Amenability of Banach algebras II 1000-1M25SAB

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.