Let R be a commutative ring and S a finite locally inverse semigroup. It is proved that the semigroup algebra R S is isomorphic to the direct product of Munn algebras M R GJ , mJ , nJ ;PJ with J ∈ S/J, where mJ is the number of R-classes in J , nJ the number of L-classes in J , and GJ a maximum subgroup of J . As applications, we obtain the sufficient and necessary conditions for the semigroup ...

Every computable universal algebra has a finitely presented expansion, but there are examples of finitely generated, computably enumerable universal algebras with no finitely presented expansions. It is natural to ask whether such examples can be found in well-known classes of algebras such as groups and semigroups. In this paper, we build an example of a finitely generated, infinite, computabl...

In [1] we introduced the concept of restricted representations for an inverse semigroup S and studied the restricted forms of some important Banach algebras on S. In this paper, we continue our study by considering the relation between the restricted positive definite functions and retricted representations. In particular, we prove restricted versions of the Godement’s characterization of the p...

In [4] B. M. Schein considered systems of the form ( ; o; n),where is a set of functions closed under the composition "o" of functions (and hence ( ; o) is a function semigroup) and the set theoretic subtraction "n" (and hence is a subtraction algebra in the sense of [1]). He proved that every subtraction semigroup is isomorphic to a di erence semigroup of invertible functions. B. Zelinka [5] d...

We investigate the relationship between the facial structure of the unit ball of an operator algebra A and its algebraic structure, including the hereditary subalgebras and the socle of A. Many questions about the facial structure of A are studied with the aid of representation theory. For that purpose we establish the existence of reduced atomic type representations for certain nonselfadjoint ...

In this article we survey some of the recent goings-on in the classification programme of C-algebras, following the interesting link found between the Cuntz semigroup and the classical Elliott invariant and the fact that the Elliott conjecture does not hold at its boldest. We review the construction of this object both by means of positive elements and via its recent interpretation using counta...

‎Let $S$ be an inverse semigroup with the set of idempotents $E$‎. We prove that the semigroup algebra $ell^{1}(S)$ is always‎ ‎$2n$-weakly module amenable as an $ell^{1}(E)$-module‎, ‎for any‎ ‎$nin mathbb{N}$‎, ‎where $E$ acts on $S$ trivially from the left‎ ‎and by multiplication from the right‎. ‎Our proof is based on a common fixed point property for semigroups‎.  

We extend the concept of amenability of a Banach algebra A to the case that there is an extra A -module structure on A, and show that when S is an inverse semigroup with subsemigroup E of idempotents, then A = l(S) as a Banach module over A= l(E) is module amenable iff S is amenable. When S is a discrete group, l(E) = C and this is just the celebrated Johnson’s theorem.

LetP be a semigroup that admits an embedding into a groupG.Assume that the embedding satisfies the Toeplitz condition recently introduced by the third named author and that the Baum–Connes conjecture holds for G. We prove a formula describing the K-theory of the reduced crossed product A α,r P by any automorphic action of P . This formula is obtained as a consequence of a result on the K-theory...

Employing the construction method of Dirichlet forms on standard forms of von Neu-mann algebras developed in Pa], we construct Dirichlet forms and associated symmetric Markovian semigroups on CCR algebras with respect to quasi-free states. More precisely, let A(h 0) be the CCR algebra over a complex separable pre-Hilbert space h 0 and let ! be a quasi-free state on A(h 0). For any normalized ad...