Subalgebras of infinite \(C^*\)-algebras with finite Watatani indices. II: Cuntz-Krieger algebras
type: Item
instance of: scholarly article
title: Subalgebras of infinite \(C^*\)-algebras with finite Watatani indices. II: Cuntz-Krieger algebras
zbMATH Open document ID: 0949.46023
DOI: 10.1215/S0012-7094-98-09118-9
author: Masaki Izumi
published in: Duke Mathematical Journal
publication date: 2000-03-27T00:00:00Z
review text: In part I [Comm. Math. Phys. 155, No. 1, 157-182 (1993; Zbl 0803.46066)], the author constructed examples of Cuntz algebras with finite Watatani indices. The present, highly technical work, gives a modification of previous constructions. Starting with biunitary connections on finite graphs, the author obtains an inclusion of the Cuntz-Krieger algebras with finite Watatani index (the construction works also for the infinite-depth case, but then the obtained algebras are a priori not necessarily Cuntz-Krieger algebras). If the graphs are of depth \(2\) and the connection is flat, the construction gives Cuntz algebras and pairs of endomorphisms giving rise to actions of multiplicative unitaries. Further, the author obtains examples of braided endomorphisms of Cuntz-Krieger algebras, and, consequently, examples of unitary representations of the braid group \(B_\infty\) with Markov traces. To this end he introduces the concept of self-conjugate structures of infinite-dimensional Kac algebras, and gives explicit equations describing braiding operators. The paper consists of 4 sections: Introduction, Combinatorial structure of index finite subfactors of infinite factors, Inclusions of Cuntz-Krieger algebras, Depth-2 inclusions and braid symmetry.
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46L37
zbMATH DE Number: 1425084
zbMATH Keywords: Cuntz algebras with finite Watatani indices
zbMATH Keywords: biunitary connections on finite graphs
zbMATH Keywords: Cuntz-Krieger algebras with finite Watatani index
zbMATH Keywords: infinite-depth case
zbMATH Keywords: braided endomorphisms
zbMATH Keywords: unitary representations of the braid group
zbMATH Keywords: Markov traces
zbMATH Keywords: infinite-dimensional Kac algebras
zbMATH Keywords: braid symmetry
reviewed by: Wiesław Tadeusz Żelazko
MaRDI profile type: Publication
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