We report on recent progress in the program to classify separable amenable C∗-algebras. Our emphasis is on the newly apparent role of regularity properties such as finite decomposition rank, strict comparison of positive elements, and Z-stability, and on the importance of the Cuntz semigroup. We include a brief history of the program’s successes since 1989, a more detailed look at the Villadsen...

We study the semigroup C?-algebra of a positive cone P weakly quasi-lattice ordered group. That is, is subsemigroup discrete group G with P?P?1={e} and such that any two elements common upper bound in also have least bound. find sufficient conditions for to be nuclear. These involve idea generalised length function, called “controlled map”, into an amenable Here we give new definition controlle...

We provide a negative solution to a question of M. Rieffel who asked if the right and left topological stable ranks of a Banach algebra must always agree. Our example is found amongst a class of nest algebras. We show that for many other nest algebras, both the left and right topological stable ranks are infinite. We extend this latter result to Popescu’s non-commutative disc algebras and to fr...

For a locally compact Hausdorff semigroup S, the L representation algebra R(S) was extensively studied by Dunkl and Ramirez. The FourierStieltjes algebra F (S) of a topological semigroup was studied by Lau. The aim of this paper is to investigate these two algebras and study the amenability of them with respect to the structure of S.

in this paper, we nd the relationships between module contractibility of abanach algebra and its ideals. we also prove that module contractibility ofa banach algebra is equivalent to module contractibility of its module uniti-zation. finally, we show that when a maximal group homomorphic image ofan inverse semigroup s with the set of idempotents e is nite, the moduleprojective tensor product ...

Let A be a finitely generated commutative algebra over a field K with a presentation A = K〈X1, . . . , Xn | R〉, where R is a set of monomial relations in the generators X1, . . . , Xn. So A = K[S], the semigroup algebra of the monoid S = 〈X1, . . . , Xn | R〉. We characterize, purely in terms of the defining relations, when A is an integrally closed domain, provided R contains at most two relati...