Units of $\mathbb{Z}/n\mathbb{Z}$-equivariant $K$-theory and bundles of UHF-algebras

Abstract

We consider infinite tensor product actions of a finite cyclic group $G$ on the UHF-algebra $D=\text{End}(V{)}^{\otimes \infty }$ for a finite-dimensional projectively faithful unitary $G$-representation $V$ and determine the equivariant homotopy type of the group $\text{Aut}(D\otimes \mathbb{K})$, where $\mathbb{K}$ are the compact operators on ${\ell }^2(G)\otimes H_0$ for a separable Hilbert space $H_0$ with $\dim (H_0)=\infty$. We show that this group carries an equivariant infinite loop space structure revealing it as the first space of a naive $G$-spectrum. We identify this spectrum with the positive units $g{l}_1(K{U}^D{)}_{+}$ of equivariant $K{U}^D$-theory, which was not possible in the case of circle actions considered by Evans and the second author. Here, $K{U}^D$ is a $G$-spectrum representing $X\mapsto K_{\ast }^G(C(X)\otimes D)$. As a consequence the first group of the cohomology theory associated to $g{l}_1(K{U}^D{)}_{+}$ classifies equivariant $D\otimes \mathbb{K}$-bundles over finite $G$-CW-complexes.

arXiv comment39 pages, we have improved the main results to treat arbitrary finite cyclic groups Z/nZ instead of just the ones of prime order