The basic properties of the cones of lower semicontinuous traces and 2-quasitraces are studied. These properties include: compactness and Hausdorffness of the given cone, continuity of the corresponding functor, and a suitable notion of dual space. These results are applied to the study of the Cuntz semigroup of some classes of C*-algebras. It is shown that if a C*-algebra absorbs the Jiang-Su ...

A class of C*-algebras is described for which the homomorphism from C0(0, 1] to the algebra may be classified by means of the Cuntz semigroup functor. Examples are given of algebras—simple and non-simple—for which this classification fails. It is shown that a suitable suspension of the Cuntz semigroup functor deals successfully with some of these counterexamples.

In this paper, we introduce a new class algebras related to BCK-algebras and semigroups, called a KS-semigroup and define an ideal of a KS-semigroups and a strong KS-semigroup, and characterizations of ideals is given. Also we define a congruence relation on a KS-semigroups and a quotient KS-semigroups and prove the isomorphism. Mathematics Subject Classification: 06F35, 03G25

In the present paper, we consider biflatness of certain classes of semigroupalgebras. Indeed, we give a necessary condition for a band semigroup algebra to bebiflat and show that this condition is not sufficient. Also, for a certain class of inversesemigroups S, we show that the biflatness of ell^{1}(S)^{primeprime} is equivalent to the biprojectivity of ell^{1}(S).

We introduce a new invariant for C?-algebras of stable rank one that merges the Cuntz semigroup information together with K1-group information. This semigroup, termed Cu1-semigroup, is constructed as equivalence classes pairs consisting positive element in stabilization given C?-algebra unitary unitization hereditary subalgebra generated by element. show Cu1-semigroup well-defined continuous fu...

Free semigroup algebras are wot-closed algebras generated by n isometries with pairwise orthogonal ranges. They were introduced in [27] as an interesting class of operator algebras in their own right. The prototype algebra, obtained from the left regular representation of the free semigroup on n letters, was introduced by Popescu [45] in connection with multi-variable non-commutative dilation t...

−→ (Ak, φk) and the operator algebras A = lim −→ (Ak, φk) obtained as limits of direct sums of matrix algebras over E with respect to star-extendible homomorphisms. The invariants in the algebraic case consist of an additive semigroup, with scale, which is a right module for the semiring VE = Homu(E⊗K, E⊗K) of unitary equivalence classes of star-extendible homomorphisms. This semigroup is refer...

Building on work of Elliott and coworkers, we present three applications of the Cuntz semigroup: (i) for many simple C∗-algebras, the Thomsen semigroup is recovered functorially from the Elliott invariant, and this yields a new proof of Elliott’s classification theorem for simple, unital AI algebras; (ii) for the algebras in (i), classification of their Hilbert modules is similar to the von Neu...

Let G ⊆ ωω be a semigroup. G polyadic algebras with equality, or simply G algebras, are reducts of polyadic algebras with equality obtained by restricting the similarity type and axiomatization of polyadic algebras to substitutions in G, and possibly weakening the axioms governing the diagonal elements. Such algebras were introduced in the context of ’finitizing’ first order logic with equality...