On normalizers of $C^{*}$-subalgebras in the Cuntz algebra $\mathcal{O}_{n}$

TL;DR

This paper studies the structure of normalizers of certain subalgebras within the Cuntz algebra, revealing finite index relations under specific conditions on the subalgebra's commutant.

Contribution

It establishes new results on the relationship between normalizers in the UHF subalgebra and the entire Cuntz algebra, especially regarding automorphism groups.

Findings

Automorphism group of the subalgebra has finite index in the larger algebra's automorphism group.

Normalizers of the subalgebra exhibit specific finite-dimensional properties.

The relative commutant condition leads to structural insights about automorphisms.

Abstract

In this paper we investigate the normalizer $\mathcal{N}_{\mathcal{O}_{n}}(A)$ of a $C^{*}$-subalgebra $A\subset \mathcal{F}_{n}$ where $\mathcal{F}_{n}$ is the canonical UHF-subalgebra of type $n^{\infty}$ in the Cuntz algebra $\mathcal{O}_{n}$. Under the assumption that the relative commutant $A'\cap \mathcal{F}_{n}$ is finite-dimensional, we show several facts for normalizers of $A$. In particular it is shown that the automorphism group $\{{\rm Ad}u|_{A}\ \ |\ u\in \mathcal{N}_{\mathcal{F}_{n}}(A)\}$ has a finite index in $\{{\rm Ad}U|_{A}\ \ |\ U\in \mathcal{N}_{\mathcal{O}_{n}}(A)\}$.