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 $g{l}_1(K{U}^D{)}_{+}$ of equivariant $K{U}^D$-theory. 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 CW-complexes.

39 pages