Involutive equivalence bimodules and inclusions of C*-algebras with Watatani index 2

The Picard groups of unital inclusions of unital $C^*$-algebras induced by involutive equivalence bimodules

Let $A$ be a unital $C^*$-algebra and $X$ an invulutive $A-A$-equivalence bimodule. Let $A\subset C_X$ be the unital inclusion of unital $C^*$-algebras induced by $X$. We suppose that $A' \cap C_X =\mathbf{C} 1$. We shall compute the Picard group of the unital inclusion $A\subset C_X$.

The strong Morita equivalence for coactions of a finite dimensional $C^*$-Hopf algebra on unital $C^*$-algebras

Following Jansen and Waldmann, and Kajiwara and Watatani, we shall introduce notions of coactions of a finite dimensional $C^*$-Hopf algebra on a Hilbert $C^*$-bimodule of finite type in the sense of Kajiwara and Watatani and define their crossed product. We shall investigate their basic properties and show that the strong Morita equivalence for coactions preserves the Rohlin property for coactions of a finite dimensional $C^*$-Hopf algebra on unital $C^*$-algebras.

更高反射与有限Watatani指数的规范移位的熵

Higher reflections and entropy of canonical shifts for inclusions of $C^*$-algebras with finite Watatani index

本文研究了C*代数包含中反射算子与规范移位熵的关系，改进了Young不等式常数并建立了与Watatani指数的联系。

Notes on the bicategory of W*-bimodules

Categories of W*-bimodules are shown in an explicit and algebraic way to constitute an involutive W*-bicategory.

The extension class and KMS states for Cuntz--Pimsner algebras of some bi-Hilbertian bimodules

For bi-Hilbertian $A$-bimodules, in the sense of Kajiwara--Pinzari--Watatani, we construct a Kasparov module representing the extension class defining the Cuntz--Pimsner algebra. The construction utilises a singular expectation which is defined using the $C^*$-module version of the Jones index for bi-Hilbertian bimodules. The Jones index data also determines a novel quasi-free dynamics and KMS states on these Cuntz--Pimsner algebras.

The Cuntz-Pimsner extension and mapping cone exact sequences

For Cuntz-Pimsner algebras of bi-Hilbertian bimodules of finite Jones-Watatani index satisfying some side conditions, we give an explicit isomorphism between the $K$-theory exact sequences of the mapping cone of the inclusion of the coefficient algebra into a Cuntz-Pimsner algebra, and the Cuntz-Pimsner exact sequence. In the process we extend some results by the second author and collaborators from finite projective bimodules to finite index bimodules, and also clarify some aspects of Pimsner's `extension of scalars' construction.