Notes on twisted equivariant $\mathrm{K}$-theory for ${\mathrm{C}}^{\ast }$-algebras

Abstract

In this paper, we study a generalization of twisted (groupoid) equivariant $\mathrm{K}$-theory in the sense of Freed-Moore for ${\mathbb{Z}}_2$-graded ${\mathrm{C}}^{\ast }$-algebras. It is defined by using Fredholm operators on Hilbert modules with twisted representations. We compare it with another description using odd symmetries, which is a generalization of van Daele's $\mathrm{K}$-theory for ${\mathbb{Z}}_2$-graded Banach algebras. In particular, we obtain a simple presentation of the twisted equivariant $\mathrm{K}$-group when the ${\mathrm{C}}^{\ast }$-algebra is trivially graded. It is applied for the bulk-edge correspondence of topological insulators with CT-type symmetries.

26 pages, minor corrections

References in corpus (3)

• Controlled topological phases and bulk-edge correspondence
• On the C*-algebraic approach to topological phases for insulators
• The stabilization theorem for proper groupoids

Cited by in corpus (3)

• Controlled topological phases and bulk-edge correspondence
• The $K$-theoretic bulk-edge correspondence for topological insulators
• Twists on the Torus Equivariant under the 2-Dimensional Crystallographic Point Groups