Strict topology over the space of measures with continuous translations on a locally compact foundation semigroup

Let $ \mathfrak{S} $ \mathfrak{S} be a locally compact semigroup, ω be a weight function on $ \mathfrak{S} $ \mathfrak{S} , and M _a ($ \mathfrak{S} $ \mathfrak{S} , ω) be the weighted semigroup algebra of $ \mathfrak{S} $ \mathfrak{S} . Let L ₀^∞ ($ \mathfrak{S} $ \mathfrak{S} ; M _a ($ \mathfrak{S} $ \mathfrak{S} , ω)) be the C*-algebra of all M _a ($ \mathfrak{S} $ \mathfrak{S} , ω)-measurable functions g on $ \mathfrak{S} $ \mathfrak{S} such that g/ω vanishes at infinity. We introduce and study a strict topology β ¹($ \mathfrak{S} $ \mathfrak{S} , ω) on M _a ($ \mathfrak{S} $ \mathfrak{S} , ω) and show that the Banach space L ₀^∞ ($ \mathfrak{S} $ \mathfrak{S} ; M _a ($ \mathfrak{S} $ \mathfrak{S} , ω)) can be identified with the dual of M _a ($ \mathfrak{S} $ \mathfrak{S} , ω) endowed with β ¹($ \mathfrak{S} $ \mathfrak{S} , ω). We finally investigate some properties of the locally convex topology β ¹($ \mathfrak{S} $ \mathfrak{S} , ω) on M _a ($ \mathfrak{S} $ \mathfrak{S} , ω).     

Approximate Amenability of Certain Semigroup Algebras

In the present paper, it is shown that a left cancellative semigroup S (not necessarily with identity) is left amenable whenever the Banach algebra ℓ¹(S) is approximately amenable. It is also proved that if S is a Brandt semigroup over a group G with an index set I, then ℓ¹(S) is approximately amenable if and only if G is amenable. Moreover ℓ¹(S) is amenable if and only if G is amenable and I is finite. For a left cancellative foundation semigroup S with an identity such that for every M_a(S)-measurable subset B of S and s ∈ S the set sB is M_a(S)-measurable, it is proved that if the measure algebra M_a(S) is approximately amenable, then S is left amenable. Concrete examples are given to show that the converse is negative.     

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.     

Strongly and weakly almost periodic linear maps on semigroup algebras

Let S be a foundation locally compact topological semigroup. Two new topologies τ _c and τ _w are introduced on M _a (S)^*. We introduce τ _c and τ _w almost periodic functionals in M _a (S)^*. We study these classes and compare them with each other and with the norm almost periodic and weakly almost periodic functionals. For f∈M _a (S)^*, it is proved that T _f ∈ℬ(M _a (S),M _a (S)^*) is strong almost periodic if and only if f is τ _c -almost periodic. Indeed, we have obtained a generalization of a well known result of Crombez for locally compact group to a more general setting of foundation topological semigroups. Finally if P(S) (the set of all probability measures in M _a (S)) has the semiright invariant isometry property, it is shown that the set of τ _w -almost periodic functionals has a topological left invariant mean.     

A mixed topology over the space of measures with continuous translations on a locally compact semigroup

In this paper, we deal with the semigroup algebra M _a (S) for a locally compact semigroup S under the topology β ¹(S) which has been recently defined in Maghsoudi, Nasr-Isfahani and Rejali (Semigroup Forum 73:367–376, 2006). We first show that the topology β ¹(S) can be considered a natural mixed topology. Then, using this fact, we obtain new results about the topology β ¹(S). As an application of our results, we show that (M _a (S),β ¹(S)), for a wide class of locally compact semigroups, is a complete semi-topological algebra with the convolution multiplication.     

Multivalued perturbations ofm-accretive differential inclusions

Given anm-accretive operatorA in a Banach spaceX and an upper semicontinuous multivalued mapF: [0,a]×X→2 ^X , we consider the initial value problemu′∈−Au+F(t,u) on [0,a],u(0)=x ₀. We concentrate on the case when the semigroup generated by—A is only equicontinuous and obtain existence of integral solutions if, in particular,X* is uniformly convex andF satisfies β(F(t,B))≤k(t)β(B) for all boundedB⊂X wherek∈L ¹([0,a]) and β denotes the Hausdorff-measure of noncompactness. Moreover, we show that the set of all solutions is a compactR _δ-set in this situation. In general, the extra condition onX* is essential as we show by an example in whichX is not uniformly smooth and the set of all solutions is not compact, but it can be omited ifA is single-valued and continuous or—A generates aC _o-semigroup of bounded linear operators. In the simpler case when—A generates a compact semigroup, we give a short proof of existence of solutions, again ifX* is uniformly (or strictly) convex. In this situation we also provide a counter-example in ℝ⁴ in which no integral solution exists. The author gratefully acknowledges financial support by DAAD within the scope of the French-German project PROCOPE.     

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,⋅).     

Stable outer conjugacy and strong Morita equivalence of group actions on pro-C *-algebras

We show that two continuous inverse limit actions α and β of a locally compact group G on two pro-C ^*-algebras A and B are stably outer conjugate if and only if there is a full Hilbert A-module E and a continuous action u of G on E such that E and E ^*(the dual module of E) are countably generated in M(E)(the multiplier module of E), respectively M(E ^*) and the pair (E, u) implements a strong Morita equivalence between α and β. This is a generalization of a result of F. Combes [Proc. London Math. Soc. 49(1984), 289–306].      

Relative invariants for representations of finite dimensional algebras

 Let M be a finite dimensional module over a finite dimensional basic K-algebra Λ, where K is an algebraically closed field. We associate with M a weight θ ^M (i.e. an element of the dual of the Grothendieck group of mod-Λ) in module theoretic terms. Let β be a dimension vector with θ ^M (β)=0. We generalize a construction of relative invariants of quivers due to Schofield [S] and define a relative invariant polynomial function d ^M _β on the variety of modules of dimension vector β, such that d ^M _β (N) = 0 for some module N if and only if there is a nonzero morphism from M to N. Assuming char (K) = 0, we conclude from the main result of Schofield-Van den Bergh [SV] that relative invariants of this form span all the spaces of relative invariants. To get algebra generators of the algebra of semi-invariants it is sufficient to take the d ^M _β with M indecomposable. Received: 31 July 2001     

On the Matlis duals of local cohomology modules and modules of generalized fractions

Let (R, m) be a commutative Noetherian local ring with non-zero identity, a a proper ideal of R and M a finitely generated R-module with aM ≠ M. Let D(−) ≔ Hom _R (−, E) be the Matlis dual functor, where E ≔ E(R/m) is the injective hull of the residue field R/m. In this paper, by using a complex which involves modules of generalized fractions, we show that, if x ₁, …, x _n is a regular sequence on M contained in α, then H _{(x1, …,xnR} ⁿ D(H _a ⁿ (M))) is a homomorphic image of D(M), where H _b ⁱ (−) is the i-th local cohomology functor with respect to an ideal b of R. By applying this result, we study some conditions on a certain module of generalized fractions under which D(H _{(x1, …,xn)R} ⁿ (D(H _a ⁿ (M)))) ⋟ D(D(M)).     

On character amenability of semigroup algebras

Let S be a foundation locally compact topological semigroup, and let M _a (S) be the space of all measures μ∈M(S) for which the maps x?|μ|?δ _x and x?|μ|?δ _x from S into M(S) are weakly continuous. The purpose of this article is to develop a notion of character amenability for semigroup algebras. The main results concern the χ-amenability of M _a (S). We give necessary and sufficient conditions for the existence of a left χ-mean on M _a (S)^?.     

Multipliers of weighted semigroups and associated Beurling Banach algebras

Given a weighted discrete abelian semigroup (S, ω), the semigroup M _ω (S) of ω-bounded multipliers as well as the Rees quotient M _ω (S)/S together with their respective weights [(w)\tilde]\tilde{\omega} and [(w)\tilde]_q\tilde{\omega}_q induced by ω are studied; for a large class of weights ω, the quotient l¹(M_w(S),[(w)\tilde])/l¹(S,w)\ell^1(M_{\omega}(S),\tilde{\omega})/\ell^1(S,{\omega}) is realized as a Beurling algebra on the quotient semigroup M _ω (S)/S; the Gel’fand spaces of these algebras are determined; and Banach algebra properties like semisimplicity, uniqueness of uniform norm and regularity of associated Beurling algebras on these semigroups are investigated. The involutive analogues of these are also considered. The results are exhibited in the context of several examples.     

Differentiable and analytic families of continuous martingales in manifolds with connection

Summary.  We prove that the derivative of a differentiable family X _t (a) of continuous martingales in a manifold M is a martingale in the tangent space for the complete lift of the connection in M, provided that the derivative is bicontinuous in t and a. We consider a filtered probability space (Ω,(ℱ _t )_{0≤ t ≤1}, ℙ) such that all the real martingales have a continuous version, and a manifold M endowed with an analytic connection and such that the complexification of M has strong convex geometry. We prove that, given an analytic family a↦L(a) of random variable with values in M and such that L(0)≡x ₀∈M, there exists an analytic family a↦X(a) of continuous martingales such that X ₁(a)=L(a). For this, we investigate the convexity of the tangent spaces T ^{( n )} M, and we prove that any continuous martingale in any manifold can be uniformly approximated by a discrete martingale up to a stopping time T such that ℙ(T<1) is arbitrarily small. We use this construction of families of martingales in complex analytic manifolds to prove that every ℱ₁-measurable random variable with values in a compact convex set V with convex geometry in a manifold with a C ¹ connection is reachable by a V-valued martingale. Received: 14 March 1996/In revised form: 12 November 1996     

Approximate Identities in Spaces of all Absolutely Continuous Measures on Locally Compact Semigroups

Let G be a locally compact group. Then M_a (G), the space of all absolutely continuous measures on G, has a bounded approximate identity. Baker and Baker proved that (S) (the space of all measures M(S) so that maps x _x *|| and x ||*_x are weak continuous from a locally compact semigroup S into M(S)) is closed under absolutely continuity and has an approximate identity. The main purpose of this paper is to show that similar results hold true for a locally compact semigroup S and M_a(S) the space of all absolutely continuous measures on S.     

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.     

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.     

Locally compact subgroups of the spectrum of the measure algebra II

The maximal ideal space Δ_G of the measure algebra M(G) of a locally compact abelian group G is a compact commutative semitopological semigroup. In this paper we show that cℓ Ĝ the closure of Ĝ, the dual of G, in Δ_G can contain maximal subgroups which are not locally compact. We have previously characterized the locally compact maximal subgroups of cℓ Ĝ as arising from locally compact topologies on G which are finer than the original topology. This research was supported in part by NSF contract number GP-19852.     

Limit theorems for convolutions of probabilities on non abelian semigroups

Let S be a locally compact semigroup. We study the sequence (λⁿ) of the convolution powers of a probability measure λ on S and their shifts by a probability measure η on S. We shall give sufficient conditions for lim ‖λⁿ−η*λⁿ‖ = 0 (where ‖.‖ denotes the norm). In particular we consider the case the η is a point measure and we study the subsemigroup L_O(λ) = {x ∈ S : lim ‖λⁿ−δ_X*λⁿ‖ = 0}. We shall give necessary and sufficient conditions for L_o(λ)=S. In this case we want to treat the problem of the convergence of the sequence (λⁿ).     

On some classes of regular semigroups

An idempotent e of a semigroup S is called right [left] principal (B.R. Srinivasan, [2]) if fef=fe [fef=ef] for every idempotent f of S. Say that S has property (LR) [(LR₁)] if every ℒ-class of S contains atleast [exactly] one right principal idempotent. There and six further properties obtained by replacing, ‘ℒ-class’ by ‘ℛ-class’ and/or ‘right principal’ by ‘left principal’ are examined. If S has (LR₁), the set of right principal elementsa of S (aa′ is right principal for some inversea′ ofa) is an inverse subsemigroup of S, generalizing a theorem of Srinivasan [2] for weakly inverse semigroups. It is shown that the direct sum of all dual Schützenberger representations of an (LR) semigroup is faithful (cf[1], Theorem 3.21, p. 119). Finally, necessary and sufficient conditions are given on a regular subsemigroup S of the full transformation semigroup on a set in order that S has each of the properties (LR), (LR₁), etc.