Subalgebras of infinite \(C^*\)-algebras with finite Watatani indices. II: Cuntz-Krieger algebras
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.
Cuntz algebras with finite Watatani indices
Cuntz-Krieger algebras with finite Watatani index
unitary representations of the braid group
scientific article; zbMATH DE number 1425084
scientific article