Banach-Saks Properties of C-Algebras and Hilbert C -Modulesmfrank/Moscow_Pontryagin_Conference_2008.pdf · Let A and B be two strongly Morita equivalent C*-algebras and E be an

Banach-Saks Properties of

C*-Algebras and Hilbert

C*-Modules

Michael FrankHTWK-Leipzig, 2008

Banach-Saks Properties of C*-Algebras and Hilbert C*-Modules

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C*-Modules

Michael Frank joint with Alexander A. Pavlov

Pontryagin Conference, Moscow, June 2008

C*-Algebras

C*C*--algebrasalgebras can be considered as self-adjoint norm-

closed subalgebras of von Neumann algebras

B(H) of all bounded linear operators on Hilbert

spaces H. (Gel‘fand-Naimark-Segal Thm.)



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spaces H. (Gel‘fand-Naimark-Segal Thm.)

They are a special class of Banach algebras. With

respect to the BanachBanach--Saks properties of Banach Saks properties of Banach

spacesspaces we use this class as one test set to obtain

good properties ...

Hilbert C*-ModulesA – a (unital) C*-algebra, E – a (left) A-module, the

complex-linear structures being given on A and Eare compatible, i.e. l(a x) =(l a)x = a(l x) for

every l in CC, a in A and x in E.

If there exists a mapping < .,. >: E µ E ØA with the

properties



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properties

• < x,x > ¥ 0 for every x in E,

• < x,x > =0 if and only if x=0,

• < x,y > = < y,x >* for every x,y in E,

• < ax,y > = a < x,y > for every a in A, every x,y in E,

• < x+y,z > = < x,z > + < y,z > for every x,y,z in E.

Hilbert C*-Modules

The pair {E, < .,. > } is called a prepre--Hilbert AHilbert A--modulemodule. The map < .,. > is said to be the Athe A--valued inner valued inner productproduct.

If the pre-Hilbert A-module {E, < .,. >} is complete

with respect to the norm ||x|| = ||< x,x >||½ , then

E is called to be HilbertHilbert.



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E is called to be HilbertHilbert.(I. Kaplansky, 1953, M. A. Rieffel, W. L. Paschke, 1973, …)

For A=CC we get Hilbert spaces, but far not all properties and structures of Hilbert spaces generalize to Hilbert C*-modules.

Operator C*-Algebras

The set EndA(E) of all bounded module operators T on E forms a Banach algebra, whereas the set EndA*(E) of all bounded module operators which possess an adjoint operator inside EndA(E) has the structure of a unital C*-algebra.

Note that these two sets do not coincide in general.



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Note that these two sets do not coincide in general.

The set KA(E) of ''compact'' operators''compact'' operators on E is defined as the norm-closure of the set of all finite linear combinations of the specific operators

{ qx,y in EndA(E) : x,y in E, qx,y (z) := < z,x >y for every z in E}.

Operator C*-Algebras

It is a C*-subalgebra and a two-sided ideal of EndA*(E), and EndA*(E) can be identified with the multiplier algebra of KA(E).

Note, that E is a right Hilbert KA(E)-module at the same time, with a K (E)-valued inner product



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same time, with a KA(E)-valued inner product < x,y >op := qx,y for x,y in E.

If E is a full Hilbert A-module then the picture is symmetric – the C*-algebras of coefficients A and KA(E) have equal rights with respect to E.

Strong Morita Equivalence

Two C*-algebras A and B are strongly Morita strongly Morita equivalentequivalent if there exists a left full Hilbert A-module E which is a right full Hilbert B-module at the same time, with the property < x,y >A z = x < y,z >B

for any x,y,z in E. [M. A. Rieffel, 1973]



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In this situation E is often called an AA--B imprimitivity B imprimitivity bimodulebimodule.

Strong Morita equivalence is a weaker equivalence notion than *-isomorphism (equiv., isometric isomorphism) or Banach isomorphism.

Strong Morita Equivalence

For the A-B imprimitivity bimodule E there exists a related C*-algebra L = KA(A ∆ E) over the ortho-

gonal sum A ∆ E of the Hilbert A-modules A and

E. The C*-algebra L is called the linking algebrathe linking algebra.

[L. G. Brown, P. Green, M. A. Rieffel, 1977;

L. G. Brown, J. Mingo, Nien-Tsu Shen, 1994]



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L. G. Brown, J. Mingo, Nien-Tsu Shen, 1994]

The multiplier algebra M(L) contains two orthogonal

projections p,q such that 11M(L)=p+q, with corners

pLp=KA(A)=A, qLq=KA(E)=B, qLp = KA(A,E) = E

and pLq = KA(E,A) = EÛ – the dual to E

B-A imprimitivity bimodule.

Second Motivation

In the theory of Hilbert C*-modules there is an

outstanding class of C*-algebras and of Hilbert

C*-modules over them – the class of dual C*-

algebras.



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A C*-algebra A is said to be a dual C*dual C*--algebraalgebra if

there exists a faithful *-representation of A in a

C*-algebra K(H) of all compact operators over

some Hilbert space H.

Motivation

A C*-algebra is a dual C*-algebra if and only if one of the following equivalent conditions hold:

�For every Hilbert A-module E every Hilbert A-submodule F Œ E is automatically orthogonally complemented, i.e. F is an orthogonal summand



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complemented, i.e. F is an orthogonal summand of E. ([B. Magajna])

�For every Hilbert A-module E each Hilbert A-submodule F Œ E that coincides with its biorthogonal complement F^^ in E is automatically orthogonally complemented in E. ([J. Schweizer])

Motivation�For every Hilbert A-module E every Hilbert A-

submodule is automatically topologically com-plemented there, i.e. it is a topological direct summand.

�For every pair of Hilbert A-modules E, F, every densely defined closed operator t: Dom(t) Œ E Ø F



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densely defined closed operator t: Dom(t) Œ E Ø Fpossesses a densely defined adjoint operator t*: Dom(t*) Œ F Ø E.

�For every pair of Hilbert A-modules E, F, every densely defined closed operator t: Dom(t) Œ E Ø Fis regular, i.e. it is adjointable and the operator 1+t*t has a dense range.

Motivation


has polar decomposition, i.e. there exists a unique

partial isometry V with initial set Ran(|t|) and final

set Ran(t) such t=V|t|.



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and its adjoint t* have generalized inverses.

�More ... all strongly Morita invariant properties!

Motivation

The question is whether there exist geometricalgeometrical--

topological backgroundstopological backgrounds for such a rich set of

good properties for this class of dual C*-algebras

and for the related Hilbert C*-modules ?!



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and for the related Hilbert C*-modules ?!

Are there “good” classes of Hilbert C*-modules?

Let us have a look on BanachBanach--Saks propertiesSaks properties ...

Banach-Saks Property

A Banach space E has the BanachBanach--Saks propertySaks property if every bounded sequence { xn }n Õ E has a

subsequence { xn(k) }k such that the derived from it

sequence of partial arithmetic means converges in

norm, i.e.



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.

(The left term can be considered as a special convex

combination of the elements of the sequence.)


It is known that Banach spaces E with the Banach-Saks property have to be reflexive as normed spaces, [Diestel, 1975, p.85]. Therefore, C*-algebras with the Banach-Saks property have to be finite-dimensional linear spaces, i.e. unital matrix algebras [Kusuda, 2007, Lemma 3.1].



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matrix algebras [Kusuda, 2007, Lemma 3.1].

In such E the weak closure of convex sets coincides with their norm closure.

The following proposition has been proved by M. Ku-suda for the unital case, [Kusuda, 2007, Thm. 3.6]:


Proposition:

Let A be a (non-unital, in general) C*-algebra and E be a full Hilbert A-module. Suppose, that E has the Banach-Saks property.

Then A has to be finite-dimensional as a linear



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Then A has to be finite-dimensional as a linear space, i.e. A is a unital matrix algebra.

In other words, if a full Hilbert A-module has the Banach-Saks property, then the C*-algebra of coefficients A has to be finite-dimensional.

Any Hilbert A-module E over a C*-algebra A with Banach-Saks property has this property, too.

Weak Banach-Saks Property

If for any given weakly null sequence { xn }n of a

Banach space E, one can extract a subsequence

{ xn(k) }k such that the derived from it sequence of

partial arithmetic means converges in norm to

zero, i.e.



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then E is said to admit the weak Banachweak Banach--Saks Saks

propertyproperty.

Weak Banach-Saks PropertyNote, that the weak Banach-Saks property inherits to

any (closed) subspace of a Banach space with

weak Banach-Saks property by definition.

Beside this, if A is a non-unital C*-algebra and

A1 =A + CC1 is its unitization, then A has the weak

Banach-Saks property if and only if A has the



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Banach-Saks property if and only if A1 has the

weak Banach-Saks property, [Cho-Ho Chu, 1994].

For C*-algebras and Hilbert C*-modules as classes

of Banach spaces we prove the following fact

relying on a key result by [Kusuda, 2007, Thm.

2.2]:


Theorem:

Let A and B be two strongly Morita equivalent C*-

algebras and E be an A-B imprimitivity bimodule.

The following three conditions are equivalent:

• A has the weak Banach-Saks property.



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• A has the weak Banach-Saks property.

• B has the weak Banach-Saks property.

• E has the weak Banach-Saks property.

• L has the weak Banach-Saks property.


The key role is played by dual C*-algebras, i.e. by

C*-algebras that admit a faithful *-representation in

some< C*-algebra KCC(H) of all linear compact

operators on some Hilbert space H, cf. [W. B.

Arveson, 1976].



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[Cho-Ho Chu, 1994] proved in that a C*-algebra A

has the weak Banach-Saks property if and only if

A admits a finite chain { Ii } of two-sided norm-closed ideals such that I0 Õ...Õ In=A, I0 = { 0 }, In=A

and any Ii+1 / Ii , i=0, ... , n-1, is a dual C*-algebra.

Uniform Weak B-S Property

A third Banach-Saks type property of Banach

spaces has been introduced by [C. Nuñez, 1989].

A Banach space E has the uniform weak Banachuniform weak Banach--Saks propertySaks property if there is a null sequence { dn }n of

positive real numbers such that, for any weakly



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positive real numbers such that, for any weakly

null sequence { xn }n in E with uniform bound || xn || 1 and for any natural number k, there exist

natural numbers n(1) <n(2) < ... < n(k) such that

.

Uniform Weak B-S Property

In [Cho-Ho Chu, 1994, Thm. 2] it has been shown that C*-algebras are Banach spaces for which the uniform weak and the weak Banach-Saks properties are equivalent.

In [M. Kusuda, 2007, Thm. 2.2, Cor. 2.3] it was found that for full Hilbert C*-modules over unital C*-



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that for full Hilbert C*-modules over unital C*-algebras both these properties are equivalent.

Moreover, for full Hilbert C*-modules over C*-algebras with weak Banach-Saks property again both these properties hold at the same time, cf. [M. Kusuda, 2001, Thm. 2.3] and [M. Kusuda, 2007, Thm. 2.2].

Uniform Weak B-S PropertyTheorem:

Let A and B be two strongly Morita equivalent C*-algebras

and E be an A-B imprimitivity bimodule. The following three

conditions are equivalent:

• A has the uniform weak Banach-Saks property.

• B has the uniform weak Banach-Saks property.



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• B has the uniform weak Banach-Saks property.

• E has the uniform weak Banach-Saks property.

• L has the uniform weak Banach-Saks property.

In particular, conditions (i)-(iv) hold in case either A or B or E or L have the weak Banach-Saks property.

Conversely, either of conditions (i)-(iv) implies A, B, E and Lto have the weak Banach-Saks property.

One outcome

We obtained two sets of C*-algebras that are

invariant under strong Morita equivalence:

�The class of dual C*-algebras



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�The class of C*-algebras with the weak Banach-

Saks property which are non-dual C*-algebras.

Hilbert C*-modules over them might have

properties which are worth to be considered.

References



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References



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Thank you very much for your attention.



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Alexander A. Pavlov

Moscow State University

Department of Geography and

Department of Mechanics and Mathematics

119 992 Moscow, Russia



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www.freewebs.com/axpavlov/

axpavlov @ mail.ru

DFG, Project „K-Theory, C*-Algebras, and Index Theory“

RFBR, grant 07-01-91555

Michael Frank

Hochschule für Technik, Wirtschaft und Kultur (HTWK) Leipzig

Fachbereich Informatik, Mathematik und Naturwissenschaften (FbIMN)

PF 301166, D-04251 Leipzig, F.R.Germany

www.imn.htwk-leipzig.de/~mfrank/



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www.imn.htwk-leipzig.de/~mfrank/

mfrank @ imn.htwk-leipzig.de

Service:

www.imn.htwk-leipzig.de/~mfrank/oasis.html

www.imn.htwk-leipzig.de/~mfrank/hilmod.html

Documents

Banach-Saks Properties of C*-Algebras and Hilbert C* -Modulesmfrank/Moscow_Pontryagin_Conference_2008.pdf · Let A and B be two strongly Morita equivalent C*-algebras and E be an

Banach-Saks Properties of C-Algebras and Hilbert C -Modulesmfrank/Moscow_Pontryagin_Conference_2008.pdf · Let A and B be two strongly Morita equivalent C*-algebras and E be an