Self-adjoint operators in Z-stable C$^*$-algebras with prescribed spectral data
Andrew S. Toms, Hao Wan
TL;DR
The paper addresses realizing prescribed spectral data for self-adjoint operators in unital simple separable $\mathcal{Z}$-stable C*-algebras via quasitrace-induced spectral measures. It constructs a Cu-morphism from a given continuous affine map $h: \mathrm{QT}(A) \to \mathcal{P}([0,1])^+$ and lifts it to a $*$-homomorphism from $C[0,1]$ into $A$ using Robert’s lifting theorem, yielding a self-adjoint element $a$ with spectrum $[0,1]$ and spectral data $f_a=h$. The work unifies measure-theoretic and Cu-theoretic approaches to classify unitary orbits by spectral data in the Z-stable setting, extending Weyl-type results to infinite-dimensional algebras. It demonstrates that all continuous, faithful spectral data maps arise from actual operators, highlighting the deep connection between quasitraces, spectral measures, and Cu-semigroup structure in this class of algebras.
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.