uniqueness theorem
DESCRIPTION
theorTRANSCRIPT
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Uniqueness theorems for combinatorialC-algebras
Sarah Reznikoff
joint work with Jonathan H. Brown, Gabriel Nagy, Aidan Sims, and Dana Williamsfunded in part by NSF DMS-1201564
Nebraska-Iowa Functional Analysis Seminar 2015Creighton University
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Introductionk -graph algebras
History of uniqueness theoremsAperiodicity
Classical theoremsRecent workWhy graph algebras?
Let G be a graph, k -graph, or groupoid, and C(G ) theuniversal C*-algebra defined from it.
Question: Under what circumstances is a -homomorphism : C(G ) B(H) injective?Classical theorems addressing this question assume either(a) the existence of intertwining gauge actions" on the
algebras (Gauge Invariant Uniqueness Theorem 1), or(b) an aperiodicity condition on the graph itself (Cuntz-Krieger
Uniqueness Theorem 2),and conclude that is injective iff it is nondegenerate, i.e.,injective on the diagonal subalgebra D.
1an-Huef, 972Fowler-Kumjian-Pask-Raeburn, 97
Sarah Reznikoff Uniqueness theorems for combinatorial C-algebras
-
Introductionk -graph algebras
History of uniqueness theoremsAperiodicity
Classical theoremsRecent workWhy graph algebras?
Theorem (Brown-Nagy-R-Sims-Williams) There is a canonicalsubalgebraM C(G ) such that a -homomorphism : C(G ) B(H) is injective iff |M is injective.Moreover,M C(G ) is a Cartan inclusion.
[NR1] Nagy and Reznikoff, Abelian core of graph algebras,J. Lond. Math. Soc. (2) 85 (2012), no. 3, 889908.[NR2] Nagy and Reznikoff, Pseudo-diagonals and uniquenesstheorems, Proc. AMS (2013).[BNR] Brown, Nagy, Reznikoff A generalized Cuntz-Kriegeruniqueness theorem for higher-rank graphs, JFA (2013).[BNRSW] Brown, Nagy, Reznikoff, Sims, and Williams, Cartansubalgebras of groupoid C-algebras (2015).
Sarah Reznikoff Uniqueness theorems for combinatorial C-algebras
-
Introductionk -graph algebras
History of uniqueness theoremsAperiodicity
Classical theoremsRecent workWhy graph algebras?
Drinen (1999): Every AF algebra is Morita equivalent to agraph algebra.
Kumjian-Pask-Raeburn (1998)I The algebra is AF iff the graph has no cycles.I All simple graph algebras are AF or purely infinite.
Hong-Szymanski (2004): the ideal structure of the algebra canbe completely described from the graph.
Generalizations and related constructions: Exel crossedproduct algebras, Leavitt path algebras (Abrams, Ruiz,Tomforde), topological graph algebras (Katsura), Ruellealgebras (Putnam, Spielberg), Exel-Laca algebras, ultragraphs(Tomforde), Cuntz-Pimsner algebras, higher-rankCuntz-Krieger algebras (Robertson-Steger), etc.
Sarah Reznikoff Uniqueness theorems for combinatorial C-algebras
-
Introductionk -graph algebras
History of uniqueness theoremsAperiodicity
Classical theoremsRecent workWhy graph algebras?
k -graph algebras (Kumjian and Pask, 2000)
developed to generalize graph algebras and higher-rankCuntz-Krieger algebras, whether simple, purely infinite, or AF can be determined fromproperties of the graph (Kumjian-Pask, Evans-Sims), are groupoid C*-algebras, include examples of algebras that are simple but neither AFnor purely infinite, and hence not graph algebras(Pask-Raeburn-Rordam-Sims), include examples that can be constructed from shift spaces(Pask-Raeburn-Weaver), can be used to construct any Kirchberg algebra (Spielberg).
Sarah Reznikoff Uniqueness theorems for combinatorial C-algebras
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Introductionk -graph algebras
History of uniqueness theoremsAperiodicity
DefinitionExample: kCuntz-Krieger families
Let k N+. We regard Nk as a category with a single object, 0,and with composition of morphisms given by addition.
A k -graph is a countable category along with a degreefunctor d : Nk satisfying the unique factorization property:
For all , and m, n Nk , if d() = m + n then there areunique d1(m) and d1(n) such that = .
I Denote the range and source maps r , s : .I Refer to objects as vertices and morphisms as paths.I We assume: for all v , n Nk ,
0 < |r1({v}) d1({n})|
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Introductionk -graph algebras
History of uniqueness theoremsAperiodicity
DefinitionExample: kCuntz-Krieger families
Example: Rectangles in NkLet k := {(l ,n) Nk Nk | l n} with d(l ,n) = n l ,s(m, l) = l = r(l ,n), and (m, l)(l ,n) = (m,n).
0
m
nl
n
m
Sarah Reznikoff Uniqueness theorems for combinatorial C-algebras
-
Introductionk -graph algebras
History of uniqueness theoremsAperiodicity
DefinitionExample: kCuntz-Krieger families
A Cuntz-Krieger -family in a C*-algebra A is a set{T, } of partial isometries in A satisfying
(i) {Tv | v d1({0}) is a family of mutually orthogonalprojections,
(ii) T = TT for all , s.t. s() = r(),(iii) T T = Ts() for all , and(iv) for all v d1({0}) and n Nk , Tv =
d()=nr()=v
TT .
C() will denote the C*-algebra generated by a universalCuntz-Krieger -family, (S, ), with P = SS.Note: C() = span{SS |, | s() = s()}Defn: The diagonal D := span{SS | }.
Sarah Reznikoff Uniqueness theorems for combinatorial C-algebras
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Introductionk -graph algebras
History of uniqueness theoremsAperiodicity
Classic theoremsNon-degeneracy does not suffice
Classic uniqueness theorems
Coburns Theorem (67) ef
Any nondegenerate representation C(Te,Tf ) is isomorphic tothe Toeplitz algebra T , generated by one non-unitary isometry.
Cuntz (77) Any nondegenerate representation isisomorphic to the Cuntz AlgebraOn, generated by n
partial isometries Si satisfying i , Si Si =n
j=1
SjSj .
n loops
Cuntz-Krieger (80) Graph with 0-1 adjacency matrix AWhen A satisfies a fullness condition (I), any nondegeneraterepresentation is isomorphic to the Cuntz-Krieger algebra OA.
Sarah Reznikoff Uniqueness theorems for combinatorial C-algebras
-
Introductionk -graph algebras
History of uniqueness theoremsAperiodicity
Classic theoremsNon-degeneracy does not suffice
Is non-degeneracy enough?
No! Consider the cycle of length three, E .The map : C(E) M3(C) given by
Svi 7 i,i Sei,j 7 j,i .
is a -homomorphism. However
(Se3,1Se2,3Se1,2) = 1,33,22,1 = 1,1 = (Sv1),
whereas Se3,1Se2,3Se1,2 6= Sv1 in C(E), so is not injective.
v1
v2 E
v3e3,1
e2,3
e1,2
Sarah Reznikoff Uniqueness theorems for combinatorial C-algebras
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Introductionk -graph algebras
History of uniqueness theoremsAperiodicity
GroupoidsThe C*-algebra of a groupoidState extensions
The infinite path space
Defn. An infinite path in a k -graph is a degree-preservingcovariant functor x : k .
k = 1 picture e1 e2x(1,3) = e1e2d(e1e2) = 2
k = 2 picture
r()
s() x((1,2), (3,3)) = d() = (2,1)
Sarah Reznikoff Uniqueness theorems for combinatorial C-algebras
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Introductionk -graph algebras
History of uniqueness theoremsAperiodicity
GroupoidsThe C*-algebra of a groupoidState extensions
An infinite path x in a k -graph is eventually periodic if there are 6= in and y such that x = y = y ; otherwise x isaperiodic.
x
y
y
Sarah Reznikoff Uniqueness theorems for combinatorial C-algebras
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Introductionk -graph algebras
History of uniqueness theoremsAperiodicity
GroupoidsThe C*-algebra of a groupoidState extensions
is aperiodic if every vertex is the range vertex of an aperiodicinfinite path. In a directed graph, cycles without entry reveal failure ofaperiodicity.
v
Clearly the only infinite path with range v , , is eventually periodic.
A k -graph is aperiodic iff D is a masa.Cuntz-Krieger Uniqueness Theorem:(Kumjian-Pask-Raeburn-Fowler, et. al. (90s))When is nondegenerate and the graph satisfies
(L) every cycle has an entrythen is injective.
Sarah Reznikoff Uniqueness theorems for combinatorial C-algebras
-
Introductionk -graph algebras
History of uniqueness theoremsAperiodicity
GroupoidsThe C*-algebra of a groupoidState extensions
Theorem Szymanski (2001), Nagy-R (2010):Condition (L) can be replaced with a condition on the spectrumof (S) for cycles without entry.
Uniqueness theorems of Raeburn-Sims-Yeend andKumjian-Pask assume aperiodicity of the k -graph.
Theorem Nagy-Brown-R (2013, JFA)A -homomorphism : C() A is injective iff it is injectiveon the subalgebraM := C(SS | = }.Theorem Yang (2014)I M = D.I For fixed m,n Nk , and 0 with d() = n, and an
element X =
SS ofM , there is at most one termS0S
0
with d(0) = m.
Sarah Reznikoff Uniqueness theorems for combinatorial C-algebras
-
Introductionk -graph algebras
History of uniqueness theoremsAperiodicity
GroupoidsThe C*-algebra of a groupoidState extensions
A groupoid G is a small category in which every element hasan inverse. When G is a topological groupoid, C(G) is definedto be a completion of Cc(G).The isotropy subgroupoid is the setIso(G) := {g G | r(g) = s(g)}Theorem (Brown-Nagy-R-Sims-Williams, 2014)Let G be a locally compact, amenable, Hausdorff, talegroupoid, with (IsoG) is closed. If : C(G) A is aC-homomorphism, then the following are equivalent.
(i) is injective.(ii) is injective onM := C((Iso(G))).
Sarah Reznikoff Uniqueness theorems for combinatorial C-algebras
-
Introductionk -graph algebras
History of uniqueness theoremsAperiodicity
GroupoidsThe C*-algebra of a groupoidState extensions
Assume G is a 2nd countable locally compact Hausdorff talegroupoid. For f Cc(G), define
f () = f (1) f g () ==
f ()g().
Define the I-norm on Cc(G) by
fI = supuG(0)
max
Gu
|f ()|,Gu
|f ()| , and let
f = sup{pi(f ) |pi is an I-norm bounded -rep. of Cc(G)}C(G) is the completion of Cc(G) with respect to this norm.
Sarah Reznikoff Uniqueness theorems for combinatorial C-algebras
-
Introductionk -graph algebras
History of uniqueness theoremsAperiodicity
GroupoidsThe C*-algebra of a groupoidState extensions
Given a k -graph , define
G = {(y ,d , y) | y , , , s() = s() = r(y),d = d() d()}
withr(x ,d , y) = x , s(x ,d , y) = y
(x ,d , y)(z,d ,w) = y (z)(x ,d + d ,w).
The cylinder sets Z (, ) = {(y ,d , z) G} form a basisfor an tale groupoid. Moreover, C() = C(G) andM = C((Iso(G))) via themap SS 7 Z (,).
Sarah Reznikoff Uniqueness theorems for combinatorial C-algebras
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Introductionk -graph algebras
History of uniqueness theoremsAperiodicity
GroupoidsThe C*-algebra of a groupoidState extensions
(Renault, 80) A masa C*-subalgebra B A is Cartan if(i) a faithful conditional expectation A B,(ii) The normalizer of B in A generates A, and(iii) B contains an approximate unit of A.Extension properties for pure states on masa B A:(UEP) Every pure state extends uniquely to A.(AEP) Densely many pure states extend uniquely.
Kadison-Singer Problem (Marcus-Spielman-Srivastava)(UEP) holds for all masa in B(`2(N)).
Thm (Nagy-R, 2011; NRBSW, 2014)M C(G) is Cartan.Thm (NRBSW, 2014) All Cartan subalgebras satisfy the AEP.
Sarah Reznikoff Uniqueness theorems for combinatorial C-algebras
-
Introductionk -graph algebras
History of uniqueness theoremsAperiodicity
GroupoidsThe C*-algebra of a groupoidState extensions
Thank you!
Sarah Reznikoff Uniqueness theorems for combinatorial C-algebras
-
Introductionk -graph algebras
History of uniqueness theoremsAperiodicity
GroupoidsThe C*-algebra of a groupoidState extensions
A. an Huef and I. Raeburn, The ideal structure ofCuntz-Krieger algebras, Ergodic Theory Dynam. Systems17 (1997), 611624.
J.H. Brown, G. Nagy, and S. Reznikoff, A generalizedCuntz-Krieger uniqueness theorem for higher-rank graphs,J. Funct. Anal. (2013),http://dx.doi/org/10.1016/j.jfa.2013.08.020.
J.H. Brown, G. Nagy, S. Reznikoff, A. Sims, andD. Williams, Cartan subalgebras of groupoid C-algebras
K.R. Davidson, S.C. Power, and D. Yang, Dilation theory forrank 2 graph algebras, J. Operator Theory.
Sarah Reznikoff Uniqueness theorems for combinatorial C-algebras
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Introductionk -graph algebras
History of uniqueness theoremsAperiodicity
GroupoidsThe C*-algebra of a groupoidState extensions
P. Goldstein, On graph C*-algebras, J. Austral. Math. Soc.72 (2002), 153160
A. Kumjian and D. Pask, Higher rank graph C*-algebras,New York J. Math. 6 (2000), 120.
A. Kumjian, D. Pask, and I. Raeburn, Cuntz-Kriegeralgebras of directed graphs, Pacific J. Math. 184 (1998)161174.
A. Kumjian, D. Pask, I. Raeburn, and J. Renault, Graphs,groupoids and Cuntz-Krieger algebras, J. Funct. Anal. 144(1997), 505541
G. Nagy and S. Reznikoff, Abelian core of graph algebras,J. Lond. Math. Soc. (2) 85 (2012), no. 3, 889908.
Sarah Reznikoff Uniqueness theorems for combinatorial C-algebras
-
Introductionk -graph algebras
History of uniqueness theoremsAperiodicity
GroupoidsThe C*-algebra of a groupoidState extensions
G. Nagy and S. Reznikoff, Pseudo-diagonals anduniqueness theorems, (2013), to appear in Proc. AMS.
D. Pask, I. Raeburn, M. Rrdam, A. Sims, Rank-twographs whose C*-algebras are direct limits of circlealgebras, J. Functional Anal. 144 (2006), 137178.
I. Raeburn, A. Sims and T. Yeend, Higher-rank graphs andtheir C*-algebras, Proc. Edin. Math. Soc. 46 (2003) 99115.
D. Robertson and A. Sims, Simplicity of C*-algebrasassociated to higher-rank graphs. Bull. Lond. Math. Soc. 39(2007), no. 2, 337344.
G. Robertson and T. Steger, Affine buildings, tiling systemsand higher rank Cuntz-Krieger algebras, J. ReineAngew. Math. 513 (1999), 115144.
Sarah Reznikoff Uniqueness theorems for combinatorial C-algebras
-
Introductionk -graph algebras
History of uniqueness theoremsAperiodicity
GroupoidsThe C*-algebra of a groupoidState extensions
A. Sims, Gauge-invariant ideals in the C*-algebras offinitely aligned higher-rank graphs, Canad. J. Math. 58(2006), no. 6, 12681290.
J. Spielberg, Graph-based models for Kirchberg algebras,J. Operator Theory 57 (2007), 347374.
W. Szymanski, General Cuntz-Krieger uniquenesstheorem, Internat. J. Math. 13 (2002) 549555.
D. Yang, Cycline subalgebras are Cartan, preprint.
T. Yeend, Groupoid models for the C*-algebras oftopological higher-rank graphs, J. Operator Theory 57:1(2007), 96120
Sarah Reznikoff Uniqueness theorems for combinatorial C-algebras
IntroductionClassical theoremsRecent workWhy graph algebras?
k-graph algebrasDefinitionExample: kCuntz-Krieger families
History of uniqueness theoremsClassic theoremsNon-degeneracy does not suffice
AperiodicityGroupoidsThe C*-algebra of a groupoidState extensions