We prove that the Cuntz semigroup is recovered functorially from the Elliott invariant for a large class of C *-algebras. In particular, our results apply to the largest class of simple C *-algebras for which K-theoretic classification can be hoped for. This work has three significant consequences. First, it provides new conceptual insight into Elliott's classification program, proving that the...

The study of semigroup algebras has a long tradition in commutative algebra. Presentation ideals of semigroup algebras are called toric ideals, in reference to their prominent role in geometry. In this paper we consider the more general class of lattice ideals. Fix a polynomial ring S = k[x1, . . . , xn] over a field k and identify monomials x in S with vectors a ∈ N. Let L be any sublattice of...

For a cancellative semigroup S and field F, it is proved that the algebra FS centrally essential if only group of fractions GS exists FGS essential. The has classical right ring which ring. There exist non-commutative algebras over fields zero characteristic (this contrasts with known fact are commutative).

The cone of lower semicontinuous traces is studied with a view to its use as an invariant. Its properties include compactness, Hausdorffness, and continuity with respect to inductive limits. A suitable notion of dual cone is given. The cone of lower semicontinuous 2-quasitraces on a (non-exact) C*-algebra is considered as well. These results are applied to the study of the Cuntz semigroup. It i...

Let $S$ be an inverse semigroup and let $E$ be its subsemigroup of idempotents. In this paper we define the $n$-th module cohomology group of Banach algebras and show that the first module cohomology group $HH^1_{ell^1(E)}(ell^1(S),ell^1(S)^{(n)})$ is zero, for every odd $ninmathbb{N}$. Next, for a Clifford semigroup $S$ we show that $HH^2_{ell^1(E)}(ell^1(S),ell^1(S)^{(n)})$ is a Banach sp...

We give a local characterization for the Cuntz semigroup of AI-algebras building upon Shen's dimension groups. Using this result, we provide an abstract AI-algebras.

We introduce the notion of the Fourier and Fouier-Stieltjes algebra of a topological ∗-semigroup and show that these are commutative Banach algebras. For a class of foundation semigroups, we show that these are preduals of von Neumann algebras. 1. Definitions and Notations Let S be a locally compact topological semigroup and M(S) be the Banach algebra of all bounded regular Borel measures μ on ...