We show that for a discrete semigroup S there exists a uniquely determined complete Boolean algebra B(S) the algebra of clopen subsets of M(S). M(S) is the phase space of the universal minimal dynamical system for S and it is an extremally disconnected compact Hausdorff space. We deal with this connection of semigroups and complete Boolean algebras focusing on structural properties of these alg...

Let K be a field of characteristic p > 0. Denote by ω(R) the augmentation ideal of either a group algebra R = K[G] or a restricted enveloping algebra R = u(L) over K. We first characterize those R for which ω(R) satisfies a polynomial identity not satisfied by the algebra of all 2× 2 matrices over K. Then, we examine those R for which ω(R) satisfies a semigroup identity (that is, a polynomial i...

We classify exactly when the toric algebras C[ST (r)] are Gorenstein. These algebras arise as toric deformations of algebras of invariants of the Cox-Nagata ring of the blow-up of n − 1 points on P, or equivalently algebras of the ring of global sections for the Plücker embedding of weight varieties of the Grassmanian Gr2(Cn), and algebras of global sections for embeddings of moduli of weighted...

Denote by (R, ·) the multiplicative semigroup of an associative algebra R over an infinite field, and let (R, ◦) represent R when viewed as a semigroup via the circle operation x ◦ y = x + y + xy. In this paper we characterize the existence of an identity in these semigroups in terms of the Lie structure of R. Namely, we prove that the following conditions on R are equivalent: the semigroup (R,...

Abstract. We introduce analogs of the Hopf algebra of Free quasi-symmetric functions with bases labelled by colored permutations. When the color set is a semigroup, an internal product can be introduced. This leads to the construction of generalized descent algebras associated with wreath products Γ ≀ Sn and to the corresponding generalizations of quasi-symmetric functions. The associated Hopf ...

The Fourier and Fourier-Stieltjes algebras A(G) and B(G) of a locally compact group G are introduced and studied in 60’s by Piere Eymard in his PhD thesis. If G is a locally compact abelian group, then A(G) ≃ L(Ĝ), and B(G) ≃ M(Ĝ), via the Fourier and Fourier-Stieltjes transforms, where Ĝ is the Pontryagin dual of G. Recently these algebras are defined on a (topological or measured) groupoid an...

For all classical groups (and for their analogs in infinite dimension or over general base fields or rings) we construct certain contractions, called homotopes. The construction is geometric, using as ingredient involutions of associative geometries. We prove that, under suitable assumptions, the groups and their homotopes have a canonical semigroup completion. Introduction: The classical group...

We continue the investigation of notions of approximate amenability that were introduced in work of the second and third authors together with R. J. Loy. It is shown that every boundedly approximately contractible Banach algebra has a bounded approximate identity, and that the Fourier algebra of the free group on two generators is not operator approximately amenable. Further examples are obtain...

We find presentations for subalgebras of invariants of the coordinate algebras of binary symmetric models of phylogenetic trees studied by Buczynska and Wisniewski in [BW]. These algebras arise as toric degenerations of rings of global sections of weight varieties of the Grassmanian of two planes associated to the Plücker embedding, and as toric degenerations of rings of invariants of Cox-Nagat...

We prove that for every group G and any two sets I, J , the Brandt semigroup algebras l(B(I,G)) and l(B(J,G)) are Morita equivalent with respect to the Morita theory of selfinduced Banach algebras introduced by Grønbæk. As applications, we show that if G is an amenable group, then for a wide class of Banach l(B(I,G))-bimodules E, and every n > 0, the bounded Hochschild cohomology groups H(l(B(I...