Tensor Products and Banach Algebras

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.

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

The purpose of this series of papers is to present a general theory of Banach modules and to give some applications of it. The applications of the theory arise from observations that certain important notions of functional analysis are very closely related to certain Banach algebras and thereby to a module structure. This holds in particular for Banach lattices (which are C(X)-modules), interpolation spaces, tensor products and operator ideals.

In order to summarize the contents of this paper it is necessary to use some of the definitions in [5]. These definitions are repeated below at the beginning of Section 1. Although this paper adds to the material in [5] we have attempted to reduce the number of references to [o] to a minimum. An approximate diagonal in a Banach algebra 31 is a bounded net {ma} in 31 a for all a in 31 in the norm topologies where 31? 31 is the usual projective tensor product [2] and -k is the continuous linear map 31?31->31 defined by ir(a?b) =ab. The net {m 31) ** with aM = Ma and ?r** (M) a = a for all a ? 31. In the algebraic cohomology of algebras the existence of an element m of A A with ma = am and ir(m) the identity of A is equivalent to the statement that H1(A, X) =0 for all 31 modules X almost never holds?in fact, in the commutative case for all A bimodules X. Among Banach algebras the statement jEP^S^X) =0 (see [3; Theorem 1] and [5; Proposition 8.1]) we can always find a Banach 31 module ? with S^1(3I,36)=t^O unless 31 zz?n. The existence of an approxi? mate diagonal is not unusual and is equivalent to the statement !M* (31,36*) = 0 for all Banach 31 modules X; thatis 31 is an emenable Banach algebra. The element M plays the same role in the theory of amenable tlgebras that the invariant mean does for amenable groups. The definitions used from [5] and the theorem connecting amenability with approximate diagonals are given in ? 1. In ? 2 we show that if 31 has a virtual diagonal with Ma? 31? St for all a in 3T then 913 (31, 3E) ? .0 for all Banach 31 modules 3?. Algebras satisfying this condition are grdup algebras of compact topological groups and the algebra c0 of convergent sequences of complex numbers. Although we do not prove it the result also applies to any annihilator B* algebra with only finite dimensional minimal ideals. In ? 3 we show that if 31 is a semi simple commutative amenable Banach algebra then there is a Banach 31 module X with 3/2(3I,3E) ^0, unless 31 is finite

Weak spectral synthesis in commutative Banach algebras. III

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

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 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

Around 1978 Max Karoubi made a striking conjecture about the K-theory of Banach algebras. His conjecture predicted that the topological K-groups K * top(B) of a unital Banach algebra B were isomorphic to the algebraic K-groups K *(B⊗r)C) of the Grothendieck’s projective tensor product of B and the ideal K of compact operators on the standard Hilbert space H = ℓ2 (ℕ). Notice that B⊗K, is a norm completion of the matrix algebra M ∞(B) = B ⊗ M ∞(ℂ) and K *(M ∞(B)) = K *(B).KeywordsSpectral SequenceCompact OperatorBanach AlgebraPseudodifferential OperatorCyclic HomologyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The paper starts a series of publications devoted to the theory of topological radicals (TRs) of normed algebras. It contains results on general properties of TRs and their domains, considers TRs close to the Jacobson radical, presents a theory of tensor TRs (they are connected with the problem of calculating the radical of a projective tensor product), introduces and studies TRs related to the notion of joint qusinilpotence.

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.

We study when certain properties of Banach algebras are stable under ultrapower constructions. In particular, we consider when every ultrapower of $\mathcal{A}$ is Arens regular, and give some evidence that this is so if and only if $\mathcal{A}$ is isomorphic to a closed subalgebra of operators on a super-reflexive Banach space. We show that such ideas are closely related to whether one can sensibly define an ultrapower of a dual Banach algebraffi We study how tensor products of ultrapowers behave, and apply this to study the question of when every ultrapower of $\mathcal{A}$ is amenable. We provide an abstract characterization in terms of something like an approximate diagonal, and consider when every ultrapower of a C*-algebra, or a group L1-convolution algebra, is amenable.

On representations of tensor products of involutive Banach algebras