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

Valerio Bianchi, Ulrich Pennig

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.

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

Abstract

We consider infinite tensor product actions of on the UHF-algebra for a finite-dimensional unitary -representation and determine the equivariant homotopy type of the group , where are the compact operators on for a separable Hilbert space with . We show that this group carries an equivariant infinite loop space structure revealing it as the first space of a naive -spectrum, which we prove to be equivalent to the positive units of equivariant -theory. Here, is a -spectrum representing . As a consequence the first group of the cohomology theory associated to classifies equivariant -bundles over finite CW-complexes.

Paper Structure

This paper contains 14 sections, 27 theorems, 153 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Preliminaries from equivariant homotopy theory
  4. Preliminaries from operator algebras.
  5. The $G$-spectrum of units of $KU^D$-theory
  6. The spectrum $KU^D$
  7. Equivariant units of $KU^D$
  8. Automorphisms of $D \otimes \mathbb{K}$
  9. The $\mathcal{I}$-$G$-space $\mathcal{G}_D$
  10. $\mathbb{Z}/p\mathbb{Z}$-equivariant homotopy type of $\text{\normalfont Aut}(D \otimes \mathbb{K})$
  11. $\mathbb{Z}/p\mathbb{Z}$-equivariant $\text{\normalfont Aut}(D \otimes \mathbb{K})$-bundles and units of $KU^D$
  12. The classifying space of equivariant $\text{\normalfont Aut}(D \otimes \mathbb{K})$-bundles
  13. Comparison between $\mathcal{G}_D$ and $\textnormal{GL}^{\bullet}_1 KU^D$
  14. The equivariant homotopy type of $\text{\normalfont Aut}(D \otimes \mathbb{K})$

Key Result

Theorem 1.1

Let $G = \mathbb{Z}/p\mathbb{Z}$ for a prime $p \in \mathbb{N}$. Let $V$ be a finite-dimensional unitary $G$-representation and let be the associated $G$-$C^*$-algebra. The group $\text{\normalfont Aut}(D \otimes \mathbb{K})$ is a $G$-equivariant infinite loop space underlying a naive $G$-spectrum $EH_{\mathcal{I}}\text{\normalfont Aut}(D \otimes \mathbb{K})$ and we have an equivalence of naive $

Figures (1)

  • Figure 1: The list of strongly self-absorbing $C^*$-algebras in the UCT class. An arrow indicates tensorial absorption (e.g. $\mathcal{O}_{\infty} \otimes \mathcal{Z} \cong \mathcal{O}_{\infty}$).

Theorems & Definitions (76)

  • Theorem 1.1: Thm. \ref{['thm:main_thm']}
  • Corollary 1.2: Cor. \ref{['cor:bdl_classification']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 66 more