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

$\{{\rm Ad}U|_{A}\ :U\in \mathcal{N}_{\mathcal{O}_{n}}(A)\}$ .

DOI: http://dx.doi.org/10.7900/jot.2011feb13.1939
Keywords:

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

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