On Endomorphisms of the Cuntz Algebra which Preserve the Canonical UHF-Subalgebra, II

TL;DR

This paper investigates specific endomorphisms of the Cuntz algebra that preserve the canonical UHF subalgebra, providing a negative answer to a question about their representation and analyzing their structural properties.

Contribution

It demonstrates that not all such endomorphisms are implementable by unitaries within the subalgebra, and characterizes their structure when the relative commutant is finite dimensional.

Findings

Counterexample to the existence of a unitary $v$ in $F_n$ such that $lambda_u|_{F_n} = lambda_v|_{F_n}$

Structural analysis of endomorphisms with finite dimensional relative commutant

Clarification of the relationship between endomorphisms and their implementing unitaries

Abstract

It was shown recently by Conti, R{\o}rdam and Szyma\'{n}ski that there exist endomorphisms $\lambda_u$ of the Cuntz algebra $\mathcal{O}_n$ such that $\lambda_u (\mathcal{F}_n)\subseteq\mathcal{F}_n$ but $u\not\in\mathcal{F}_n$, and a question was raised if for such a $u$ there must always exist a unitary $v\in\mathcal{F}_n$ with $\lambda_u|_{\mathcal{F}_n} = \lambda_v|_{\mathcal{F}_n}$. In the present paper, we answer this question to the negative. To this end, we analyze the structure of such endomorphisms $\lambda_u$ for which the relative commutant $\lambda_u(\mathcal{F}_n)'\cap\mathcal{F}_n$ is finite dimensional.