Self-adjoint operators in Z-stable C$^*$-algebras with prescribed spectral data

TL;DR

This paper characterizes how spectral measures induced by quasitraces determine self-adjoint operators in Z-stable C*-algebras, showing any continuous spectral measure map can be realized by an operator.

Contribution

It proves that any continuous map from quasitraces to spectral measures on [0,1] can be realized by a self-adjoint operator in the algebra.

Findings

Spectral measures determine unitary equivalence for connected spectra.

Any continuous spectral measure map is realizable by an operator.

The results apply to operators with spectrum equal to [0,1].

Abstract

We consider the variety of spectral measures that are induced by quasitraces on the spectrum of a self-adjoint operator in a simple separable unital and Z-stable C$^*$-algebra. This amounts to a continuous map from the simplex of quasitraces of the C$^*$-algebra into regular Borel probability measures on the spectrum of the operator under consideration. In the case of a connected spectrum this data determines the unitary equivalence class of the operator, and may be reduced to to the case of an operator with spectrum equal to the closed unit interval. We prove that any continuous map from the simplex of quasitraces with the topology of pointwise convergence into regular faithful Borel probability measures on $[0,1]$ with the Levy-Prokhorov metric is realized by some self-adjoint operator in the C$^*$-algebra.