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Journal of Operator Theory

Volume 69, Issue 2, Spring 2013  pp. 525-533.

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

Authors Tomohiro Hayashi
Author institution: Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya, Aichi, 466-8555, Japan

Summary:  In this paper we investigate the normalizer $\mathcal{N}_{\mathcal{O}_{n}}(A)$ of a $C^{*}$-sub\-algebra $A\subset \mathcal{F}_{n}$ 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 $\{{\rm Ad}U|_{A}\ :U\in \mathcal{N}_{\mathcal{O}_{n}}(A)\}$.

Keywords:  $C^*$-algebra, Cuntz algebra, normalizers

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