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Banach-Saks Properties of
C*-Algebras and Hilbert
C*-Modules
Michael FrankHTWK-Leipzig, 2008
Banach-Saks Properties of C*-Algebras and Hilbert C*-Modules
18.06.2008/1
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.)
Michael FrankHTWK-Leipzig, 2008
Banach-Saks Properties of C*-Algebras and Hilbert C*-Modules
18.06.2008/2
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
Michael FrankHTWK-Leipzig, 2008
Banach-Saks Properties of C*-Algebras and Hilbert C*-Modules
18.06.2008/3
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.
Michael FrankHTWK-Leipzig, 2008
Banach-Saks Properties of C*-Algebras and Hilbert C*-Modules
18.06.2008/4
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.
Michael FrankHTWK-Leipzig, 2008
Banach-Saks Properties of C*-Algebras and Hilbert C*-Modules
18.06.2008/5
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
Michael FrankHTWK-Leipzig, 2008
Banach-Saks Properties of C*-Algebras and Hilbert C*-Modules
18.06.2008/6
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]
Michael FrankHTWK-Leipzig, 2008
Banach-Saks Properties of C*-Algebras and Hilbert C*-Modules
18.06.2008/7
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]
Michael FrankHTWK-Leipzig, 2008
Banach-Saks Properties of C*-Algebras and Hilbert C*-Modules
18.06.2008/8
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.
Michael FrankHTWK-Leipzig, 2008
Banach-Saks Properties of C*-Algebras and Hilbert C*-Modules
18.06.2008/9
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
Michael FrankHTWK-Leipzig, 2008
Banach-Saks Properties of C*-Algebras and Hilbert C*-Modules
18.06.2008/10
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
Michael FrankHTWK-Leipzig, 2008
Banach-Saks Properties of C*-Algebras and Hilbert C*-Modules
18.06.2008/11
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
�For every pair of Hilbert A-modules E, F, every densely defined closed operator t: Dom(t) Œ E Ø F
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|.
Michael FrankHTWK-Leipzig, 2008
Banach-Saks Properties of C*-Algebras and Hilbert C*-Modules
18.06.2008/12
�For every pair of Hilbert A-modules E, F, every densely defined closed operator t: Dom(t) Œ E Ø F
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 ?!
Michael FrankHTWK-Leipzig, 2008
Banach-Saks Properties of C*-Algebras and Hilbert C*-Modules
18.06.2008/13
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.
Michael FrankHTWK-Leipzig, 2008
Banach-Saks Properties of C*-Algebras and Hilbert C*-Modules
18.06.2008/14
.
(The left term can be considered as a special convex
combination of the elements of the sequence.)
Banach-Saks Property
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].
Michael FrankHTWK-Leipzig, 2008
Banach-Saks Properties of C*-Algebras and Hilbert C*-Modules
18.06.2008/15
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]:
Banach-Saks Property
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
Michael FrankHTWK-Leipzig, 2008
Banach-Saks Properties of C*-Algebras and Hilbert C*-Modules
18.06.2008/16
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.
Michael FrankHTWK-Leipzig, 2008
Banach-Saks Properties of C*-Algebras and Hilbert C*-Modules
18.06.2008/17
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
Michael FrankHTWK-Leipzig, 2008
Banach-Saks Properties of C*-Algebras and Hilbert C*-Modules
18.06.2008/18
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]:
Weak Banach-Saks Property
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.
Michael FrankHTWK-Leipzig, 2008
Banach-Saks Properties of C*-Algebras and Hilbert C*-Modules
18.06.2008/19
• 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.
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].
Michael FrankHTWK-Leipzig, 2008
Banach-Saks Properties of C*-Algebras and Hilbert C*-Modules
18.06.2008/20
[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
Michael FrankHTWK-Leipzig, 2008
Banach-Saks Properties of C*-Algebras and Hilbert C*-Modules
18.06.2008/21
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*-
Michael FrankHTWK-Leipzig, 2008
Banach-Saks Properties of C*-Algebras and Hilbert C*-Modules
18.06.2008/22
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.
Michael FrankHTWK-Leipzig, 2008
Banach-Saks Properties of C*-Algebras and Hilbert C*-Modules
18.06.2008/23
• 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
Michael FrankHTWK-Leipzig, 2008
Banach-Saks Properties of C*-Algebras and Hilbert C*-Modules
18.06.2008/24
�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
Michael FrankHTWK-Leipzig, 2008
Banach-Saks Properties of C*-Algebras and Hilbert C*-Modules
18.06.2008/25
References
Michael FrankHTWK-Leipzig, 2008
Banach-Saks Properties of C*-Algebras and Hilbert C*-Modules
18.06.2008/26
Thank you very much for your attention.
Michael FrankHTWK-Leipzig, 2008
Banach-Saks Properties of C*-Algebras and Hilbert C*-Modules
18.06.2008/27
Alexander A. Pavlov
Moscow State University
Department of Geography and
Department of Mechanics and Mathematics
119 992 Moscow, Russia
Michael FrankHTWK-Leipzig, 2008
Banach-Saks Properties of C*-Algebras and Hilbert C*-Modules
18.06.2008/28
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/
Michael FrankHTWK-Leipzig, 2008
Banach-Saks Properties of C*-Algebras and Hilbert C*-Modules
18.06.2008/29
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