Permanent Weak Module Amenability of Semigroup Algebras

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

In [1 ] and [2] a construction of a bijection M3+->M1 X M2 is given, where MZi is the set of maximal modular (two-sided) ideals in the Banach algebraAi (i=1, 2, 3) and where A3=Al0,A2 is the greatest cross-norm tensor product of A1 and A2. In a recent correction [3] it is shown that there is indeed a closed 1-1 mapping MlXM2--+M3 when hull-kernel topologies are used. However, it is an open question when this mapping is surjective. In this note we show that the mapping is onto when one of the Banach algebras A1 and A2 is commutative. Also we give a correct proof of a theorem in [5], the original proof depended on [2]. The methods employed are adaptions of those in [6]. Suppose A1 and A2 are Banach algebras and suppose A1 is commutative. Let Sti be the set of maximal modular (two-sided) ideals. Each hEz-Tl is a continuous C-valued homomorphism and induces a homomorphismn

We apply the G-algebra theory to the tensor product of algebras. These considerations are applied to extend the results of Alghamdi and Khammash [1], Khammash [4] and Külshammer [5, Proposition 1.2] on the tensor product of group algebras and modules over an algebraically closed field to lattices over a complete discrete valuation ring. This places these results in the standard integral finite group modular representation theory of G-algebras as pioneered by Puig (cf. [8]). We also study some aspects of covering homomorphisms and the Green correspondence in this context (cf. [8, Sections 20 and 25]).

Polynomial identities for graded tensor products of algebras

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.

Let [Ufr ] be an associative Banach algebra. Given a set S, we write l∞(S, [Ufr ]) for the Banach algebra of all bounded functions f: S→[Ufr ] with the usual norm ∥f∥∞ = sups∈S∥f(s)∥[Ufr ] and pointwise multiplication. When S is countable, we simply write l∈([Ufr ]).In this short note, we exhibit examples of amenable (resp. weakly amenable) Banach algebras [Ufr ] for which l∈(S, [Ufr ]) fails to be amenable (resp. weakly amenable), thus solving a problem raised by Gourdeau in [7] and [8]. We refer the reader to [4, 9, 10] for background on amenability and weak amenability. For basic information about the Arens product in the second dual of a Banach algebra the reader can consult [5, 6].

For a Banach algebra ${\cal A}$, we introduce and investigate weak amenability of ${\cal A}$ with respect to a character. We give some necessary conditions for the weak amenability of $\cal A$ with respect to a character and describe a class of Banach algebras that are not weakly amenable with respect to characters. Finally, we give examples of Banach algebras which are weakly amenable with respect to characters but neither weakly amenable nor amenable with respect to characters.

Let X X be a Banach space and A \mathcal A be the Banach algebra B ( X ) \mathcal {B}(X) of bounded (i.e. continuous) linear transformations (to be called operators) on X X to itself. Let E \mathcal E be the set of idempotents in A \mathcal A and S \mathcal S be the semigroup generated by E \mathcal E under composition as multiplication. If T ∈ S T\in \mathcal S with 0 ≠ T ≠ I X 0\ne T\ne I_{X} then T T has a local block representation of the form ( T 1 a m p ; T 2 0 a m p ; 0 ) \begin {pmatrix} T_1 & T_2 \\ 0 & 0 \end {pmatrix} on X = Y ⊕ Z X=Y\oplus Z , a topological sum of non-zero closed subspaces Y Y and Z Z of X X , and any A ∈ A A\in \mathcal A has the form ( A 1 a m p ; A 2 A 3 a m p ; A 4 ) \begin {pmatrix} A_1 & A_2 \\ A_3 & A_4 \end {pmatrix} with T 1 , A 1 ∈ B ( Y ) T_1,A_1 \in \mathcal {B}(Y) , T 2 , A 2 ∈ B ( Z , Y ) , A 3 ∈ B ( Y , Z ) T_2,A_2\in \mathcal {B}(Z,Y), A_3\in \mathcal {B}(Y,Z) , and A 4 ∈ B ( Z ) A_4 \in \mathcal {B}(Z) . The purpose of this paper is to study conditions for A A to be in S \mathcal {S} .

Tensor products of quantum logics and effect algebras with some known problems are reviewed. It is noticed that although tensor products of effect algebras having at least one state exist, in the category of complex Hilbert space effect algebras similar problems as with tensor products of projection lattices occur. Nevertheless, if one of the two coupled physical systems is classical, tensor product exists and can be considered as a Boolean power. Some applications of tensor products (in the form of Boolean powers) to quantum measurements are reviewed.

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

A well-known result of Haagerup from 1983 states that every C*-algebra A is weakly amenable, that is, every (associative) derivation from A into its dual is inner. A Banach algebra B is said to be ternary weakly amenable if every continuous Jordan triple derivation from B into its dual is inner. We show that commutative C*-algebras are ternary weakly amenable, but that B(H) and K(H) are not, unless H is finite dimensional. More generally, we inaugurate the study of weak amenability for Jordan–Banach triples, focussing on commutative JB*-triples and some Cartan factors.

In this paper, we concentrate on the generalized Jones result in Kennedy and Jaish (2021) which says that [Formula: see text], the tensor product of m-partition algebras is a centralizer algebra of the action of the direct product of symmetric groups, [Formula: see text], on the k-folder tensor products [Formula: see text], where [Formula: see text]. In particular, we restrict the action of the direct products of symmetric groups, [Formula: see text], to the action of the direct product of alternating groups, [Formula: see text]. Herein, we determine the basis for the centralizer algebra and exhibit that when the centralizer is isomorphic to the tensor product of m-partition algebras, [Formula: see text].

0. Introduction. In [4], [5], [6] the structures A1 0 ) A2, where A1 and A2 are Banach algebras, are discussed. Actually, a proper parallel to the algebraic situation is: three commutative Banach algebras A, B, C, where A and B are C-bimodules (in the sense described below), and some Banach-algebraic version of A 03cB. In the first part of the following, we shall give a general discussion of A ?cB, a natural Banach-algebraic version of the (algebraic) tensor product of A and B over C. Thereafter, we shall discuss a special case in which A, B, C are group algebras of locally compact abelian groups. Finally we shall handle the even more special problem in which A and B are group algebras of locally compact abelian groups and C is a group algebra of a compact abelian group. In the last two parts, particularly the last, a connection will be established between the theory of tensor products and the theory of group extensions. In this connection, the author thanks L. Auslander for indicating the point in question. The author is also indebted to Dr. B. Natzitz for pointing out errors and the need for clarifying several items of an earlier draft of this paper.

This paper is concerned with two notions of cohomological triviality for Banach algebras, weak amenability and cyclic amenability. The first is defined within Hochschild cohomology and the latter within cyclic cohomology. Our main result is that where ℱ is a Banach algebraic free product of two Banach algebras and ℬ. It follows that cyclic amenability is preserved under the formation of free products.

The weakly multiplicative bilinear maps among R-algebras are introduced and the corresponding to them category of pointed algebras is defined. By the resolution of the respective universal problem, a new tensor product of two R-algebras is realized and its connection with the usual one is pointed out. Furthermore, after establishing the existence of an infinite (new) tensor product algebra, as a special colimit, it is shown that coproducts there always exist in the category Al of noncommutative (associative, unitary) R-algebras. Certain canonical decompositions of a given algebra in some particular cases are also investigated.