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

Understand

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 $\mathrm{C}^*$-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 $\mathrm{C}^*$-algebra is trivially graded.