Path-connected Closures of Unitary Orbits

TL;DR

This paper investigates conditions under which the closure of a unitary orbit of a representation between certain C*-algebras is path-connected, extending previous results to broader classes of algebras.

Contribution

It generalizes earlier findings by establishing path-connectedness of unitary orbit closures for all separable A and specific classes of B.

Findings

Path-connected closures for all separable A.

Extension to AF and homogeneous A with von Neumann B.

Results for ASH A and finite von Neumann B.

Abstract

Suppose A and B are unital C*-algebras and A is separable. Let Rep(A,B) denote the set of all unital *-homomorphisms from A to B with the topology of pointwise convergence. We consider the problem of when the closure of the unitary orbit of a single representation in Rep(A,B) is path-connected. An affirmative answer was given by the first author when A is singly generated and B is the algebra of all operators on a separable Hilbert space. We extend this result for all separable A. We also give an affirmative answer when A is AF or homogeneous and B is a von Neumann algebra or when A is ASH and B is a finite von Neumann algebra.