Lifting Commutation Relations in Cuntz Algebras

TL;DR

This paper investigates conditions under which the quotient map from free products of unital C*-algebras to their tensor product splits, revealing rare cases involving Cuntz algebras and identifying K-theoretic obstructions.

Contribution

It demonstrates the existence of splitting in specific cases involving Cuntz algebras and describes K-theoretic obstructions to such splittings.

Findings

Splitting occurs when both algebras are Cuntz algebras O_2 or O_infinity.

Splitting is generally rare and not explicit.

K-theoretic obstructions prevent splitting in many cases.

Abstract

We examine splitting of the quotient map from the full free product $A*B$, or the unital free product $A*_{\mathbb C}B$, to the (maximal) tensor product $A\otimes B$, for unital C*-algebras $A$ and $B$. Such a splitting is very rare, but we show there is one if $A$ and $B$ are both the Cuntz algebra $O_2$ or $O_\infty$, and in a few other cases. The splitting is not explicit (and in principle probably cannot be). We also describe severe $K$-theoretic obstructions to a splitting.