Projections on weak*-closed subspace of dual Banach algebras

Our first purpose in this paper is to provide necessary conditions for a weak*-closed translation invariant subspace in the semigroup algebra of a locally compact topological foundation semigroup to be completely complemented. We give conditions when a weak*-closed left translation invariant subspace in Ma(S)* of a compact cancellative foundation semigroup S is the range of a weak*-weak* continuous projection on M _a (S)* commuting with translations. Let G be a locally compact group and A be a Banach G-module. Our second purpose in this paper is to study some projections on A* and B(A*) which commutes with translations and convolution.     

Preduals of semigroup algebras

For a locally compact group G, the measure convolution algebra M(G) carries a natural coproduct. In previous work, we showed that the canonical predual C ₀(G) of M(G) is the unique predual which makes both the product and the coproduct on M(G) weak^*-continuous. Given a discrete semigroup S, the convolution algebra ℓ ¹(S) also carries a coproduct. In this paper we examine preduals for ℓ ¹(S) making both the product and the coproduct weak^*-continuous. Under certain conditions on S, we show that ℓ ¹(S) has a unique such predual. Such S include the free semigroup on finitely many generators. In general, however, this need not be the case even for quite simple semigroups and we construct uncountably many such preduals on ℓ ¹(S) when S is either ℤ₊×ℤ or (ℕ,⋅).     

On character amenability of semigroup algebras

We study the character amenability of semigroup algebras. We work on general semigroups and certain semigroups such as inverse semigroups with a finite number of idempotents, inverse semigroups with uniformly locally finite idempotent set, Brandt and Rees semigroup and study the character amenability of the semigroup algebra l¹(S) in relation to the structures of the semigroup S. In particular, we show that for any semigroup S, if ?¹(S) is character amenable, then S is amenable and regular. We also show that the left character amenability of the semigroup algebra ?¹(S) on a Brandt semigroup S over a group G with index set J is equivalent to the amenability of G and J being finite. Finally, we show that for a Rees semigroup S with a zero over the group G, the left character amenability of ?¹(S) is equivalent to its amenability, this is in turn equivalent to G being amenable.     

On the heredity of Hun and Hungarian property

Denote byD(S) the convolution semigroup of compact-regular probability measures on a topological semigroupS. Hincin's classical decomposition theorems are extended to finite point processes on a completely regular topological space and to the convolution semigroupsD(D(G)), D(D(D(G))),... whereG is a locally compact Hausdorff group. The paper applies the Hun-Hungarian semigroup theory approach of Ruzsa and Székely; the proofs also follow this abstract setting.     

Metrizability of Clifford topological semigroups

We prove that a countably compact Clifford topological semigroup S is metrizable if and only if the set E={e∈S:ee=e} of idempotents of S is a metrizable G _δ -set in S.     

Universal Objects in Some Classes of Clifford Topological Inverse Semigroups

We construct universal objects in various classes of Clifford topological inverse semigroups. In particular, we show that each compact Abelian topological inverse semigroup with Lawson maximal semilattice embeds into a power of the cone over the circle.     

The continuity of the inversion and the structure of maximal subgroups in countably compact topological semigroups

We search for conditions on a countably compact (pseudocompact) topological semigroup under which: (i) each maximal subgroup H(e) in S is a (closed) topological subgroup in S; (ii) the Clifford part H(S) (i.e. the union of all maximal subgroups) of the semigroup S is a closed subset in S; (iii) the inversion inv: H(S) → H(S) is continuous; and (iv) the projection π: H(S) → E(S), π: x ↦ xx ⁻¹, onto the subset of idempotents E(S) of S, is continuous.      

Amenability constants for semilattice algebras

For any finite commutative idempotent semigroup S, a semilattice, we show how to compute the amenability constant of its semigroup algebra ℓ ¹(S). This amenability constant is always of the form 4n+1. We then show that these give lower bounds to amenability constants of certain Banach algebras graded over semilattices. We also give example of a commutative Clifford semigroups G _n whose semigroup algebras ℓ ¹(G _n ) admit amenability constants of the form 41+4(n−1)/n. We also show there is no commutative semigroup whose semigroup algebra has an amenability constant between 5 and 9. N. Spronk’s research was supported by NSERC Grant 312515-05.     

Topological Brandt semigroups

In the present paper, the properties of a locally compact Hausdorff topological Brandt semigroup, and the relation between its semigroup algebras and ? ¹-Munn algebras over group algebras are investigated. It is proved that for each locally compact Hausdorff topological group G, and each index set I, there exists a locally compact Hausdorff topological Brandt semigroup S=B(G,I) such that the Banach algebras $\mathcal {LM}_{I}(M(G))$ and $\mathcal{LM}_{I}(L^{1}(G))$ are isometrically isomorphic to M(S)/? ¹({0}) and M _a (S)/? ¹({0}), respectively.     

On the factor sets of measures and local tightness of convolution semigroups over Lie groups

It is shown that for a large class of Lie groups (called weakly algebraic groups) including all connected semisimple Lie groups the following holds: for any probability measure on the Lie group the set of all two-sided convolution factors is compact if and only if the centralizer of the support of inG is compact. This is applied to prove that for any connected Lie groupG, any homomorphism of any real directed (submonogeneous) semigroup into the topological semigroup of all probability measures onG is locally tight.     

Semiprimitivity of Orthodox Semigroup Algebras

Let S be a finite orthodox semigroup or an orthodox semigroup where the idempotent band E(S) is locally pseudofinite. In this paper, by using principal factors and Rukolaǐne idempotents, we show that the contracted semigroup algebra R₀[S] is semiprimitive if and only if S is an inverse semigroup and R[G] is semiprimitive for each maximal subgroup G of S. This theorem strengthens previous results about the semiprimitivity of inverse semigroup algebras.     

Uniform semigroups and fixed point properties

LetS be a uniform semigroup (this includes all topological groups and affine semigroups). We show that a certain space of uniformly continuous functions onS has a left invariant mean iffS has the fixed point property for uniformly continuous affine actions ofS on compact convex sets. This is closely related to but independent of the results of T. Mitchell in [13] and A. Lau in [10]. Interesting examples and consequences are given for the special cases of topological groups and affine convolution semigroups of probability measures on a locally compact semigroup or group. Research Supported by NSERC of Canada Grant No. A8227.     

Pseudo-contractibility and pseudo-amenability of semigroup algebras

In this paper, we characterize pseudo-contractibility of ℓ ¹(S), where S is a uniformly locally finite inverse semigroup. As a consequence, we show that for a Brandt semigroup S=M⁰(G,I),{S={\mathcal{M}}^{0}(G,I),} the semigroup algebra ℓ ¹(S) is pseudo-contractible if and only if G and I are finite. Moreover, we study the notions of pseudo-amenability and pseudo-contractibility of a semigroup algebra ℓ ¹(S) in terms of the amenability of S.     

Isometric Isomorphisms Between Banach Algebras Related to a Certain Class of Clifford Topological Semigroups

In the present paper it is shown that if S₁ and S₂ are two Clifford topological semigroups satisfying certain conditions and T is an isometric isomorphism of LUC(S₁*) onto LUC(S₂*), then T maps S₁ topologically isomorphically onto S₂. Furthermore, T maps M ^l _n(S₁) (M(S₁), respectively) isometrically isomorphically onto M ^l _n(S₂) (M(S₂), respectively). Indeed, we have obtained a generalization of a well-known result of Ghahramani, Lau and Losert for locally compact groups to a more general setting of Clifford topological semigroups.     

Undirected power graphs of semigroups

The undirected power graph G(S) of a semigroup S is an undirected graph whose vertex set is S and two vertices a,b∈S are adjacent if and only if a≠b and a ^m =b or b ^m =a for some positive integer m. In this paper we characterize the class of semigroups S for which G(S) is connected or complete. As a consequence we prove that G(G) is connected for any finite group G and G(G) is complete if and only if G is a cyclic group of order 1 or p ^m . Particular attention is given to the multiplicative semigroup ℤ _n and its subgroup U _n , where G(U _n ) is a major component of G(ℤ _n ). It is proved that G(U _n ) is complete if and only if n=1,2,4,p or 2p, where p is a Fermat prime. In general, we compute the number of edges of G(G) for a finite group G and apply this result to determine the values of n for which G(U _n ) is planar. Finally we show that for any cyclic group of order greater than or equal to 3, G(G) is Hamiltonian and list some values of n for which G(U _n ) has no Hamiltonian cycle.     

Regular Semigroups Each of Whose Least Completely Simple Congruence Classes Has a Greatest Element

A regular semigroup S is termed locally F-regular, if each class of the least completely simple congruence ξ contains a greatest element with respect to the natural partial order. It is shown that each locally F-regular semigroup S admits an embedding into a semidirect product of a band by S/ξ. Further, if ξ satisfies the additional property that for each s ∈ S and each inverse (sξ)′ of sξ in S/ξ the set (sξ)′ ∩ V(s) is not empty, we represent S both as a Rees matrix semigroup over an F-regular semigroup as well as a certain subsemigroup of a restricted semidirect product of a band by S/ξ.     

Involutions and trivolutions on second dual of algebras related to locally compact groups and topological semigroups

We investigate involutions and trivolutions in the second dual of algebras related to a locally compact topological semigroup and the Fourier algebra of a locally compact group. We prove, among the other things, that for a large class of topological semigroups namely, compactly cancellative foundation \(*\)-semigroup S when it is infinite non-discrete cancellative, \(M_a(S)^{**}\) does not admit an involution, and \(M_a(S)^{**}\) has a trivolution with range \(M_a(S)\) if and only if S is discrete. We also show that when G is an amenable group, the second dual of the Fourier algebra of G admits an involution extending one of the natural involutions of A(G) if and only if G is finite. However, \(A(G)^{**}\) always admits trivolution.     

Classification of real Nash fibre bundles

Let G be a compact subgroup of an orthogonal group and X an affine, real, semialgebraic Nash variety. A principal Nash G-bundle over X is said to be strongly Nash if it is induced, up to Nash equivalences, of some universal bundle under a Nash map. Not all Nash bundles are strongly Nash and we denote by S(X, G) the class of strongly Nash G-bundles over X. The principal aim of this paper is to prove the following classification theorem: two bundles of S(X, G) are Nash equivalent if and only if they are topologically equivalent; more,there exists a bijection between the family of the classes of Nash equivalent bundles of S(X, G) and , where is the sheaf of germs of the continous maps from X to G. This result leads to find the largest class of principal Nash G-bundles over X in which the topological equivalence always implies the Nash one. Well, we prove that this class is exactly S(X, G). Research partially supported by M.I.U.R.     

Almost maximal spaces

A topological space X is called almost maximal if it is without isolated points and for every x∈X, there are only finitely many ultrafilters on X converging to x. We associate with every countable regular homogeneous almost maximal space X a finite semigroup Ult(X) so that if X and Y are homeomorphic, Ult(X) and Ult(Y) are isomorphic. Semigroups Ult(X) are projectives in the category F of finite semigroups. These are bands decomposing into a certain chain of rectangular components. Under MA, for each projective S in F, there is a countable almost maximal topological group G with Ult(G) isomorphic to S. The existence of a countable almost maximal topological group cannot be established in ZFC. However, there are in ZFC countable regular homogeneous almost maximal spaces X with Ult(X) being a chain of idempotents.