On normalizers of $C^{\ast }$-subalgebras in the Cuntz algebra ${\mathcal{O}}_n$

Abstract

In this paper we investigate the normalizer ${\mathcal{N}}_{{\mathcal{O}}_n}(A)$ of a $C^{\ast }$-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 $A^{\prime }\cap {\mathcal{F}}_n$ is finite-dimensional, we show several facts for normalizers of $A$. In particular it is shown that the automorphism group $\{\mathrm{Ad}u{|}_A|u\in {\mathcal{N}}_{{\mathcal{F}}_n}(A)\}$ has a finite index in $\{\mathrm{Ad}U{|}_A|U\in {\mathcal{N}}_{{\mathcal{O}}_n}(A)\}$.

arXiv comment9 pages

References in corpus (1)

• Automorphisms of the Cuntz algebras