N-Weak Module Amenability of Triangular Banach Algebras

In this paper we define the module topological center of the second dual \(\mathcal{A}^{**}\) of a Banach algebra \(\mathcal{A}\) which is a Banach \(\mathfrak{A}\)-module with compatible actions on another Banach algebra \(\mathfrak{A}\). We calculate the module topological center of ℓ 1(S)**, as an ℓ 1(E)-module, for an inverse semigroup S with an upward directed set of idempotents E. We also prove that ℓ 1(S)** is ℓ 1(E)-module amenable if and only if an appropriate group homomorphic image of S is finite.

Geometry of C⁎-algebras, and the bidual of their projective tensor product

Let [Formula: see text] and [Formula: see text] be Banach algebras such that [Formula: see text] is a Banach [Formula: see text]-bimodule with compatible actions. We define the product [Formula: see text], which is a strongly splitting Banach algebra extension of [Formula: see text] by [Formula: see text]. After characterization of the multiplier algebra, topological center, (maximal) ideals and spectrum of [Formula: see text], we restrict our investigation to the study of semisimplicity, regularity, Arens regularity of [Formula: see text] in relation to that of the algebras [Formula: see text], [Formula: see text] and the action of [Formula: see text] on [Formula: see text]. We also compute the first cohomology group [Formula: see text] for all [Formula: see text] as well as the first-order cyclic cohomology group [Formula: see text], where [Formula: see text] is the [Formula: see text]th dual space of [Formula: see text] when [Formula: see text] and [Formula: see text] itself when [Formula: see text]. These results are not only of interest in their own right, but also they pave the way for obtaining some new results for Lau products and module extensions of Banach algebras as well as triangular Banach algebra. Finally, special attention is devoted to the cyclic and [Formula: see text]-weak amenability of [Formula: see text]. In this context, several open questions arise.

In this paper we introduce two symmetric variants of amenability, symmetric module amenability and symmetric Connes amenability. We determine symmetric module amenability and symmetric Connes amenability of some concrete Banach algebras. Indeed, it is shown that $ell^1(S)$ is a symmetric $ell^1(E)$-module amenable if and only if $S$ is amenable, where $S$ is an inverse semigroup with subsemigroup $E(S)$ of idempotents. In symmetric connes amenability, we have proved that $M(G)$ is symmetric connes amenable if and only if $G$ is amenable.

In this paper we discuss the Arens regularity of projective tensor product of\nSchatten p-class operators. We use the biregularity condition given by \\"Ulger\nto prove that $S_p(\\mathcal H)\\otimes^\\gamma S_q(\\mathcal H)$ is not Arens\nregular. We further prove that $B(S_2(\\mathcal H))\\otimes^\\gamma S_2(\\mathcal\nH)$ is not Arens regular(with respect to usual multiplication) while it is\nregular with respect to Schur product. Thus we demonstrate the importance of\nbiregularity condition given in \\cite{Ulger} and the convenience of its use to\nprove Arens regularity or irregularity through some concrete examples.\n

Let A be a weakly sequentially complete Banach algebra containing a bounded approximate identity that is an ideal in its second dual A∗∗. We refer to such an algebra as a WeSeBai algebra. In the present paper, we examine the Arens regularity properties of closed ideals of algebras in this class and observe that, although WeSeBai algebras themselves are always strongly Arens irregular, a variety of Arens regularity properties can be found among their closed ideals. After characterizing Arens regular ideals and strongly Arens irregular ideals, we focus on the main examples of WeSeBai algebras: the convolution group algebras L1(G), G compact, and the Fourier algebras A(Γ), Γ discrete and amenable. We find examples of Arens regular ideals in L1(G) and A(Γ), reflexive and nonreflexive, and examples of strongly Arens irregular ideals that are not in the WeSeBai class. For this, we introduce a new class of Riesz sets that are not Λ(p), for any p>1, and show how to obtain them in a broad family of noncommutative groups. As a consequence of our approach, we prove that every infinite Abelian group contains a Rosenthal set that is not Λ(p), for any p>0.

In a recent paper [3] Dales and Pandey have shown that the class Sp of Segal algebras is weakly amenable. In this paper, for various classes of Segal algebras, we characterize derivations and multipliers from a Segal algebra into itself and into its dual module. In particular, we prove that every Segal algebra on a locally compact abelian group is weakly amenable and an abstract Segal subalgebra of a commutative weakly amenable Banach algebra is weakly amenable. We also introduce the Lebesgue–Fourier algebra of a locally compact group G and study its Arens regularity when G is discrete or compact.

SynopsisThe aim of this paper is to describe the free product of a pair G, H of groups in the category of inverse semigroups. Since any inverse semigroup generated by G and H is a homomorphic image of this semigroup, this paper can be regarded as asking how large a subcategory, of the category of inverse semigroups, is the category of groups? In this light, we show that every countable inverse semigroup is a homomorphic image of an inverse subsemigroup of the free product of two copies of the infinite cyclic group. A similar result can be obtained for arbitrary cardinalities. Hence, the category of inverse semigroups is generated, using algebraic constructions by the subcategory of groups.The main part of the paper is concerned with obtaining the structure of the free product G inv H, of two groups G, H in the category of inverse semigroups. It is shown in section 1 that G inv H is E-unitary; thus G inv H can be described in terms of its maximum group homomorphic image G gp H, the free product of G and H in the category of groups, and its semilattice of idempotents. The second section considers some properties of the semilattice of idempotents while the third applies these to obtain a representation of G inv H which is faithful except when one group is a non-trivial finite group and the other is trivial. This representation is used in section 4 to give a structure theorem for G inv H. In this section, too, the result described in the first paragraph is proved. The last section, section 5, consists of examples.

In this paper, we extend some propositions of Banachalgebras into module actions and establish the relationships betweentopological centers of module actions. We introduce some newconcepts as -property and -property for Banach modules and obtain some conclusions inthe topological center of module actions and Arens regularity of Banachalgebras.

Let [Formula: see text] and [Formula: see text] be two Banach algebras such that [Formula: see text] is a Banach [Formula: see text]-bimodule with the left and right compatible actions of [Formula: see text] on [Formula: see text]. Let [Formula: see text] be a strongly splitting Banach algebra extension of [Formula: see text] by [Formula: see text]. In this paper, we investigate some homological aspects such as injectivity, projectivity and flatness of [Formula: see text] and give some necessary and sufficient conditions for injectivity, projectivity and flatness of [Formula: see text].

We employ the fact that L1(G) is n-weakly amenable for each n ≥ 1 to show that for an inverse semigroup S with the set of idempotents E, ℓ1(S) is n- weakly module amenable as an ℓ1(E)-module with trivial left action. We study module amenability and weak module amenability of the module projective tensor products of Banach algebras.

Let 𝒜 be a Banach algebra and let J be a closed ideal of 𝒜 such that φ|J≠0 for some nonzero character φ on 𝒜. We obtain some relations between the existence of compact and weakly compact multipliers on J and on 𝒜 in some sense. Then we apply these results to hypergroup algebra L1(K) when K is a locally compact hypergroup. In particular, for a closed ideal J in L1(K) we prove that K is compact if and only if there is f∈J such that φ1(f)≠0 and the multiplication operator λf:g↦g∗f is weakly compact on J. Using this, we study Arens regularity of J whenever it has a bounded left approximate identity. Finally, we apply these results on some abstract Segal algebras with respect to the L1(K).

Let A and B be two Banach algebras. On the projective tensor product A s B of A and B there exists a natural algebra structure. In this note we study Arens regularity of the Banach algebra A s B.

In this paper, we define the notion of sigma-approximate module amenability of Banach algebras and give some properties about this notion. Also for Banach A-bimodule A, and J, the closed ideal of A generated by elements of form ( α · a ) b - a ( b · α ) , ( a , b ∈ A , α ∈ A ) , we considered some corollaries about σ ^ -approximate amenability of A J as a Banach A-bimodule, where σ ^ : A J → A J by σ ^ ( a + J ) = σ ( a ) + J has a dense range.

An inverse semigroup S is called E-unitary if the equations ea = e = e2 together imply a2 = a. In a previous paper the author showed that every inverse semigroup is an idempotent separating homomorphic image of an E-unitary inverse semigroup. The main question considered in this paper is the following. Given an inverse semigroup S give a method for constructing E-unitary inverse semigroups P and idempotent separating homomorphisms φ P → S in such a way that the structure of P as a P-semigroup is evident.