Amenability and weak containment for actions of locally compact groups on C*-algebras / Alcides Buss, Siegfried Echterhoff, Rufus Willett.

Format:

Book

Author/Creator:

Buss, Alcides, 1979- author.

Echterhoff, Siegfried, 1960- author.

Willett, Rufus, 1983- author.

Series:

Memoirs of the American Mathematical Society ; v. 1513.

Memoirs of the American Mathematical Society ; v. 1513

Language:

English

Subjects (All):

Operator algebras.

Dynamics.

Group theory.

Physical Description:

v, 88 pages ; illustrations ; 26 cm

Place of Publication:

Providence, Rhode Island : American Mathematical Society, 2024.

Summary:

In this work we introduce and study a new notion of amenability for actions of locally compact groups on C*-algebras. Our definition extends the definition of amenability for actions of discrete groups due to Claire Anantharaman-Delaroche. We show that our definition has several characterizations and permanence properties analogous to those known in the discrete case. For example, for actions on commutative C*-algebras, we show that our notion of amenability is equivalent to measurewise amenability. Combined with a recent result of Alex Bearden and Jason Crann, this also settles a long standing open problem about the equivalence of topological amenability and measurewise amenability for a second countable G-space X. We use our new notion of amenability to study when the maximal and reduced crossed products agree. One of our main results generalizes a theorem of Matsumura: we show that for an action of an exact locally compact group G on a locally compact space X the full and reduced crossed products C₀(X)⋊ max G and C₀(X)⋊ red G coincide if and only if the action of G on X is amenable. We also show that the analogue of this theorem does not hold for actions on noncommutative C*-algebras. Finally, we study amenability as it relates to more detailed structure in the case of C*-algebras that fibre over an appropriate G-space X, and the interaction of amenability with various regularity properties such as nuclearity, exactness, and the (L)LP, and the equivariant versions of injectivity and the WEP.

Notes:

"September 2024 ; vol. 301 number 1513 (seventh of 7 numbers)".

Includes bibliographical references (pages 85-88).

ISBN:

1470471523

9781470471521

OCLC:

1467672259