Amenability of Banach Algebras 1000-1M24SAB
1. Amenable groups, paradoxical decompositions, Følner’s condition, connections with the Ulam-type stability problem for group homomorphisms.
2. Group C*-algebras and their connections with amenability of groups.
3. The notion of amenability for Banach algebras; B.E. Johnson’s theorem on characterization of amenable Banach algebras in terms of approximate and virtual diagonals.
4. Amenability of Banach algebras in terms of Hochschild cohomology groups.
5. C.J. Read’s theorem on non-amenability of the algebra of bounded linear operators on ℓ_1 and on isometric embedding of finite-dimensional, nilpotent, commutative algebras into radical, commutative, 1-amenable Banach algebras.
6. Kazhdan’s theorem on Ulam-type stability for ɛ-representations of amenable groups on Hilbert spaces.
7. Theorems of B.E. Johnson and Choi-Horváth-Laustsen on Ulam stability for almost multiplicative maps acting between the algebras of bounded linear operators on a broad class of Banach spaces.
Main fields of studies for MISMaP
Type of course
Mode
Prerequisites
Prerequisites (description)
Course coordinators
Learning outcomes
The student knows and understands the notion of amenability, both in the context of group theory and Banach algebras, in particular C*-algebras. The student understands connections between amenability and topics like: the Banach-Tarski paradox, cohomologies of Banach algebras, Ulam-type stability.
Assessment criteria
The final grade is determined by the outcome of an oral exam and points gained for activity during classes.
Bibliography
1. B. Blackadar, K-theory for operator algebras, Mathematical Sciences Research Institute Publications, Springer-Verlag, New York 1986.
2. N.P. Brown, N. Ozawa, C*-algebras and finite-dimensional approximations, Graduate Studies in Mathematics, vol. 88, American Mathematical Society, Providence, R.I. 2008.
3. K.R. Davidson, C*-algebras by example, Fields Institute Monographs, American Mathematical Society, Providence, R.I. 1996.
4. I. Farah, Combinatorial set theory of C*-algebras, Springer Monographs in Mathematics, Springer 2019.
5. V. Runde, Lectures on amenability, Lecture Notes in Mathematics 1774, Springer-Verlag, Berlin Heidelberg 2002.
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
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