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

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.