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Labeled Trees and Localized Automorphisms of the Cuntz Algebras
, 2008
"... We initiate a detailed and systematic study of automorphisms of the Cuntz algebras On which preserve both the diagonal and the core UHFsubalgebra. A general criterion of invertibility of endomorphisms yielding such automorphisms is given. Combinatorial investigations of endomorphisms related to per ..."
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We initiate a detailed and systematic study of automorphisms of the Cuntz algebras On which preserve both the diagonal and the core UHFsubalgebra. A general criterion of invertibility of endomorphisms yielding such automorphisms is given. Combinatorial investigations of endomorphisms related to permutation matrices are presented. Key objects entering this analysis are labeled rooted trees equipped with additional data. Our analysis provides insight into the structure of Aut(On) and leads to numerous new examples. In particular, we completely classify all such automorphisms of O2 for the permutation unitaries in ⊗ 4 M2. We show that the subgroup of Out(O2) generated by these automorphisms contains a copy of the infinite dihedral group Z ⋊ Z2.
RELATIVE POSITIONS OF MATROID ALGEBRAS
, 1998
"... A classification is given for regular positions D ⊕ D ⊆ D of Jones index 4 where D = alg lim (C) is an even matroid algebra and where the individual summands have index 2. A similar classification is obtained for positions of direct sums of 2symmetric algebras and, in the odd case, for the positi ..."
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A classification is given for regular positions D ⊕ D ⊆ D of Jones index 4 where D = alg lim (C) is an even matroid algebra and where the individual summands have index 2. A similar classification is obtained for positions of direct sums of 2symmetric algebras and, in the odd case, for the positions of sums of 2symmetric C*algebras in matroid C*algebras. The approach relies on an analysis of intermediate nonselfadjoint operator algebras and the classifications are given in terms of K0 invariants, partial isometry homology and scales in the composite invariant K0(−) ⊕ H1(−).
ON LOCALIZED AUTOMORPHISMS OF THE CUNTZ ALGEBRAS WHICH PRESERVE THE DIAGONAL SUBALGEBRA
"... ABSTRACT. 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 i ..."
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ABSTRACT. 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 ktuples $\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 multiindices 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].
Subalgebras of graph C*algebras
, 2005
"... We prove a spectral theorem for bimodules in the context of graph C∗algebras. A bimodule over a suitable abelian algebra is determined by its spectrum (i.e., its groupoid partial order) iff it is generated by the Cuntz– Krieger partial isometries which it contains iff it is invariant under the gaug ..."
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We prove a spectral theorem for bimodules in the context of graph C∗algebras. A bimodule over a suitable abelian algebra is determined by its spectrum (i.e., its groupoid partial order) iff it is generated by the Cuntz– Krieger partial isometries which it contains iff it is invariant under the gauge automorphisms. We study 1cocycles on the Cuntz–Krieger groupoid associated with a graph C∗algebra, obtaining results on when integer valued or bounded cocycles on the natural AF subgroupoid extend. To a finite graph with a total order, we associate a nest subalgebra of the graph C∗algebra and then determine its spectrum. This is used to investigate properties of the nest subalgebra. We give a characterization of the partial isometries in a graph C∗algebra which normalize a natural diagonal subalgebra and use this to show that gauge invariant generating triangular subalgebras are classified by their