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

Abstract

We consider infinite tensor product actions of $G = \mathbb{Z}/p\mathbb{Z}$ on the UHF-algebra $D = \text{End}(V)^{\otimes \infty}$ for a finite-dimensional 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, which we prove to be equivalent to the positive units $gl_1(KU^D)_+$ of equivariant $KU^D$-theory. Here, $KU^D$ is a $G$-spectrum representing $X \mapsto K_*^G(C(X) \otimes D)$. As a consequence the first group of the cohomology theory associated to $gl_1(KU^D)_+$ classifies equivariant $D \otimes \mathbb{K}$-bundles over finite CW-complexes.