21.01.2016, 11¹⁵ room 105
Speaker: Damian Sobota (IM PAN)
Title: The Nikodym property of von Neumann algebras

Abstract: "The Nikodym property of C*-algebras comes from the theory of Boolean algebras and is closely related to the classical Banach-Steinhaus theorem for Banach spaces (also called the Uniform Boundedness Principle). A Boolean algebra A has the Nikodym property if every elementwise bounded sequence of finitely additive measures on A is uniformly bounded. Classical examples of algebras with the Nikodym property contain σ-complete algebras, algebras with Haydon's Subsequential Completeness Property and the algebra of Jordan-measurable subsets of the unit interval [0,1]. The Nikodym property for C*-algebras is defined similarly. A C*-algebra A has the Nikodym property if every sequence of linear functionals on A which is bounded on each projection in A is bounded on each element of A (and consequently norm-bounded). In the first part of the talk, we shall prove the Nikodym theorem stating that all σ-complete Boolean algebras have the Nikodym property and provide several basic facts concerning the property for C*-algebras. After this, using the representation of commutative von Neumann algebras in terms of L_∞(μ)-spaces and the Nikodym theorem, we shall obtain the similar result of Darst for von Neumann algebras. The second part will be devoted to the proof of the above-mentioned representation theorem of commutative von Neumann algebras."

14.01.2016, 11¹⁵ room 105
Speaker: Maciej Malicki (SGH)
Title: Amenable groups

Abstract: "The talk will present an overview of some aspects of classical and contemporary studies of amenable groups. I will start with basic definitions, and say a little bit about the Banach-Tarski paradox. Then, using a characterization of the notion of amenability that can be applied to all topological groups (the existence of invariant measures on flows), I will formulate certain variants of it, such as extreme amenability, and unique ergodicity. I will also briefly discuss recent research in this area, in connection with Fraissé limits, Ramsey theory, and Stone-Èech compactifications."

19.11.2015, 11¹⁵ room 105
Speaker: Tomasz Kochanek (IMPAN)
Title: A non-separable reflexive Banach space without the Elton-Odell property

Abstract: "We shall discuss a construction of a non-separable, reflexive Banach space X for which there is no e > 0 and an uncountable subset A of the unit ball of X so that ||x - y|| > 1 + e for all distinct x,y from A. In other words, X fails the "uncountable" version of the Elton-Odell property; recall that a non-reflexive example, namely c₀(ω₁), was given by Elton and Odell themselves. The space in question can be described as a "quasi-symmetric version of a long Tsirelon space with weights". A crucial role in the construction is played by certain Todorcevic's pseudo-metric on ω₁ which we shall also discuss. The construction moreover shows that the two theorems obtained jointly with T. Kania (discussed during this seminar on October 1) are, in a sense, sharp.

05.11.2015, 11¹⁵ room 105
Speaker: Eva Pernecka (IMPAN)
Title: Uniformly differentiable mappings from l_∞

Abstract: "We will discuss the rigidity of l_∞ and l_∞ⁿ with respect to uniformly differentiable mappings. Our main result is a non-linear analogy of the result on rigidity of l_∞ with respect to non-weakly compact linear operators by Rosenthal, and it generalises the theorem on non-complementability of c₀ in l_∞ due to Phillips."

29.10.2015, 11¹⁵ room 105
Speaker: Saeed Ghasemi (IMPAN)
Title: The rigidity of isomorphisms between some corona algebras - continuation

15.10.2015, 11¹⁵ room 105
Speaker: Saeed Ghasemi (IMPAN)
Title: The rigidity of isomorphisms between some corona algebras - continuation

08.10.2015, 11¹⁵ room 105
Speaker: Saeed Ghasemi (IMPAN)
Title: The rigidity of isomorphisms between some corona algebras

Abstract: "The rigidity question was first studied for the automorphisms of Boolean algebras. A groundbreaking result of Shelah shows that it is consistent with the usual axioms of mathematics (ZFC) that all automorphisms of the Boolean algebra P(N)/Fin are trivial, in the sense that they are implemented by almost permutations of the natural numbers N. While assuming the Continuum Hypothesis, a transfinite induction due to Rudin shows that there are many nontrivial automorphisms of P(N)/Fin. Using the Stone duality, the results of this kind can be translated in the category of (zero-dimensional locally compact and Hausdorff) topological spaces. Motivated by a question of Brown-Douglas-Fillmore, the rigidity question was studied in the non-commutative settings for the corona of C*-algebras. It is proved by Philips-Weaver that assuming the Continuum Hypothesis the Calkin algebra has outer automorphisms. On the other hand Farah showed that the Open Coloring Axiom implies that all the automorphisms of the Calkin algebras are inner (implemented by almost unitary elements of B(H)). In my talks we will take a closer look at Farah's result and study the rigidity question for the isomophisms between the corona of Finite-Dimensional Decomposition algebras (reduced products of matrices) and its relevance to Farah's result. We will also look at the later question in more general setting, motivated by the more recent results of Farah-Shelah about the Boolean algebra counterparts."

01.10.2015, 11¹⁵ room 105
Speaker: Tomasz Kochanek (IMPAN/UW)
Title: Uncountable sets of unit vectors that are separated by more than 1

Abstract: "Kottman proved that in the unit ball of any infinite-dimensional Banach space one can find an infinite subset such that the distance between any two distinct elements is larger than 1. Elton and Odell, employing Ramsey theory, strengthened this result by showing that it is possible to have all distances at least equal to 1+c for some c>0 depending only on the given space. Both these results are far-reaching generalizations of the Riesz lemma. We will investigate their non-separable versions focusing on two results: (1) the analogue of Kottman's theorem is valid for every non-separable reflexive Banach space (the set from the assertion is then uncountable); (2) the analogue of the Elton-Odell theorem is valid for every non-separable superreflexive Banach space."

24.09.2015, 11¹⁵ room 105
Speaker: Antonio Aviles (Universidad de Murcia)
Title: Compact subsets of the first Baire class.

Abstract: "Compact spaces consisting of Baire first class functions in the pointwise topology have been extensively studied with a number of remarkable results by Rosenthal, Bourgain, Fremlin, Talagrand, Todorcevic and others. They play a role in the category of compact spaces that is analogous to that of Borel sets in the class of all subsets of the real line. We shall discuss some history and some recent results about them in collaboration with Stevo Todorcevic."

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.