Semgroupoid with Tracially Z-Absorbing and Non –Stable K- Theory with non Commutative Amir- Combern Theorem on C^*Algebras

Abstract

The semigoroupoid C^*-algebra is shown to be isomorphic to the algebras usually attached to the corresponding combinatorial object, namely the Cuntz- Krieger algebras and the higher-rank graph C^*-algebras, respectively. In the case of a higher-rank graph, it follows that the dimension function is superfluous for defining the corresponding C^*-algebra. We study a tracial notion of Z- absorption for simple, unital C^*-algebras. We also show that weak cancellation implies the properties for extremally rich C^*-algebras and that the class of extremally rich C^*-algebras with weak cancellation is closed under extensions. Moreover, we consider analogous properties which replace the group K_1 (A) with the extremal K-set K_e (A) as well as two versions of K_0-surjectivity. We study that von Neumann algebras and separable nuclear C^*-algebras are stable for the Banach-Mazur cb- distance. A technical step is to show for the unital almost completely isometric maps between C^*-algebras are almost multiplicative and almost self adjoint.