∈ ∂D, we denote by Mλ(QC)the set of all functionals x in M(QC) such that x(f0) = λ, that is

Mλ(QC) := x ∈ M(QC) : x(f0) = f0(λ) = λ.

In other words, x belongs to Mλ(QC) if the restriction of x to thecontinuous functions is the evaluation functional at the point λ.

Definition 2.6. [5, Page 822] We let M+λ (QC) denote the set of x ∈

Mλ(QC) with the property that f(x) = 0 whenever f inQC is a functionsuch that lim

t→λ+f(t) = 0. M−

λ (QC) is defined in an analogous way.


Let f be bounded function on ∂D. The harmonic extension of f tothe unit disk is denoted by fH and is given by the formula

(2) fH(z) := fH(r, θ) :=1

2π

∫

∂D

Pr(θ − λ)f(λ)dλ,

where

Pr(θ) := Re

(1 + reiθ

1− reiθ

)=

1− r2

1− 2rcos(θ) + r2

is the Poisson kernel for the unit disk.For every point z in D we define a functional in QC by the following

rule: z(f) = fH(z), so, we consider D as a subset of the dual space ofQC. Under this identification we have that the weak-star closure of Dcontains M(QC) [5, Lemma 7].

Lemma 2.7. Let f be a function in QC which is continuous at thepoint λ0. Then x(f) = f(λ0) for every functional x in Mλ0(QC).

Proof. Consider the case where the function f is continuous at λ0 andsuch that f(λ0) = 0. Let x be a point in Mλ0(QC). For ε > 0 there isδ0 > 0 such that |f(λ)| < ε for all λ in the arc Vλ0 = (λ0 − δ0, λ0 + δ0).The values taken by the Poisson extension of f should be small if weevaluate points in D of an open disk with center at λ0, i.e, there is a δ1such that |fH(z)| < ε/2 if dist(z, λ0) < δ1 and z ∈ D.

Using ε1 = minδ0, δ1, ε we construct a neighbourhood Vx in QC∗

with parameters f, f0, ε1.By Lemma 7 in [5], there is a z in D such that z ∈ Vx, that is

|fH(z)− x(f)| < ε1 < ε and |f0(z)− f0(λ0)| = |z − λ0| < ε1 < δ1.

This implies that dist(z, λ0) ≤ δ1 and then |fH(z)| < ε/2.Now we estimate x(f),

|x(f)| ≤ |x(f)− fH(z)|+ |fH(z)| < ε,

consequently x(f) = 0.In the general case, when f(λ0) = 0, we apply the previous argument

to the function g = f − f(λ0). For g we obtain 0 = x(g) = x(f)− f(λ0)and then x(f) = f(λ0) for all x ∈ Mλ0(QC).

For z = 0 in D, we let Iz denote the closed arc of ∂D whose centeris z/|z| and whose length is 2π(1− |z|). For completeness, I0 = ∂D.

24 Breitner Ocampo

Lemma 2.8. [5, Lemma 5] For f in QC and any positive number ε,there is a positive number δ such that |fH(z) − Iz(f)| < ε whenever1− |z| < δ.

The average of a function f over an arc I defines a linear functionalon QC. Let us identify each arc I with the “averaging” functional inQC. The set of all these functionals is denoted by G. By Lemma 7 in[5] and Lemma 2.8 we come to the following lemma.

Lemma 2.9. [5, Page 822] M(QC) is the set of points in the weak-starclosure of G ( denoted here by G∗

) which does not belong to G.

For λ ∈ ∂D we denote by G0λ the set of all arcs I in G with center

at λ. Let M0λ(QC) be the set of functionals in Mλ(QC) that lie in the

weak-star closure of G0λ. By Lemma 2.8, the set M0

λ(QC) coincides withthe set of functionals in Mλ(QC) that lie in the weak-star closure of theradius of D terminating at λ.

In [5], D. Sarason splits the space Mλ(QC) into three sets:

M+λ (QC) \M0

λ(QC), M−λ (QC) \M0

λ(QC) and M0λ(QC).

These three sets are mutually disjoint due to the next lemma:

Lemma 2.10. [5, Lemma 8] M+λ (QC) ∪ M−

λ (QC) = Mλ(QC) andM+

λ (QC) ∩M−λ (QC) = M0

λ(QC).

The result in Lemma 2.10 allows us to draw the maximal ideal spaceM(QC). We consider the unit circle as the interval [0, 2π], where thepoints 0 and 2π represent the same point. At each point λ in [0, 2π] wedraw a segment representing the fiber Mλ(QC). The segment Mλ(QC)is splitted into two parts, the upper part M+

λ (QC) and the lower partM−

λ (QC). Their intersection is M0λ(QC), the central part of the fiber.

0 2π

Mλ(QC)

M0λ(QC)

M+λ (QC)

M−λ (QC)

Figure 1: The maximal ideal space of QC.


3 Toeplitz operators with piecewise quasicon-tinuous symbols on the Bergman space

This section deals with Toeplitz operators with symbols in certain ex-tension of PQC acting on A2(D). The C∗-algebra PQC is generatedby both, the space PC of piecewise continuous functions, and QC, thespace of quasicontinuous functions, both extended from ∂D to the unitdisk D. The main result of this section (Theorem 3.15) describes theCalkin algebra TPQC := TPQC/K as the C∗-algebra of continuous sec-tions over a bundle ξ constructed from the operator algebra TPQC .

Definition 3.1. Let Λ := λ1, λ2, . . . , λn be a fixed set of n differentpoints on ∂D. Define PC := PCΛ as the set of continuous functions on∂D \Λ with one sided limits at every point λk in Λ. For a function a inPC we set

a+k := limλ→λ+

k

a(λ) and a−k := limλ→λ−

k

a(λ),

following the standard positive orientation of ∂D.

Definition 3.2. PQC is defined as the C∗-algebra generated by PCand QC.

Our interest is to describe a certain Toeplitz operator algebra actingon the Bergman space A2(D). For this we need to extend the functionsin PQC to the whole disk. There are two most natural ways of suchextensions

• the harmonic extension gH , given by the Poisson formula 2,

• the radial extension gR, defined by gR(r, θ) = g(θ).

In this section we use the radial extension, however, we emphasizethat the main result of this paper does not depend on the extensionsmentioned above (Theorem 4.11).

Recall that the Bergman space A2(D) is the closed subspace of L2(D)which consists of all functions analytic in D. Being closed, the spaceA2(D) has the orthogonal projection BD : L2(D) → A2(D), called theBergman projection. Let K denote the ideal of compact operators actingon A2(D).

Given a function g in L∞(D), the Toeplitz operator Tg : A2(D) →A2(D) with generating symbol g is defined by Tg(f) = BD(gf).

26 Breitner Ocampo

In [7], K. Zhu describes the largest C∗-algebra Q ⊂ L∞(D) such thatthe map

ψ : Q → B(A2(D))/Kf → Tf +K,

is a C∗-algebra homomorphism. This algebra is closely related to QCbecause both can be described using spaces of vanishing mean oscillationfunctions.

Definition 3.3. [7, Page 633] Consider

Γ := f ∈ L∞(D) : TfTg − Tfg ∈ K for all g ∈ L∞(D).

Let Q := Γ ∩ Γ.

For z in D, we define

Sz := w ∈ D : |w| ≥ |z| and | arg(z)− arg(w)| ≤ 1− |z|,

The area of Sz, denoted by |Sz|, is π(1− |z|)2(1 + |z|).

Definition 3.4. [7, Page 621] A function f in L1(D) belongs toVMO∂(D), the space of functions with vanishing mean oscillation nearthe boundary of D, if

lim|z|→1−

1

|Sz|

∫

Sz

∣∣∣∣f(w)−1

|Sz|

∫

Sz

f(u)dA(u)

∣∣∣∣ dA(w) = 0.

Theorem 3.5. [7, Theorem 13] The algebra Q is the set of boundedfunctions with vanishing mean oscillation near the boundary, i.e.,

Q = VMO∂(D) ∩ L∞(D).

For the proof we refer the reader to [7].

Lemma 3.6. Let f be a function in QC. Then, the function fR belongsto Q.

Proof. According to Theorem 3.5 and Definition 3.4, we need to esti-mate

(3)1

|Sz|

∫

Sz

∣∣∣∣f(w)−1

|Sz|

∫

Sz

f(u)dA(u)

∣∣∣∣ dA(w).

Using polar coordinates we get that this quantity is equal to


(4)2

|Iz|2

∫

Iz

∫

Iz

|f(θ)− f(φ)|dA(θ)dA(φ).

If z is close to the boundary, then the measure of |Iz| is small. Hence,the expresion in (4) goes to zero because f is in QC. This implies thatthe expresion in (3) goes to zero if |z| goes to 1, thus fR is in Q asrequired.

In Lemma 4.5 we prove that the harmonic extension fH also belongsto Q, but the tools needed for the proof of this fact are not stablishedyet.

From now on, and until further notice, we use only the radial ex-tension of a function in PQC. To simplify the notation, we use PQCto denote functions defined on ∂D as well as radial extensions of suchfunctions. Moreover g will denote both, the function on ∂D and itsradial extension to D.

By TPQC we denote the C∗-algebra generated by Toeplitz operators

with symbols in PQC. We use TPQC to denote the Calkin algebraTPQC/K. The main goal of this paper is to describe the C∗-algebra

TPQC .We use the Douglas-Varella Local Principle (DVLP for short) to

describe the C∗-algebra TPQC . A complete description of this principlecan be found, for example, in [6, Chapter 1].

Let A be a C∗-algebra with identity, Z be some of its central C∗-subalgebras with the same identity, T be the compact of maximal idealsof Z. Furthermore, let Jt be the maximal ideal of Z corresponding tothe point t ∈ T , and J(t) := Jt · A be the two sided closed ideal in thealgebra A generated by Jt .

We define Et := A/J(t) as the local algebra at the point t. [a]t standsfor the class of the element a in the quotient algebra Et. Two elementsa, b of A are say locally equivalents at the point t ∈ T if [a]t = [b]t inEt.

Using the spaces

E :=⋃t∈T

Et

and T , there is a standard procedure to construct the C∗-bundle ξ =(p,E, T ), where p : E → T is a projection such that p|Et = t. Thisprocedure gives to E a compatible topology such that the function a :T → E with a(t) = [a]t ∈ Et is continuous for each a in A.

28 Breitner Ocampo

A function γ : T → E is called a section of the C∗-bundle ξ, ifp(γ(t)) = t. Let Γ(ξ) denote the C∗-algebra of all continuous sectionsdefined on ξ.

Theorem 3.7 (Douglas-Varela Local Principle). The C∗-algebra A isisomorphic and isometric to the C∗-algebra Γ(ξ), where ξ is the C∗-bundle constructed from A and its central algebra Z.

Lemma 3.6 and the results in [7] imply that the quotient TQC =

TQC/K is a commutative C∗-subalgebra of TPQC . Thus we use TQC

as the central algebra needed to apply the DVLP in the description ofTPQC . The algebra TQC can be identified with QC

TQC = Tf +K|f ∈ QC,

hence we localize by points in M(QC). We first construct the systemof ideals parametrized by points x in M(QC).

Definition 3.8. For every point x ∈ M(QC), we define the maximalideal of TQC , Jx := f ∈ QC : f(x) = 0 = Tf + K|f(x) = 0. Theideal J(x) is defined as the set Jx · TPQC/K.

We set the notation TPQC(x) := TPQC/J(x) for the local algebra at

the point x. The class of the element Tf +K ∈ T in the quotient algebra

T (x) shall be denoted by [Tf ]x, in order to simplify the notation we say

“Tf is locally equivalent...” instead of “the class of [Tf ]x is loccalyequivalent...”

Lemma 3.9. Let f be a function in QC and x a point of M(QC). TheToeplitz operator Tf is locally equivalent, at the point x, to the complexnumber f(x) (realized as the operator f(x)I).

Proof. Let x be a point in M(QC) and f be a function in QC. Thefunction f − f(x) belongs to J(x), thus, the operator Tf − Tf(x) =

Tf−f(x) is zero in TPQC(x). This means that the operator Tf is locallyequivalent to the operator Tf(x) = f(x)I and then, the operator Tf islocally equivalent to the complex number f(x).

Lemma 3.10. Let x be a point of Mλ(QC) with λ /∈ Λ and a be afunction in PC. Then, the Toeplitz operator Ta, in the local algebraTPQC(x), is equivalent to the complex number a(λ) (realized as the op-erator a(λ)I).


The proof is very similar to the proof of Lemma 3.9 and is omitted.For the case when x ∈ Mλk

(QC), we use Lemma 2.10 to split the fiberMλk

(QC) into three disjoints sets: M+λk(QC) \ M0

λk(QC), M−

λk(QC) \

M0λk(QC) and M0

λk(QC).

Lemma 3.11. Let x be a point of M+λk(QC) \ M0

λk(QC) and a be a

function in PC. Then, the Toeplitz operator Ta, in the local algebraTPQC(x), is equivalent to the complex number a+k (realized as the oper-ator a+k I).

Proof. Let a be a function for which a+k = 0. If x belongs to

M+λk(QC) \M0

λk(QC),

then x belongs to M+λk(QC) and does not belong to M−

λk(QC). This

implies the existence of a function g in QC such that

limλ→λ−

k

g(λ) = 0

and g(x) = 1.The product ag is continuous at λk and ag(λk) = 0. The difference

Ta−Tag can be rewritten as T(1−g)a = T1−gTa+K where K is a compactoperator. Since the function 1 − g vanishes at x, T1−g belongs to Jx,and then Ta − Tag belongs to J(x).

From this we conclude that the Toeplitz operator with symbol ais locally equivalent to the Toeplitz operator with symbol ag. At thesame time, the Toeplitz operator Tag is locally equivalent to the complexnumber 0 = ag(λk), hence, the operator Ta is locally equivalent to thecomplex number a+k = 0.

For the general case, if the function a in PC has limit a+k = 0, weconstruct the function b(λ) = a(λ)−a+k . The function b has lateral limitb+k = 0, fulfilling the initial assumption of the proof. By the first part ofthe proof, the Toeplitz operator Tb = Ta−a+k I is locally equivalent to thecomplex number 0, thus the Toeplitz operator Ta is locally equivalentto the complex number a+k .

Similarly the following lemma holds:

Lemma 3.12. Let x be a point of M−λk(QC) \ M0

λk(QC) and a be a

function in PC. Then, the Toeplitz operator Ta, in the local algebraTPQC(x), is equivalent to the complex number a−k (realized as the oper-ator a−k I).

30 Breitner Ocampo

Now we analize the case when x belongs to central part of the fiberMλk

(QC), i.e, x ∈ M0λk(QC). For this case we use some results regard-

ing Toeplitz operators with zero-order homogeneous symbols defined inthe upper half plane Π.

We consider A2(Π) as the Bergman space of Π, that is, the (closed)space of square integrable and analytic functions on Π. Let BΠ standsfor the Bergman projection BΠ : L2(Π) → A2(Π).

Denote by A∞ the C∗-algebra of bounded mesureable homogeneousfunctions on Π of zero-order, or functions depending only in the po-lar coordinate θ. We introduce the Toeplitz operator algebra T (A∞)generated by all Toeplitz operators

Ta : φ ∈ A2(Π) → BΠ(aφ) ∈ A2(Π)

with defining symbols a(r, θ) = a(θ) ∈ A∞.

Theorem 3.13. [6, Theorem 7.2.1] For a = a(θ) ∈ A∞, the Toeplitzoperator Ta acting in A2(Π) is unitary equivalent to the multiplicationoperator γaI acting on L2(R). The function γa(s) is given by

γa(s) =2s

1− e−2sπ

∫ π

0a(θ)e−2sθdθ.

Let ∂D+k denote the upper half of the circunference and D+

k the upperhalf of the disk D both determined by the diameter passing through λk

and −λk. Denote by ∂D−k and D−

k the complement of ∂D+k and D+

k ,respectively.

Let H be a function in PC with lateral limits H+k and H−

k , weconstruct the function h in PC such that h = H+

k in ∂D+ and h = H+k

in ∂D−. The function H − h is continuous at λk and (H − h)(λk) = 0.For any point x ∈ M0

λk, the Toeplitz operator TH−h belongs to J(x)

and thus TH and Th are locally equivalent at the point x.

The previous paragraph implies that the C∗-algebra generated byTH in TPQC(x) depends only on the values H+

k and H−k . To describe

the local algebra at x, we need to analize the algebra generated by theToeplitz operator which symbol h is constant on both ∂D+ and ∂D−.The radial extension of such function h is the function which is constantin D+ with value A and constant in D− with value B for some complexconstants A and B.

Let φ be a Mobius transformation which sends the upper half planeto the unit disk and such that: φ(0) = λk, φ(i) = 0 and φ(∞) = −λk.


Using the function φ we construct a unitary transformation W whichsends L2(D) onto L2(Π). Under the unitary transformation W , theToeplitz operator with symbol h, acting on A2(D), is unitary equivalentto the Toeplitz operator Th(φ(w)) acting on A2(Π). The correspondingsymbol h(φ(w)) is a homogeneous function of zero-order.

By Theorem 3.13, the Toeplitz operator with symbol h(φ(w)), actingon A2(Π), is unitary equivalent to the multiplication operator by thefunction γh(φ(w)), acting on L2(R). Following the unitary equivalenceswe deduce that Th is unitary equivalent to γh(φ(w)).

By Corollary 7.2.2 in [6], the function γh(φ)(s) is continuous in R =R ∪ −∞,+∞, the two point compactification of R; furthermore, for

the function h = χD+k

we have γh(φ)(s) = 1−e−sπ

1−e−2sπ = 11+e−2sπ . The

function γh(φ)(s) and the identity function 1 generate the algebra ofcontinuous functions on R [6, Corollary 7.2.6].

Recall that all piecewise constant functions are generated by linearcombinations of the identity and the function χD+

k. Thus, using the

change of variables

t =1

1 + e−2sπ,

which is a homeomorphism between [0, 1] and R, we conclude that thelocal algebra TPQC(x) is isomorphic to C[0, 1] for every x ∈ M0

λk(QC);

further, such isomorphism, denoted here by ψ, acts on the generatorTχD+

k

as follows:

TχD+k

→ t.

This implies that the Toeplitz operator with symbol a in PC is sent toC([0, 1]), via ψ, to the function a−k (1 − t) + a+k t. Thus we come to thefollowing lemma.

Lemma 3.14. If x belongs to M0λk(QC), then the local algebra gener-

ated by the Toeplitz operators with symbols in PQC is isometric andisomorphic to the algebra of all continuous functions in [0, 1].

With the set M(QC), we construct the C∗-bundle

ξPQC := (p,E,M(QC)).

We use the description of the local algebras given by Lemmas 3.10, 3.11,3.12 and 3.14 to construct the bundle

E :=⋃

x∈M(QC)

Ex

32 Breitner Ocampo

where

• Ex = C, if x ∈ Mλ(QC) with λ /∈ Λ,

• Ex = C, if x ∈ M+λk(QC) \M0

λk(QC), λk ∈ Λ,

• Ex = C, if x ∈ M−λk(QC) \M0


• Ex = C([0, 1]), if x ∈ M0λk(QC), λk ∈ Λ.

The function p is the natural projection from E to M(QC).Let Γ(ξPQC) denote the algebra of all continuous sections of the

bundle ξPQC . Applying the DVLP (Theorem 3.7) we get the followingtheorem:

Theorem 3.15. The C∗algebra TPQC is isometric and isomorphic tothe C∗-algebra of continuous sections over the C∗-bundle ξPQC.

As a corollary of Theorem 3.15, the algebra TPQC is commuta-

tive, thus there exists a compact space X = M(TPQC), such that

TPQC∼= C(X) = C(M(TPQC)). The compact space M(TPQC) can

be constructed using the irreducible representations of TPQC .

Let ∂D be the set ∂D cut by the points λk of Λ. The pair of pointsof ∂D which correspond to the point λk will be denoted by λ+

k and λ−k ,

following the positive orientation of ∂D. Let In := ni=1[0, 1]k be the

disjoint union of n copies of the interval [0, 1].Denote by Σ the union of ∂D and In with the point identification

λ−k ≡ 0k λ+

k ≡ 1k,

where 0k and 1k are the boundary points of [0, 1]k, k = 1, . . . , n.Let M(TPQC) :=

⋃λ∈Σ

Mλ(TPQC) where each fiber corresponds to

Mλ(TPQC) :=Mλ(QC) if λ ∈ ∂D, λk /∈ Λ

Mλ+k(TPQC) :=

(M+

λk(QC) \M−

λk(QC)

)∪M0


Mλ−k(TPQC) :=

(M−

λk(QC) \M+

λk(QC)

)∪M0


Mt(TPQC) :=M0λk(QC) if t ∈ (0, 1)k, k = 1, . . . , n.

With the help of Figure 1, we draw the maximal ideal space forTPQC . The idea is to duplicate the set M0

λ(QC) and then connect thistwo copies by the interval [0, 1].


Mλ(TPQC) = Mλ(QC) Mλ+i(TPQC)

Mt(TPQC)

Mλ−i(TPQC)

M0λ(QC)

M+λi(QC)\M−

λi(QC)

M−λi(QC)\M+

λi(QC)

Figure 2: The maximal ideal space of TPQC .

We use the topology of M(QC) in order to describe the topologyof M(TPQC). We only describe the topology of the fibers Mλ±

k(TPQC)

and Mt(TPQC), since the topology on the other fibers corresponds tothe topology of Mλ(QC). For x in M(QC), let Ω(x) denote the familyof open neighbourhoods of x. For x ∈ Mλ(QC) and N in Ω(x), letNλ = N ∩Mλ(QC), and let Nλ+ and Nλ− denote the sets of points inN that lie above the semicircles ∂D+

k and ∂D−k , respectively.

Consider the fiber Mλ+k(TPQC). The sets N in Ω(x) satisfying N =

Nλk∪Nλ+

kform neighbourhoods of x ∈ M+

λk(QC)\M−

λk(QC). Let Ω+(x)

be the set of neighbourhoods N in Ω(x) satisfying N = Nλ ∪Nλ+ . Thesets

(Nλk× (1− ε, 1]) ∪Nλ+

kN ∈ Ω+(x), and 0 < ε < 1,

form open neighbourhoods of points x in M0λk(QC).

The open neighbourhoods for points in the fiber Mλ−k(TPQC) are

constructed analogously.The sets N in Ω(x) satisfying N = Nλk

∪Nλ−kform neighbourhoods

of x ∈ M−λk(QC) \M+

λk(QC).

The sets

(Nλk× [0, ε)) ∪Nλ−

kN ∈ Ω−(x), and 0 < ε < 1,

form open neighbourhoods of points x in M0λk(QC).

Each set M0λk(QC) × (0, 1) is open in M(TPQC) and carries the

product topology.

34 Breitner Ocampo

Theorem 3.16. Let X := M(TPQC) as described above. The algebra

TPQC is isomorphic to the algebra of continuous functions over X, theisomorphism acts on the generators in the following way:

• For generators which symbols is a function a in PC

Φ(Ta)(x) =

a(λ), if x ∈ Mλ(TPQC) with λ = λk;

a+k , if x ∈ Mλ+k(TPQC);

a−k , if x ∈ Mλ−k(TPQC);

a−k (1− t) + a+k t, if x ∈ Mt(TPQC).

• For generators which symbols are functions f in QC, Φ(Tf )(x) =f(x).

4 Independence of the result on the extensionchosen

In this section we prove that the description of the algebra TPQC doesnot depend of the extension chosen for functions in PQC. Recall thatPQC is the algebra generated by PC and QC. This algebra is definedon ∂D and then extended to the whole disk by two different ways:

• the harmonic extension gH given by the Poisson formula 2,

• the radial extension gR, defined by gR(r, θ) = g(θ).

Let a be a function in PC. At the point x ∈ Mλ(QC), for λ /∈ Λ;the Toeplitz operator TaR is locally equivalent to the complex numbera(λ). The same still true if we use the harmonic extension aH . Forpoints x in M+

λk(QC) \M0

λk(QC) (respectively M−

λk(QC) \M0

λk(QC)),

the Toeplitz operators TaR and TaH are equivalent to the number a+k(Respectively a−k ), and then, the local algebras are the same.

Now, we analize the case when x belongs to M0λk(QC). Let a be a

function in PC, we construct a function a such that a = a+k in ∂D+k

and a = a−k in ∂D−k . The Toeplitz operator with symbol aH is locally

equivalent to TaH .As in Section 2, we use a Mobius transformation φ to generate a

unitary operator between L2(D) and L2(Π). For the function a in PCdescribed earlier, the function aH(φ(z)) is harmonic in Π and corre-sponds to the harmonic extension of a(φ(t)).


The harmonic extension of a(φ(t)) is aΠH := θπ (a

−k −a+k )−a+k , which is

a zero-order homogeneous function on Π. By Theorem 3.13, the Toeplitzoperator TaΠH

is unitary equivalent to the multiplication operator γaΠH.

The function γaΠHis given by

γaΠH= A

(1

2sπ− 1

e−2sπ − 1

)+B,

for suitable complex constants A and B. Corollary 7.2.7 of [6] showsthat the algebra generated by γaΠH

and the identity is the algebra of

continuous functions on R.Following the unitary equivalences from TaH to γaΠH

and makinga change of variables, we have that the algebra generated by TaH isisomorphic to the algebra of continuous functions over the segment [0, 1].

We already know, from Theorem 3.14, that the Toeplitz operatorwith symbol aR generates the algebra of continuous functions over [0, 1]as well, so, locally, the algebras genereated by TaH and TaR are thesame. We have thus proved

Theorem 4.1. Consider the algebra PC defined on ∂D and its exten-sions PCR and PCH . The local algebras TPCR

(x) and TPCH(x) are the

same.

To show the same theorem for functions f in QC we need to stab-lish some definitions related to the space Q in Definition 3.3. Furtherinformation on the theorems and definitions below can be found in [7].

Definition 4.2. For a function g ∈ L∞(D) we define its Berezin trans-form g by the formula

g(z) :=

∫

D

g(w)1− |w|2

(1− zw)2dA(w).

Note that g belongs to L∞(D) and ‖g‖∞ ≤ ‖g‖∞.

Definition 4.3. Define B as the set of bounded functions on D suchthat its Berezin transform goes to zero as z approaches to the boundaryof D, that is,

B := f ∈ L∞(D) : lim|z|→1

f(z) = 0.

36 Breitner Ocampo

In [1], S. Axler and D. Zheng proved that a Toeplitz operator Tg,with bounded symbol g, is compact if and only if g is in B. The nextlemma is due to D. Sarason and is a combination of some results in [5].

Theorem 4.4. The set Q in Definition 3.3 is described as

Q = f ∈ L∞(D) : lim|z|→1

|f |2(z)− |f(z)|2 = 0.

The set B ∩Q is an ideal of Q and, for f ∈ Q, the Toeplitz operator Tf

is compact if and only if f belongs to B ∩Q.

Lemma 4.5. For a function f in QC, the function fH belongs to Q.

Proof. For this proof we use two facts:

1. The Berezin transform of a harmonic function is the function itself,in our case, fH = fH .

2. By [3], the harmonic extension is asymptotically multiplicative inQC, that is

lim|z|→1−

|f |2H(z)− |fH(z)|2 = 0.

Now we proceed with the proof:

|fH |2(z)− |fH(z)|2 ≤∣∣∣|fH |2(z)− |f |2H(z)

∣∣∣+ ∣∣|f |2H(z)− |fH(z)|2∣∣

≤∣∣∣∣ ˜|fH |2(z)− |f |2H(z)

∣∣∣∣+∣∣|f |2H(z)− |fH(z)|2

∣∣ ,

the last two sumands goes to zero as z approaches to the boundary ∂D;the later because of item 2, and the former is due to items 2 and 1.Finally, using Theorem 4.4, we have that fH is in Q.

Definition 4.6 (page 626, [7]). For each point z in D we define

S′z :=

w ∈ D : |w| ≥ |z| and | arg(z)− arg(w)| ≤ 1− |z|

2

.

Definition 4.7 (page 627, [7]). For a function f in L∞(D) define

f(z) :=1

|S′z|

∫

S′z

f(w)dA(w).


Definition 4.8 (page 626, [7]). Let f be in L∞(D, dA). We say f is inESV (D) if and only if for any ε > 0, and σ ∈ (0, 1), there exists δ0 > 0such that |f(z)−f(w)| < ε whenever w ∈ S′

z and |z|, |w| ∈ [1−δ, 1−δσ],with δ < δ0.

The notation ESV (D) means eventually slowly varying and was in-troduced by C. Berger and L. Coburn in [2].

Theorem 4.9. [7, Theorem 5] Q = ESV +Q∩B. A decomposition isgiven by f = f + (f − f). Moreover

ESV (D) ∩B = f ∈ L∞(D) | f(z) → 0 as |z| → 1−.

We calculate fR and get fR(z) = Iz(f). Then, Theorem 4.9 gives usthe decomposition fR(z) = Iz(f)+(fR(z)− Iz(f)), where Iz(f) belongsto ESV (D) and fR(z)− Iz(f) belongs to Q ∩B.

Lemma 4.10. Consider the function f in QC. The Toeplitz operatorwith symbol fR − fH is compact.

Proof. We write fR(z)−fH(z) = (Iz(f)−fH(z))+(fR(z)−Iz(f)). Thefirst summand goes to zero as |z| goes to 1 by Theorem 2.8. Then byTheorem 4.9, the function Iz(f) − fH(z) belongs to ESV (D) ∩ B. Bythe decomposition of Q as ESV (D)+Q∩B we have that (fR(z)−Iz(f))belongs to Q ∩B.

In summary, the function fR(z)− fH(z) belongs to Q∩B and thenthe Toeplitz operator with symbol fR − fH is compact.

Now we stablish the main result of this section: the algebra de-scribed in Theorem 3.15 does not depend on the extension chosen forthe symbols in PQC.

Theorem 4.11. Let PQCR and PQCH denote, respectively, the radialand the harmonic extension to the disk of functions in PQC. Then, theCalkin algebras TPQCR

/K and TPQCH/K are the same.

Proof. The proof follows from Theorem 4.1 and Lemma 4.10.

Ocampo, B.Departmento de Matematicas,Centro de Investigacion y de Estudios Avanzados del IPN,Apartado Postal 14-740,07360 Mexico, [email protected]

38 Breitner Ocampo

References

[1] Axler S.; Zheng D., Compact operators via the Berezin transform,Indiana Univ. Math. J. 47:2 (1998), 387–400.

[2] Berger C.A.; Coburn L. A., Toeplitz operators and quantum mechan-ics, J. Funct. Anal. 68:3 (1986), 273–299.

[3] Sarason D., Algebras of functions on the unit circle, Bull. Amer.Math. Soc. 79 (1973), 286–299.

[4] Sarason D., Functions of vanishing mean oscillation, Trans. Amer.Math. Soc. 207 (1975), 391–405.

[5] Sarason D., Toeplitz operators with piecewise quasicontinuous sym-bols, Indiana Univ. Math. J. 26:5 (1977), 817–838.

[6] Vasilevski N. L., Commutative Algebras of Toeplitz Operators onthe Bergman Space, vol. 185 of Operator Theory: Advances andApplications, Birkhauser Verlag, Basel, 2008.

[7] Zhu K. H., VMO, ESV, and Toeplitz operators on the Bergmanspace, Trans. Amer. Math. Soc. 302:2 (1987), 617–646.

Morfismos se imprime en el taller de reproduccion del Departamento de Matema-ticas del Cinvestav, localizado en Avenida Instituto Politecnico Nacional 2508, Colo-nia San Pedro Zacatenco, C.P. 07360, Mexico, D.F. Este numero se termino deimprimir en el mes de julio de 2015. El tiraje en papel opalina importada de 36kilogramos de 34 × 25.5 cm. consta de 50 ejemplares con pasta tintoreto color verde.

Apoyo tecnico: Omar Hernandez Orozco.

morfismos, vol 19, no 1, 2015

Documents