Equivariant homologies for operator algebras; a survey

Subject Areas : Functional analysis

A. Shirinkalam ¹

1 - Department of mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Iran

Received: 2022-05-02 Accepted : 2022-06-29 Published : 2022-06-30

Keywords: group action, Equivariant Hochschild cohomology, equivariant $L^2$-cohomology, graded $ L^2 $-Betti number, graded algebra,

Abstract :

‎This is a survey of a variety of equivariant (co)homology theories for operator algebras‎. ‎We briefly discuss a background on equivariant Hochschild cohomology‎. ‎We discuss a notion of equivariant $ L^2 $-cohomology and equivariant $ L^2 $-Betti numbers for subalgebras of a von Neumann algebra‎. ‎For graded $C^*$-algebras (with grading over a group) we elaborate on a notion of graded $ L^2 $-cohomology and its relation to equivariant $L^2$-cohomology‎.

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