# Equivariant homologies for operator algebras

## TL;DR

This paper surveys various equivariant (co)homology theories for operator algebras, focusing on equivariant L^2-cohomology and Betti numbers for subalgebras of von Neumann algebras, and their graded versions.

## Contribution

It introduces and discusses the notion of equivariant L^2-cohomology and Betti numbers for operator algebras, including graded cases, providing a comprehensive overview.

## Key findings

- Introduction of equivariant L^2-cohomology for subalgebras of von Neumann algebras
- Relation between graded L^2-cohomology and equivariant L^2-cohomology
- Summary of equivariant K-theory and cyclic homology in operator algebras

## Abstract

This is a survey of a variety of equivariant (co)homology theories for operator algebras. We briefly discuss a background on equivariant theories, such as equivariant $K$-theory and equivariant cyclic homology. As the main focus, 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.