non-archimedean operator algebrasp-adics2015.matf.bg.ac.rs/slides/kochubei.pdf · an operator with...

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Non-Archimedean Operator Algebras

Anatoly N. Kochubei

Institute of Mathematics,National Academy of Sciences of Ukraine

Anatoly N. Kochubei Non-Archimedean Operator Algebras




Publications

Spectral theorem. Let A be a bounded linear operator on a Banachspace B over a complete algebraically closed non-Archimedeanvalued field K with a nontrivial valuation; | · | will denote theabsolute value in K , O is the ring of integers in K . We denote by

‖ · ‖ both the norm in B and the operator norm ‖A‖ = supx 6=0

‖Ax‖‖x‖

.

We will denote by K̃ the residue field of K .Denote by LA the commutative Banach algebra generated by Aand the unit operator I . LA is the closure of the algebra K [A] ofpolynomials in A, with respect to the norm of operators; thus LA isa Banach subalgebra of the algebra L(B) of all bounded linearoperators. Elements λ ∈ K are identified with the operators λI .





Publications

The spectrum M(LA) of the algebra LA is defined as the set of allbounded multiplicative seminorms | · | on LA. M(LA) is providedwith the weakest topology, with respect to which all functions| · | 7→ |B|, B ∈ LA, are continuous. In this topology, it is anonempty Hausdorff compact topological space. If the algebra LAis uniform, that is

‖T 2‖ = ‖T‖2 for any T ∈ LA,and all the characters take their values in K , then (Berkovich) thespace M(LA) is totally disconnected, and LA is isomorphic to thealgebra C (M(LA),K ) of continuous functions on M(LA) withvalues in K . In this case the above isomorphism transforms thecharacteristic functions ηA of nonempty open-closed subsetsΛ ⊂M(LA) into idempotent operators E (Λ) ∈ LA, ‖E (Λ)‖ = 1.





Publications

These operators form a finitely additive norm-boundedprojection-valued measure on the Boolean algebra of open-closedsets, with the non-Archimedean orthogonality property

‖f ‖ = supΛ‖E (Λ)f ‖, f ∈ B.

An operator with the above properties is called normal. It is calledstrongly normal, if its spectrum σ(A) is a nonempty totallydisconnected compact subset of K , and M(LA) = σ(A).





Publications

For a strongly normal operator A, we have the spectraldecomposition

A =

∫σ(A)

λE (dλ).

For any ϕ ∈ C (σ(A),K ) we can define the operator

ϕ(A) =

∫σ(A)

ϕ(λ)E (dλ).

The operator ϕ(A) is strongly normal.





Publications

The number operator. The model of p-adic representation of thecanonical commutation relations of quantum mechanics (K.,1996): B = C (Zp,Cp),

(a+f )(x) = xf (x − 1), (a−f )(x) = f (x + 1)− f (x), x ∈ Zp.

These operators are bounded and satisfy the relation [a−, a+] = I .Let A = a+a−, so that (Af )(x) = x{f (x)− f (x − 1)}. ThenAPn = nPn, n ≥ 0, where

Pn(x) =x(x − 1) · · · (x − n + 1)

n!, n ≥ 1; P0(x) ≡ 1,

is the Mahler basis, an orthogonal basis in C (Zp,Cp) ∼= c(Z+,Cp).Thus A is normal. An equivalent, though more complicated,construction was given a little earlier by Albeverio and Khrennikov.





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Orthoprojections. A projection on a Banach space B is such alinear bounded operator P that P is idempotent: P2 = P. It isobvious that either P = 0, or ‖P‖ ≥ 1. Kernels of P 6= 0 andI − P complement each other having a trivial intersection; if theyare orthogonal (in the non-Archimedean sense), then P is called anorthoprojection. In this and only this case, ‖P‖ = 1. For anorthoprojection P different from 0 and I , ‖P‖ = ‖I − P‖.

Theorem

A projection P is an orthoprojection, if and only if it is stronglynormal.





Publications

Multiplication operators. Let us consider the Banach spaceB = C (M,K ) where M is a compact totally disconnectedHausdorff topological space. It is known that B possesses anorthonormal basis.

Theorem

Let A be an operator of multiplication on B by a function a ∈ B.Then A is strongly normal.





Publications

An operator U on a Banach space B over a completenon-Archimedean valued field K with a nontrivial valuation will becalled unitary, if U = I + V where ‖V ‖ < 1 and V is stronglynormal. A unitary operator admits a spectral decomposition

U =

∫σ(V )

(1 + λ)EV (dλ) =

∫σ(U)

µEU(dµ).

Here EV is the spectral measure of the operator V , the mappingϕ(λ) = 1 + λ transforms the spectrum of V into that of U,EU(M) = EV (ϕ

−1(M)) for any open-closed subset of σ(U).There is an analog of Stone’s theorem about one-parameter groupsof unitary operators.





Publications

Commutative algebras. Classically, if a ∗-algebra A of boundedoperators on a Hilbert space is commutative, then AA∗ = A∗A forany A ∈ A, so that all the operators from A are normal. Thereforein our situation it is reasonable to consider a commutative algebraA of normal operators on a Banach space B over the field K . Weassume that A is complete with respect to the norm of operatorsand contains the unit operator I .





Publications

Theorem

Under the above assumptions, the algebra A is isomorphic to thealgebra C (M,K ) of K -valued continuous functions on a compacttotally disconnected Hausdorff topological space M. Under thisisomorphism, characteristic functions of open-closed subsetsΛ ⊂ M correspond to orthoprojections E (Λ) forming anorthoprojection-valued finitely additive measure on the algebra ofopen-closed subsets of M. For an operator F ∈ A corresponding toa function f ∈ C (M,K ), there is an integral representation

F =

∫M

f (λ)E (dλ)

convergent with respect to the norm of operators.





Publications

Algebras with Baer reductions. In order to introduce a possiblenon-Archimedean counterpart for the class of von Neumannalgebras, we need the following reduction procedure. We assumethat the space B is infinite-dimensional and possesses anorthonormal basis (in the non-Archimedean sense). Then B isisomorphic to the space c0(J,K ) of sequencesx = (x1, x2, . . . , xi , . . .), i ∈ J, xi ∈ K , xi → 0, by the filter ofcomplements to finite subsets of an infinite set J. We may assumethat B = c0(J,K ).





Publications

Let A be an algebra of linear bounded operators on c0(J,K ).Denote by A1 the closed unit ball in A – the set of all operatorsfrom A with norm ≤ 1. A1 is an algebra over the ring O, just asits ideal A0 consisting of operators of norm < 1. The reducedalgebra Ã = A1/A0 can be considered as a K̃ -algebra. Now we canlook for a class of K̃ -algebras, for which there is a (purelyalgebraic) theory parallel to the theory of von Neumann algebras.Then the class of algebras A corresponding to Ã from that classwill be the desired one.





Publications

An algebraic theory of the above kind is the theory of Baer ringsand algebras developed by Kaplansky:

I. Kaplansky, Rings of Operators, W. A. Benjamin, New York,1968.

A unital ring R is called the Baer ring, if each left (or, equivalently,each right) annihilator in R is generated by an idempotentelement. This property was proved by Baer for the ring of allendomorphisms of a vector space of an arbitrary dimension.Kaplansky proved it for any AW ∗-algebra (the class ofAW ∗-algebras is wider than the class of von Neumann algebras).





Publications

A Baer ring R is called Abelian, if all its idempotents are central,and Dedekind finite, if xy = 1 implies yx = 1. An idempotente ∈ R is called Abelian (finite), if the Baer ring eRe is Abelian(respectively, Dedekind finite). If u and v are central idempotents,we write u ≤ v , if vu = u. An idempotent e is called faithful, if thesmallest of the central idempotents v satisfying ve = e is equal to1.Kaplansky introduced the following types of Baer rings. A Baerring R is of type I, if it has a faithful Abelian idempotent. It is oftype II, if it has a faithful finite idempotent, but no nonzeroAbelian idempotents, and of type III, if it has no nonzero finiteidempotents. These classes are subdivided further into finite andinfinite ones.





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A typical example of a type I Baer ring is the ring of all lineartransformations of a vector space of countable dimension. Undersome additional conditions, an arbitrary Baer factor of type I is ofthis form (Wolfson, 1999).The main result of the theory of Baer rings is the uniquedecomposition of every Baer ring into a direct sum of rings of theabove types.





Publications

Let us call an operator algebra A an algebra with the Baerreduction, if the reduced algebra Ã is a Baer ring. It is well knownthat a finite system of elements of norm 1 in a non-Archimedeannormed space is orthonormal if and only if their reductions arelinearly independent. Therefore the operator ring A1 with theBaer reduction is an orthogonal sum of rings with reductionsof types I, II, and III.The simplest example of algebras with the Baer reduction of type Iis the algebra of all bounded operators on a Banach space ofcountable type (that is, c0(J,K ) with a countable set J). The veryexistence of other operator algebras (moreover, factors) with thisproperty is far from obvious. We will present a class of suchalgebras.





Publications

Weak Baer reduction:Suppose Ã is not a Baer ring. There is a ring monomorphismÃ→ Q(Ã) where Q(Ã) is the maximum right quotient ring. IfQ(Ã) is a Baer ring, then its type decomposition also produces anorthogonal decomposition of the ring A1.The study of various classes of operator algebras with the Baerreduction or the weak Baer reduction seems a huge problemcomparable with the whole theory of von Neumann algebras.





Publications

Analysis on product spaces. Let S be a totally disconnectedcompact Hausdorff topological space, G be an Abelian infinitesecond-countable totally disconnected compact Hausdorfftopological group acting transitively on S by homeomorphisms.The action will be denoted as x 7→ xa, x ∈ S , a ∈ G (the groupoperation is written multiplicatively). Here we construct and studysome operators on C (S × G ,Cp), the space of continuousfunctions on S × G with values in Cp. Here p is a prime number,Cp is the completion of an algebraic closure of the field Qp ofp-adic numbers. We will denote | · |p the absolute value in Cp.Some features of our approach follow the seminal work,

F. J. Murray and J. von Neumann, On rings of operators, Ann.Math. 37 (1936), 116–229,

though the actual meaning of our objects is different.





Publications

We assume that the group G is p-compatible. This notion isdefined as follows. Denote by o(G ) the set of all such naturalnumbers n, for which there exists a subgroup H ⊂ G with theproperty that G/H has an element of order n. The p-compatibilitymeans that p /∈ o(G ). A typical example is G = S = Zl where l isa prime different from p.p-Compatible groups possess nice properties resembling thoseappearing in classical harmonic analysis. G possesses a Cp-valuedHaar measure µ. In the Banach space C (G ,Cp), there is anorthonormal basis {gj} consisting of Cp-valued characters, that ismaps G → Cp, such that gj(ab) = gj(a)gj(b). It is convenient toindex the characters not by natural numbers but by elements ofthe dual group Ĝ consisting of all Cp-valued characters or,equivalently, of all continuous homomorphisms G → Tp where Tpis the set of all roots of 1 in Cp of orders prime to p.





Publications

For groups of this type, Ĝ is isomorphic also to the Pontryagindual. The second-countability property of G implies itsmetrizability and the countability of Ĝ . The group Ĝ (with thediscrete topology) is torsional, that is every finite subset of G liesin a finite subgroup.

The characters are orthonormal not only in the non-Archimedeansense, but also in the integral sense, with respect to the Cp-valuedHaar measure µ:∫

G

gj(a)gn(a−1)µ(da) = δj ,n, j , n ∈ Ĝ ,

where δj ,n is the Kronecker symbol.





Publications

Dealing with a function F ∈ C (S × G ,Cp) we can write

F (x , a) =∑n∈Ĝ

ϕn(x)gn(a), x ∈ S , a ∈ G ,

where

ϕn(x) =

∫G

F (x , a)gn(a−1)µ(da).

The functions ϕn are continuous. We use the notationF ∼ 〈ϕn(x)〉n∈Ĝ where ‖ϕn‖ → 0 by the filter of complements tofinite sets in Ĝ (in such cases we will write n→∞). Thisrepresentation implies a matrix representation for operators, themain technical tool of this work.





Publications

The crossed product construction

. On the Banach space B = C (S × G ,Cp), we consider theoperators

Ua0F (x , a) = F (xa0, aa0);

V a0F (x , a) = F (x , a−10 a);

WF (x , a) = F (xa−1, a−1);

LϕF (x , a) = ϕ(x)F (x , a);

MϕF (x , a) = ϕ(xa−1)F (x , a),

x ∈ S , a ∈ G . Here a0 ∈ G is a fixed element, ϕ ∈ C (S ,Cp) is afixed function.All the above operators are bounded. It is easy to check that

W = W−1, W Ua0W = V a0 , W LϕW = Mϕ.





Publications

Denote by I the set of all the operators Ua0 and Lϕ, and by J theset of all the operators V a0 and Mϕ. The closed linear hull (withrespect to the strong operator topology) of the set I will bedenoted R(I). Similarly, R(J) is the strongly closed linear hull of J.The mapping A 7→W AW is a spatial isomorphism of the algebrasR(I) and R(J). These algebras are the non-Archimedean crossedproduct algebras, at least for the case of p-compatible groups.

Commutants.

Theorem

(i) R(J) = I′; R(I) = J′;

(ii) R(J) = R(J)′′; R(I) = R(I)′′;

(iii) If the action of G on S is free, then R(I) and R(J)are factors.





Publications

Reduction.

Let Ã be the reduced algebra (over the algebraic closure F of thefinite field Fp) corresponding to the algebra A = R(J).

Theorem

Ã is a type I Baer ring.





Publications

Group algebrasLet G be a discrete group. We consider operators on the spacec0(G ) of sequences over a non-Archimedean field K indexed byelements from G decaying at infinity by the filter of complementsto finite sets.Operators of the right and left regular representations:

Ua0 [xa; a ∈ G ] = [xaa0 ; a ∈ G ];

Va0 [xa; a ∈ G ] = [xa−10 a; a ∈ G ];

a0 ∈ G .





Publications

Let I be the set of all Ua0 , J be the set of all Va0 , R(I) and R(J)their closed hulls with respect to strong (equivalently, uniform)operator topology.

Theorem

R(I) = J′, R(J) = I′.I′′ = R(J), J′′ = R(I).These closed algebras are factors if and only if ∀a ∈ G , a 6= 1, theset of conjugate elements

La = {c−1ac, c ∈ G}

is infinite. The reduction Ã for the algebra A = R(I) is the groupalgebra FG over the residue field F .





Publications

A(G ) is an analog, simultaneously, of the group von Neumannalgebra and the reduced group C ∗-algebra. An explicitdescription of A = A(G ):A(G ) is isomorphic to the set of sequences η = [η(d) : d ∈ G ]tending to zero by the filter of complements to finite sets, with thenatural structure of a K-Banach space (= c0(G )), with themultiplication

(ηζ)(d) =∑l∈G

η(l)ζ(l−1d).

The unit in A(G ) is the sequence e(d) =

{1, if d = 1,

0, otherwise.





Publications

Examples (with brief references to the substantiating results fromalgebra).1) If G is a finite group, then A has a type I Baer reduction.2) If G is a torsion-free polycyclic-by-finite group, then A has atype I Baer reduction (Cliff, 1980; Faith, 2003).3) Let G be prime (with no elements of order p), locally finite,with countable classes of conjugate elements. Then A is of weaktype III Baer reduction (Hannah and O’Meara, 1977).4) If G is prime (with no elements of order p), solvable, andcontains a nontrivial locally finite normal subgroup, then A is ofweak type III Baer reduction (Hannah, 1977, 1979).5) Let G be a finitary symmetric group, that is a group ofpermutations of an infinite set X moving only its finite subsets.Then A is of weak type III Baer reduction (O’Meara, 1980).





Publications

A(G ) as a Banach-Hopf algebraAn orthonormal basis in A(G ) (in the non-Archimedean sense):

ex(d) =

{1, if d = x ,

0, otherwise.

Comultiplication ∆ : A→ A⊗̂A. For u =∑x∈G

u(x)ex ,

∆(u) =∑x∈G

u(x)(ex ⊗ ex).

The counit:ε(u) =

∑x∈G

u(x), u ∈ A(G ).

The antipode: S(ex) = ex−1 , x ∈ G , with the extension bylinearity and continuity.





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There is a duality theory – the dual object is a multiplierBanach-Hopf algebra of discrete type.

A(G ) is an example of a non-Archimedean compact quantumgroup.





Publications

Publications

1 A. N. Kochubei, Non-Archimedean unitary operators,Methods of Functional Analysis and Topology, 17, No. 3(2011), 219–224.

2 A. N. Kochubei, On some classes of non-Archimedeanoperator algebras, Contemporary Math. 596 (2013), 133–148.

3 A. N. Kochubei, Non-Archimedean group algebras with Baerreductions, Algebras and Represent. Theory 17, No. 6 (2014),1861–1867.


Normal OperatorsUnitary OperatorsCommutative AlgebrasAlgebras with Baer ReductionsNon-Archimedean Crossed ProductsGroup algebrasPublications

non-archimedean operator algebrasp-adics2015.matf.bg.ac.rs/slides/kochubei.pdf · an operator with...

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