Previous talks this semester:
December 1, 2016, 10^{15}12, room 105, Saeed Ghasemi (IM PAN)
Title: "A noncommutative Mrówka C*algebra"
Abstract:
"For an almost disjoint family of subsets of natural numbers,
one can define the associated Ψspace, which is a locally compact,
separable and scattered space of the CantorBendixson height 2. In
1977 Mrowka constructed a maximal almost disjoint family of size
continuum for which the associated Ψspace has the property that
it's CechStone compactification and onepoint compactification
coincide. By the GelfandNaimark duality this translates to a
commutative C*algebra A which contains c_{0} as an essential ideal such that A/c_{0} is isomorphic to
c_{0}(2^{ω}) and has the property that its multiplier algebra is *isomorphic to its
(minimal) unitization. In a joint work with Piotr Koszmider we
constructed a noncommutative analog of this example. Namely a
C*algebra A which contains the algebra K(l_{2}) of compact operators on the
separable Hilbert space l_{2} as an essential ideal such that A/K(l_{2}) is *isomorphic
to K(l_{2}(2^{ω})) and has the property
that the multiplier algebra of A is *isomorphic to the unitization of
A. In particular this algebra can not be isomorphic to the tensor
product of itself with the algebra of compact operators on l_{2}, i.e.,
it is not stable. The C*algebra A is a nonseparable scattered
C*algebra of the CantorBendixson height 2, and in particular it is
approximately finite and type I.
In my talks I will present the ideas behind the commutative
construction of Mrowka Ψspace and show how they can be
generalized to the noncommutative setting using "systems of
almost matrix units" instead of almost disjoint families. Moreover I will
talk about the relation between our example with the context of
extensions of stable C*algebras"
17.11.2016, No seminar due to conference: Topological quantum groups and Hopf algebras
24.11.2016, No seminar due to conference:
Structure and Classification of C*algebras
November 10, 2016, 10^{15}12, room 105, Alessandro Vignati (York University, Canada)
Title: "Set theory and automorphisms of C*algebras"
Abstract:
"The influence of set theory on the homeomorphisms group of StoneCech remainders of locally compact
spaces has been noted since the work of Rudin and of Shelah on the automorphisms group of P(ω)/Fin.
Motivated by the search of Ktheoretical reversing automorphisms of the Calkin algebra, Phillips and Weaver,
and then Farah, showed that the assumption of the Continuum Hypothesis on one hand, and of forcing axioms
on the other, has influence on the automorphisms structure of certain corona C*algebras.
It is conjectured that under the assumption of CH, whenever A is a separable nonunital C*algebra,
its corona has a large (and wild) group of automorphisms, while forcing axioms (such as PFA), provide a strong rigidity for these groups.
In this talk, after a brief introduction of the main concepts involved and recalling past results,
we present two recent results. We show that under CH the automorphisms group of C(βXX)
(which is the corona of C_{0}(X)) has size larger than continuum, whenever X is a noncompact manifold.
Working on the other side of the conjecture we sketch the argument that shows that if A is a nuclear
separable C*algebra with plenty of projections then PFA implies that the corona of A has only trivial (i.e., Borel)
automorphisms. The latter result is joint work with McKenney.
November 3, 2016, 10^{15}12, room 105, Tristan Bice (IM PAN/WCMCS)
Title: "The AkemannWeaver counterexample to Naimark's Problem"
Abstract:
"In 1948, Naimark observed that the compact operators K(H) on a Hilbert space H
have a unique nonzero irreducible representation, up to unitary equivalence.
He then asked if this in fact characterises K(H) among C*algebras, which
became the longstanding open question known as Naimark's problem.
In 2004, Akemann and Weaver constructed a (necessarily nonseparable)
counterexample using the diamond principle (a wellknown combinatorial principle independent of ZFC),
which we outline in this talk.
October 27, 2016, 10^{15}12, room 105, Tomasz Kochanek (IM PAN/Uw)
Title: "Ulam's stability problem for disjointness preserving operators on C*algebras"
Abstract:
"Ulam's general stability problem asks whether given some mathematical object
satisfying a certain property approximately, there must exist an object close to
the given one and satisfying the given property exactly.
Ulam himself posed this question in 1940 in the context of almost
homomorphisms between metric groups (solved positively by Hyers)
and since then this stability problem has been widely investigated for various
classes of maps and proved to be of crucial importance in many parts of functional analysis.
Just to name a few: the socalled quasilinear maps play an important role in the theory of
twisted sums of Banach spaces; approximate homomorphisms on Banach algebras are crucial
in the formulation of B.E. Johnson's theory of derivations and cohomologies of Banach algebras.
We shall deal with disjointness preserving maps between C*algebras which
are building blocks in the WinterZacharias theory of nuclear dimension
(and are rather called order zero maps in this context).
A bounded linear operator T: A → B acting between C*algebras A and B is
called εdisjointness preserving,
provided that T(x*) = T(x)* for every x in A and T(x)T(y) ≤ εxy
for all selfadjoint x,y in A with xy = 0.
The aim of the talk is to prove a stability theorem for almost
disjointness preserving operators defined on a nuclear C*algebra
and taking values in a C*algebra which is isomorphic to a dual
Banach bimodule over itself (or, more generally, is a closed twosided ideal in such an algebra).
The key step of the proof is to reduce this problem to a corresponding stability question concerning
Jordan homomorphisms.
The first part of the talk should have a preparatory character.
We will recall some basic facts about nuclear dimension, in particular,
how almost order zero approximations arise in this context.
We will also need a little bit of the theory of cohomology of Jordan triples which
is required to understand the approximation procedure applied to almost Jordan homomorphisms.
The second part will be devoted to the proof of the announced stability result."
October 20, 2016, 10^{15}12, no seminar due to Scientific Council of the Institute,
instead we invite the participants of the seminar to the Doctoral defence (in Polish) of
Damian Sobota (IM PAN) at 11.15 on 19.10, room 106. The title of the thesis written under the guidance of
Piot Koszmider is "Cardinal invariants of the continuum and
convergence of measures on compact spaces.
October 12, 2016, 15^{15}17, room 403, Saeed Ghasemi (IM PAN)
Title: "An introduction to scattered C*algebras"  Continuation
Abstract:
"The techniques and constructions of compact, Hausdorff
scattered spaces, or equivalently (by the Stone duality) superatomic
Boolean algebras, have been used in the literature of Banach spaces
for many fundamental results in the forms of Banach spaces C(K), or
more generally Asplund spaces. Scattered C*algebras were introduced
as C*algerbas which are Asplund as Banach spaces. However, the
analogues of the commutative tools and constructions were not
developed for these C*algebras. In a joint work with Piotr Koszmider (S. Ghasemi, P.
Koszmider; Noncommutative CantorBendixson
derivatives and scattered C*algebras)
we investigated these tools and constructions parallel to the ones in
settheoretic topology. I will introduce the CantorBendixson
derivatives for C*algebras, obtained by using the ideal generated by
the minimal projections of these algebras, and present some of the
basic properties of these ideals. I will also show how these notions
can be used to construct exotic C*algebras. In particular, I will
show the existence of a nonseparable AFalgebra which is an inductive
limit of stable AFideals, yet it has no maximal stable ideal."
October 5, 2016, 15^{15}17, room 403, Saeed Ghasemi (IM PAN)
Title: "An introduction to scattered C*algebras"
Abstract:
"The techniques and constructions of compact, Hausdorff
scattered spaces, or equivalently (by the Stone duality) superatomic
Boolean algebras, have been used in the literature of Banach spaces
for many fundamental results in the forms of Banach spaces C(K), or
more generally Asplund spaces. Scattered C*algebras were introduced
as C*algerbas which are Asplund as Banach spaces. However, the
analogues of the commutative tools and constructions were not
developed for these C*algebras. In a joint work with Piotr Koszmider (S. Ghasemi, P.
Koszmider; Noncommutative CantorBendixson
derivatives and scattered C*algebras)
we investigated these tools and constructions parallel to the ones in
settheoretic topology. I will introduce the CantorBendixson
derivatives for C*algebras, obtained by using the ideal generated by
the minimal projections of these algebras, and present some of the
basic properties of these ideals. I will also show how these notions
can be used to construct exotic C*algebras. In particular, I will
show the existence of a nonseparable AFalgebra which is an inductive
limit of stable AFideals, yet it has no maximal stable ideal."
September 27, 2016, 15^{15}17, room 106, Piotr Koszmider (IM PAN)
Title: "A nonseparable scattered C*algebra without
a nonseparable commutative subalgebra"
Abstract:
"This talk is based on a paper T. Bice, P. Koszmider, A note on the AkemannDoner and FarahWofsey constructions,
To appear in PAMS where we removed an additional assumption of the continuum hypothesis from a previous
construction of Akemann and Doner of an algebra like in the title (A nonseparable C*algebra with only separable abelian C*subalgebras.
Bull. London Math. Soc. 11 (1979), no. 3, 279–284). The main combinatorial "trick" is to use Luzin's almost disjoint family,
so first, we will describe this notion."
September 20, 2016, 15^{15}17, room 106, Tristan Bice (IM PAN/WCMCS)
Title: "Locally Compact Stone Duality"
Abstract:
"Almost all wellstudied real rank zero C*algebras can be constructed from inverse semigroups.
We focus on just the first part of this construction, where a zero dimensional compact (Hausdorff)
topological space comes from Exel's tight spectrum of the idempotent semilattice.
First we show how this can be generalized to a kind of Stone duality between separative posets
and 'pseudobases' of zero dimensional locally compact spaces.
This is closely related to a wellknown set theoretic construction of a Boolean algebra from a poset.
Next, we consider bases of general locally compact spaces, how these can be axiomatized and how the space can be reconstructed as a generalized Stone space.
Time permitting, we will outline how this should allow more general (e.g. projectionless)
C*algebras to be constructed from inverse semigroups."
Talks in the second semester of 201516.
Talks in the first semester of 201516.
Talks in the second semester of 201415.
Talks in the first semester of 201415.
Talks in the second semester of 201314.
Talks in the first semester of 201314.
Talks in the second semester of 201213.
Talks in the first semester of 201213.
Talks in the second semester of 201112.
Talks in the first semester of 201112.
