On Banach modules I

The classical approach to studying operator ideals using tensor norms mainly focuses on those tensor norms and operator ideals defined by means of ℓp spaces. In a previous paper, an interpolation space, defined via the real method and using ℓp spaces, was used to define a tensor norm, and the associated minimal operator ideals were characterized. In this paper, the next natural step is taken, that is, the corresponding maximal operator ideals are characterized. As an application, necessary and sufficient conditions for the coincidence of the maximal and minimal ideals are given. Finally, the previous results are used in order to find some new metric properties of the mentioned tensor norm.

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.

Tensor Norms and Operator Ideals

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

This paper deals with the method of hyperbolic summation of tensor product orthogonal spline functions onId. The spaces, defined in terms of the order of the best approximation by the elements of the space spanned by the tensor product functions with indices from a given hyperbolic set, are described both in terms of the coefficients in some basis and as interpolation spaces. Moreover, the hyperbolic modulus of smoothness is studied, and some relations between hyperbolic summation and hyperbolic modulus of smoothness are established.

C. Le Merdy in (Proc. Amer. Math. Soc. 126 (1998), 715- 719) proved the following complex interpolation formula for injective tensor products: ('2 ˜ '1,'2 ˜ '1) 1 = S4. We investigate whether related formulas hold when considering arbitrary 0 <

Let $E$ and $F$ be locally convex spaces and $G$ their completed $\varepsilon$-tensor product. It is shown that if $S$ and $T$ are weakly almost periodic equicontinuous semigroups of operators on $E$ and $F$ respectively, then, under mild restrictions on $E$ or $F, S \otimes T$ is a weakly almost periodic equicontinuous semigroup of operators on $G$, and the almost periodic and flight vector subspaces of $G$ are related in a natural way to the corresponding subspaces of $E$ and $F$ via the $\varepsilon$-tensor product. Furthermore, if $E$ and $F$ both decompose into a direct sum of these subspaces then so does $G$.

Tensor Products and Banach Algebras

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

Recent research on Dunford-Pettis operators associated with the group algebra L 1 (G) and the Fourier algebra A(G) of a locally compact group has motivated the study of the structure of the Dunford-Pettis space for a Banach module ϵ over a Banach algebra . In this paper, we study conditions under which (ϵ) coincides with the entire Banach right -module ϵ, and investigate its connections with some classical geometric properties. We analyze the structure of Dunford-Pettis elements for various classical Banach spaces and study their behavior under continuous homomorphisms and quotient maps. Moreover, we explore the geometric properties of certain ideals in commutative Banach algebras.

This is an introductory text on the theory of modules over rings or, equivalently, over algebras, addressed to beginning graduate students. It stresses the importance of the categorical perspective and the use of homological tools. Modules are studied both from a classical point of view and a categorical one. For this purpose, rudiments of category theory, homological algebra and representations of quivers are introduced and applied in the study of module categories. After introducing such fundamental tools as the Hom functor and the tensor product comes the study of projective, injective and flat modules. The classical results, such as the classification of modules over principal ideal domains, the Jordan-Hölder, Wedderburn-Artin, and Morita theorems, are proven, as well as the Eilenberg-Watts theorems. Radicals (of modules, algebras and categories) are thoroughly studied. Homological invariants, such as the extension and torsion modules or the homological dimensions are introduced using derived functors and are applied in module theory. The book stresses the construction and analysis of examples.

The main topics of the second chapter is an introduction to module theory, aiming at application to the structure theorems about finitely generated modules over Dedekind domain (and general results about reductions of endomorphisms of vector spaces as a byproduct). On the way, the main tools of module theory (including tensor products) are introduced, as well as the rudiments of the language of homological algebra (exact sequences, left exactness of Hom, right exactness of tensor product).A particular emphasis is given to finitely generated modules, with detailed applications to rings of integers, Noetherian and Jacobson rings. Localization and projectivity of modules are carefully treated in order to be applied to Dedekind domains.

Quantum Harmonic Analysis and Geometric Invariants

On the stability of the quadratic mapping in Banach modules

For a scattered, locally compact Hausdorff space , we prove that the essential norm on the Calkin algebra is a minimal algebra norm. The proof relies on establishing a quantitative factorization for the identity operator on through noncompact operators , where is any Banach space that does not contain a copy of or whose dual unit ball is sequentially compact. It follows that, for every ordinal , the algebras and have a unique algebra norm.