Amenability of Banach algebras II

elective monographs

Functional Analysis 1000-135AF

(in Polish) Student powinien znać najważniejsze metody i twierdzenia analizy funkcjonalnej; opanowanie materiału zawartego w podstawowym kursie analizy funkcjonalnej jest wystarczające. Zaliczenie pierwszej części wykładu “Średniowalność algebr Banacha” będzie oczywiście pomocne, ale nie jest warunkiem koniecznym.

Classroom

This course is a continuation of the first part of the lecture

"Amenability of Banach Algebras". Its primary goal will be to discuss

results that complement the list of examples of (non-)amenable Banach

algebras presented during the first part. This will include, among

others, the measure algebra on a locally compact group and the algebra

of operators on the space l_p for p∈[1,∞]. A significant part of the

lecture will be devoted to the development of Ulam stability theory in

operator algebras, in the context of ϵ-homomorphisms and almost

multiplicative maps on C*-algebras and general Banach algebras. We will

also discuss certain generalizations of the concept of amenability,

formulated in terms of both cohomology and derivations with values in

bimodules over Banach algebras. The main emphasis will be on the study

of algebras of bounded linear operators on Banach spaces.

* The Dales-Ghahramani-Helemskii theorem on the amenability of the

measure algebra on a locally compact group.

* Basic facts about Kazhdan's property (T) for discrete groups.

* The application of property (T) of the group SL(3,Z) in the proof of

the Read-Ozawa-Runde theorem on the non-amenability of the algebra

B(l_p) for p∈[1,∞].

* The Choi-Horváth-Laustsen theorems on Ulam stability for almost

multiplicative operators acting between operator algebras on a broad

class of Banach spaces.

* The application of Kazhdan's theorem on ϵ-representations to the

proof of Farah's theorem on ϵ-homomorphisms between finite-dimensional

C*-algebras; connections with automorphisms of the Calkin algebra.

* Generalizations of the concept of amenability of Banach algebras:

weakly, pseudo-, and approximately amenable algebras. Blanco's theorem

on the weak amenability of the algebras B(X) for classical Banach spaces X.