ON LOCALIZED AUTOMORPHISMS OF THE CUNTZ ALGEBRAS WHICH PRESERVE THE DIAGONAL SUBALGEBRA (New development of Operator Algebras)

In 1978, Cuntz raised the problem of classifying automorphisms of $O_{\mathfrak{n}}$ which leave both the diagonal and the core UHF subalgebra invariant. In this note, we start developing a machinery that might be useful towards this goal. In particular, we give a practical criterion of invertibility of endomorphisms of $O_{n}$ corresponding to unitaries in the normalizer of the diagonal inside the UHF subalgebra. We also analyze the action of such localized automorphisms on the spectrum of the diagonal thus obtaining criteria of outerness. If $n$ is an integer greater than 1, then the Cuntz algebra $O_{n}$ is a unital, simple C’-algebra generated by $n$ isometries $S_{1},$ $\ldots$ , $S_{n}$ satisfying $\sum_{j=1}^{n}S_{1}S_{j}^{*}=1[5]$ . As in [5], we denote by $W_{n}^{k}$ the set of k-tuples $\alpha=(\alpha^{1}, \ldots, \alpha^{k})$ with $\alpha^{m}\in\{1, \ldots, n\}$ , and we denote by $W_{n}$ the union $\bigcup_{k=0}^{\infty}W_{n}^{k}$ , where $W_{n}^{0}=\{0\}$ . Elements of $W_{n}$ are called multi-indices and if $\alpha\in W_{n}^{k}$ then $l(\alpha)=k$ , the length of $\alpha$ . If $\alpha=(\alpha^{1}, \ldots, \alpha^{k})\in W_{n}^{k}$ , then $S_{\alpha}=S_{\alpha^{1}}\cdots S_{\alpha^{k}}$ , with $S_{0}=1$ by convention. Each $S_{\alpha}$ is an isometry and its range projection is $S_{\alpha}S_{\alpha}^{*}$ . Every word in $\{S_{1}, S_{i}^{*} : i=1, \ldots, n\}$ can be uniquely expressed as $S_{\alpha}S_{\beta}$ for some $\alpha,\beta\in W_{n}[5$ , Lemma 1.3]. The C’-subalgebra of $O_{n}$ generated by $\{S_{\alpha}S_{\beta}^{*} : l(\alpha)=l(\beta)\}$ is isomorphic to $M_{n^{k}}(\mathbb{C})$ and denoted $\mathcal{F}_{n}^{k}$ . The norm closure of the union $\bigcup_{k=0}^{\infty}\mathcal{F}_{n}^{k}$ is a UHF-algebra of type $n^{\infty}$ and is denoted $\mathcal{F}_{n}$ . It is called the core UHF-subalgebra of $O_{n}$ . There exists a faithful conditional expectation from $O_{n}$ onto $\mathcal{F}_{n}[5]$ . The $C$“-subalgebra of $O_{n}$ generated by all projections $S_{\alpha}S_{\alpha}^{*},$ $\alpha\in W_{n}$ , is denoted $\mathcal{D}_{n}$ and called the diagonal subalgebra of $O_{n}$ . It is a maximal abelian subalgebra of $\mathcal{O}_{n}$ , regular both in $\mathcal{F}_{n}$ and in $O_{n}[8]$ . The spectrum of $\mathcal{D}_{n}$ is naturally identified with $X_{n}$ , the collection of infinite words on the alphabet $\{1, \ldots, n\}$ [8]. With the product topology, $X_{n}$ is homeomorphic to the Cantor set. There exists a faithful conditional expectation from $\mathcal{F}_{n}$ onto $D_{n}$ and whence from $\mathcal{O}_{n}$ onto $\mathcal{D}_{\mathfrak{n}}$ as well. We denote $\mathcal{D}_{n}^{k}=D_{n}\cap \mathcal{F}_{n}^{k}$ . Let End $(\mathcal{O}_{n})$ be the semigroup (with composition) of endomorphisms of $O_{n}$ , that is unital $*$-homomorphisms of $O_{n}$ into itself. Since $O_{n}$ is simple, each endomorphism is injective and it is invertible (automorphism) if and only if it is surjective. Let $\mathcal{U}(O_{n})$ be the group of all unitaries in $\mathcal{O}_{n}$ . As shown in [6], there is abijective map $\lambda$ : $\mathcal{U}(O_{n})arrow End(O_{n})$ determined by (1) $\lambda_{u}(S_{1})=u^{*}S_{1}$ , $i=1,$ $\ldots,$ $n$ . The inverse of $\lambda$ is the map $\psirightarrow\sum_{i=1}^{n}S_{1}\psi(S_{i}^{*})$ . The map $\lambda$ becomes a semigroup isomorphism once $\mathcal{U}(O_{n})$ is equipped with the convolution multiplication (2) $u*w=u\lambda_{u}(w)$ . Endomorphisms of Cuntz algebras have been studied extensively by many authors and in variety of contexts. In particular, they appear in connection with Jones index theory for subfactors. We would only like to mention papers [7, 9, 12, 2, 10, 11] which are closest in spirit to the present note. In these and other works, a prominent role is played by Date: January 16, 2008. 数理解析研究所講究録 第 1587巻 2008年 109-115 109