Kaplansky's problem and unitary orbits in matrix amplifications

TL;DR

This paper investigates how the distances between unitary orbits of elements in certain C*-algebras behave under matrix amplifications, revealing invariance, non-monotonicity, and topological obstructions.

Contribution

It demonstrates that unitary orbit distances are invariant under amplification in UHF-stable C*-algebras, shows non-monotonic behavior in general, and identifies K-theoretic obstructions in purely infinite cases.

Findings

Distances remain unchanged under amplification in UHF-stable C*-algebras.

Examples show distances can decrease after amplification.

K-theory provides obstructions in purely infinite C*-algebras.

Abstract

We study the distances between the unitary orbits of matrix amplifications of elements in certain C*-algebras. In particular, we show that the distance between unitary orbits of arbitrary elements in unital, separable, UHF-stable C*-algebras remains unchanged when amplifying to certain matrix sizes. We further exhibit examples of elements in C*-algebras where the distance between unitary orbits becomes strictly smaller after amplifying by a certain matrix size, and we demonstrate that distances between unitary orbits of amplifications are not monotone in the multiplicity of the amplifications, even in the setting of matrix algebras. Lastly, we show that topological K-theory provides obstructions in the purely infinite setting.