impan seminar:

working group in applications of set theory



December 1, 2016, 1015-12, room 105, Saeed Ghasemi (IM PAN)

Title: "A non-commutative Mrówka C*-algebra"

Abstract: "For an almost disjoint family of subsets of natural numbers, one can define the associated Ψ-space, which is a locally compact, separable and scattered space of the Cantor-Bendixson height 2. In 1977 Mrowka constructed a maximal almost disjoint family of size continuum for which the associated Ψ-space has the property that it's Cech-Stone compactification and one-point compactification coincide. By the Gelfand-Naimark duality this translates to a commutative C*-algebra A which contains c0 as an essential ideal such that A/c0 is isomorphic to c0(2ω) and has the property that its multiplier algebra is *-isomorphic to its (minimal) unitization. In a joint work with Piotr Koszmider we constructed a non-commutative analog of this example. Namely a C*-algebra A which contains the algebra K(l2) of compact operators on the separable Hilbert space l2 as an essential ideal such that A/K(l2) is *-isomorphic to K(l2(2ω)) 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 l2, i.e., it is not stable. The C*-algebra A is a non-separable scattered C*-algebra of the Cantor-Bendixson 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 Ψ-space and show how they can be generalized to the non-commutative 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"



Previous talks this semester:

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, 1015-12, 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 Stone-Cech 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 K-theoretical 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(βX-X) (which is the corona of C0(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, 1015-12, room 105, Tristan Bice (IM PAN/WCMCS)

Title: "The Akemann-Weaver counterexample to Naimark's Problem"

Abstract: "In 1948, Naimark observed that the compact operators K(H) on a Hilbert space H have a unique non-zero irreducible representation, up to unitary equivalence. He then asked if this in fact characterises K(H) among C*-algebras, which became the long-standing open question known as Naimark's problem. In 2004, Akemann and Weaver constructed a (necessarily non-separable) counterexample using the diamond principle (a well-known combinatorial principle independent of ZFC), which we outline in this talk.


October 27, 2016, 1015-12, 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 so-called quasi-linear 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 Winter-Zacharias 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 ε-disjointness preserving, provided that T(x*) = T(x)* for every x in A and ||T(x)T(y)|| ≤ ε||x||||y|| for all self-adjoint 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 two-sided 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, 1015-12, 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, 1515-17, 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 Cantor-Bendixson derivatives and scattered C*-algebras) we investigated these tools and constructions parallel to the ones in set-theoretic topology. I will introduce the Cantor-Bendixson 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 non-separable AF-algebra which is an inductive limit of stable AF-ideals, yet it has no maximal stable ideal."


October 5, 2016, 1515-17, 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 Cantor-Bendixson derivatives and scattered C*-algebras) we investigated these tools and constructions parallel to the ones in set-theoretic topology. I will introduce the Cantor-Bendixson 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 non-separable AF-algebra which is an inductive limit of stable AF-ideals, yet it has no maximal stable ideal."


September 27, 2016, 1515-17, 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 Akemann-Doner and Farah-Wofsey 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, 1515-17, room 106, Tristan Bice (IM PAN/WCMCS)

Title: "Locally Compact Stone Duality"
Abstract: "Almost all well-studied 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 well-known 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 2015-16.

Talks in the first semester of 2015-16.

Talks in the second semester of 2014-15.

Talks in the first semester of 2014-15.

Talks in the second semester of 2013-14.

Talks in the first semester of 2013-14.

Talks in the second semester of 2012-13.

Talks in the first semester of 2012-13.

Talks in the second semester of 2011-12.

Talks in the first semester of 2011-12.

SPECIAL NONCOMMUTATIVE SEMESTER

During the fall semester 2016/17 the seminar is devoted to applictaions of set theory in noncommutative mathematics. This is a part of satellite events to Simons semester: Noncommutative geometry the next generation which takes place at IM PAN during this semester.

Time and place: Thursdays 10.15-12.00 am, room 105, Sniadeckich 8

Due to lecture courses at Simon's semester temporary time and place of the seminar is Tuesdays 15.15-17.00, room 106, Sniadeckich 8

The scope of the seminar: Set-theoretic combinatorial and topological methods in diverse fields of mathematics, with a special emphasis on abstract analysis like Banach spaces, Banach algebras, C*-algebras, Here we include both the developing of such methods as forcing, descriptive set theory, Ramsey theory as well as their concrete applications in the fields mentioned above.

Working group style: We will make efforts so that this seminar has more a working character rather than the presentation style. This means that we encourage long digressions, discussions, background preparations and participation of everyone. We would like to immerse ourselves into the details of the mathematical arguments studied. Also the talks are usualy devoted to research in progress or fascinating results leading to some project not yet resolved. While ready final results could be presented at other seminars at IM PAN or UW.

Participants this semester so far:

  • Tristan Bice (IM PAN/WCNM)
  • Saeed Ghasemi (IM PAN/WCNM)
  • Clayton Suguio Hida (Ph. D. student, São Paulo)
  • Paweł Józiak (Ph. D. student, IMPAN/WCNM)
  • Tomasz Kochanek (IM PAN/UW)
  • Piotr Koszmider (IM PAN)
  • Yulia Kuznetsova (Besançon)
  • Eva Pernecká (IM PAN)
  • Adam Skalski (IM PAN)
  • Karen Strung (IM PAN)
  • Mateusz Wasilewski (Ph. D. student, IMPAN/WCNM)
Forthcoming talks :

  • 8.12.2016, Karen Strung will speak on continuous transformations of the Cantor set and C*-algebras
  • 15.12.2016, Saeed Ghasemi; continuation from 1.12.
  • The last seminar this semester will take place on 15.12.