[Paper Review] Unitary orbits of self-adjoint operators in simple Z-stable C*-algebras

Bhishan Jacelon, Karen R. Strung+1 more|arXiv (Cornell University)|Feb 18, 2015

Advanced Operator Algebra Research22 references3 citations

TL;DR

This paper establishes that in simple, unital, exact, Z-stable C*-algebras of stable rank one, the distance between unitary orbits of self-adjoint elements with connected spectrum is completely determined by spectral data, specifically via the Lévy–Prokhorov distance. The key result is the equality $ d_U(a,b) = d_P(a,b) = d_W(a,b) $, where $ d_U $ is the unitary orbit distance, $ d_P $ is the spectral measure distance, and $ d_W $ is the Cuntz equivalence pseudometric.

ABSTRACT

We prove that in a simple, unital, exact, Z-stable C*-algebra of stable rank one, the distance between the unitary orbits of self-adjoint elements with connected spectrum is completely determined by spectral data. This fails without the assumption of Z-stability.

Motivation & Objective

• To determine when two self-adjoint elements in a C*-algebra are approximately unitarily equivalent using spectral data.
• To investigate whether the unitary orbit distance $ d_U $ can be computed purely from spectral measures in regular C*-algebras.
• To establish the equivalence of three pseudometrics: $ d_U $, $ d_P $ (Lévy–Prokhorov), and $ d_W $ (Cuntz subequivalence) under Z-stability and stable rank one.
• To clarify the necessity of Z-stability and connected spectra for such spectral characterizations.
• To extend known results from finite-dimensional matrices and semifinite factors to infinite-dimensional, simple, Z-stable C*-algebras.

Proposed method

• Use the Cuntz semigroup framework to define the pseudometric $ d_W $ via Cuntz subequivalence of positive elements.
• Establish $ d_W(a,b) = d_P(a,b) $ in stably finite, simple, Z-stable C*-algebras with connected spectrum using strict comparison and trace conditions.
• Prove $ d_U(a,b) = d_W(a,b) $ by lifting the equality from matrix algebras through classification theory to the class of simple, Z-stable C*-algebras of stable rank one.
• Define $ d_P(a,b) $ as the infimum over $ r > 0 $ such that spectral measure inequalities hold for all open sets under every trace $ \tau \in T(A) $.
• Apply classification results and stability properties to extend the equality $ d_U = d_P = d_W $ beyond finite algebras to the full class of simple, unital, Z-stable, exact, stably finite C*-algebras.
• Use functional calculus and spectral projections to relate $ d_P $ to the distribution of eigenvalues and traces of continuous functions on the spectrum.

Experimental results

Research questions

• RQ1 Can the unitary orbit distance $ d_U(a,b) $ between self-adjoint elements be completely determined by spectral data in simple, Z-stable C*-algebras?
• RQ2 Does the equality $ d_U = d_P $ hold for self-adjoint elements with connected spectrum in stably finite, Z-stable C*-algebras?
• RQ3 Is Z-stability necessary for the spectral determination of $ d_U $, or does it fail in non-Z-stable algebras?
• RQ4 To what extent can the equality $ d_U = d_P = d_W $ be extended to normal operators or more general compact spectra?
• RQ5 Can the results be generalized to order zero maps from finite-dimensional C*-algebras or other domains?

Key findings

• In simple, unital, exact, Z-stable C*-algebras of stable rank one, the unitary orbit distance $ d_U(a,b) $ is completely determined by the spectral data of self-adjoint elements with connected spectrum.
• The equality $ d_U(a,b) = d_P(a,b) = d_W(a,b) $ holds for positive contractions $ a,b $ with $ \sigma(a) = \sigma(b) = [0,1] $ in such algebras.
• The result fails without Z-stability: there exist non-approximately unitarily equivalent positive contractions $ a,b $ with $ d_P(a,b) = 0 $ but $ d_U(a,b) > 0 $ in non-Z-stable AH algebras.
• The equality $ d_W(a,b) = d_P(a,b) $ relies on strict comparison and the absence of Cuntz-equivalent projections, which holds under Z-stability.
• The assumption of connected spectrum is necessary; counterexamples exist in monotracial AF algebras where projections with equal trace are not unitarily equivalent.
• The result extends to the broader class of pure C*-algebras of stable rank one, not just Z-stable ones, due to shared structural properties in the Cuntz semigroup.