Some classes of multilinear operators on C(K) spaces

In this paper we study n-inverse pairs of operators on the tensor product of Banach spaces. In particular we show that an n-inverse pair of elementary tensors of operators on the tensor product of two Banach spaces can arise only from l- and m-inverse pairs of operators on the individual spaces. This gives a converse to a result of Duggal and Muller, and proves a conjecture of the second named author. Our proof uses techniques from algebraic geometry, which generalize to other relations among operators in a tensor product. We apply this theory to obtain results for n-symmetries in a tensor product as well.

This volume provides a self-contained introduction to the theory of tensor products of Banach spaces. It is written for graduate students in analysis or for researchers in other fields who wish to become acquainted with this area. The only prerequisites are a basic knowledge of functional analysis and measure theory. Features of particular interest include: - A full treatment of the Grothendieck theory of tensor norms; - Coverage of the Chevet-Saphar norms and their duals, along with the associated classes of nuclear, integral and summing operators; - Chapters on the approximation property and the Radon-Nikodym property; - Topics such as the Bochner and Pettis integrals, the principle of local reflexivity and the Grothendieck inequality placed in a natural setting; - The classes of operators generated by a tensor norm and connections with the theory of operator ideals. Each chapter is accompanied by worked examples and a set of exercises, and two appendices provide essential material on summability in Banach spaces and properties of spaces of measures that may be new to the beginner.

It is shown that the above sequences always determine linear transformations and if the sequences are bounded under the least cross norm, that the transformations are continuous. Such operators are characterized to within algebraic isomorphism with the weak‐star sequential closure of the tensor product space in its second dual, and consequently certain classes of weakly sequentially complete projective tensor products are exhibited.

Two classes of random linear operators, w- and s-operators, are defined, using cylindrical measures and F-cylindrical probabilities on tensor products of Banach spaces. Various Radonifying theorems for such measures are used in order that the composition vAw of a w- or s-operator A and deterministic operators w, v be a bounded one.

Infinite tensor products of Banach spaces I

We define the basic concepts of the theory of M-structure and investigate the M-structure properties of the ∈-tensor product. Our main result generalizes a theorem due to author. It describes how the centralizer of the tensor product can be constructed from the centralizers of the factors. In the last sections we investigate some applications and indicate some open problems.

In this paper, we discuss the atomic solutions of the abstract Cauchy problem in two variables. The proposed abstract Cauchy problem is assumed to be non-homogeneous and in tensor product form. We utilize some results from the theory of tensor product of Banach spaces to obtain solutions in the atom operator form. The approach is applied for both ordinary and α-conformable fractional derivatives.

It was shown by M.I. Zelikin (2007) that the spectrum of a nuclear operator in a Hilbert space is central-symmetric iff the traces of all odd powers of the operator equal zero. B. Mityagin (2016) generalized Zelikin’s criterium to the case of compact operators (in Banach spaces) some of which powers are nuclear, considering even a notion of so-called \( \mathbb Z_d \)-symmetry of spectra introduced by him. We study α-nuclear operators generated by the tensor elements of so-called α-projective tensor products of Banach spaces, introduced in the paper (α is a quasi-norm). We give exact generalizations of Zelikin’s theorem to the cases of \( \mathbb Z_d \)-symmetry of spectra of α-nuclear operators (in particular, for s-nuclear and for (r, p)-nuclear operators). We show that the results are optimal.

Let dent A A denote the set of denting points of a subset A A of some Banach space. We prove \[ dent cl co( K ⊗ L ) = dent K ⊗ dent L {\text {dent cl co(}}K \otimes L) = {\text {dent }}K \otimes {\text {dent }}L \] for closed, bounded, absolutely convex subsets K K and L L of Banach spaces X X and Y Y . Here the closure refers to the completion of X ⊗ Y X \otimes Y w.r.t. some reasonable crossnorm.

In this paper the projective tensor products of approximation spaces associated with positive operators in Banach spaces are characterized. We show that the tensor products of approximation spaces can be considered as the interpolation spaces generated by $K$-method of real interpolation. The inequalities that provide a sharp estimates of best approximations by analytic vectors of positive operators on projective tensor products are established. Application to spectral approximations of the regular elliptic operators on projective tensor products of Lebesgue spaces is shown.

In this work, we introduce the notion of tensor product of spaces with a generalized 2-inner product (see Definition 5), and we establish several interesting properties (see Proposition5), thereby generalizing the classical properties of the tensor product of inner product spaces. Moreover, we equip this tensor product with a mapping that defines a generalized 2-inner product(see Theorem 3) and, consequently, endow it with a generalized 2-norm (see Theorem 1). In this context, we also define the tensor product of linear operators (see Definition 9) and prove a series of results for example, that the tensor product of two 2-bounded linear operators is again 2-bounded under the tensor product (see Proposition 10).

A theorem of Eilenberg-Watts type for tensor products of Banach spaces

That the classical interpolation theorems of Riesz-Thorin and Marcinkiewicz are of great importance in harmonic analysis has long since been realized. The functional analytic generalization of these interpolation theorems — abstract interpolation space theory — also is now proving to be of importance in harmonic analysis (cf., for instance, [1]). In this paper we reverse the flow in the sense that a circle of ideas from Lp-multiplier theory and tensor products of Banach spaces is used to provide answers to a number of questions that arise naturally in the general theory of abstract interpolation spaces. While some of these questions have been answered before (though not always published), the unified approach used in this paper introduces techniques which may well be of wider utility in interpolation space theory. We shall be interested in both the real interpolation spaces (X o, X 1)θq;J, (X o, X 1)θ, q;k of Peetre and the complex interpolation spaces (X o,X 1)θ of Lions and Calderon. For all unexplained notation and terminology see [2], [5], [6] or [11].

An application of the DR-duality theory for compact groups to endomorphism categories of C*-algebras with nontrivial center by H. Baumgartel and F. Lledo Modular invariants and subfactors by J. Bockenhauer and D. E. Evans On the PCT-theorem in the theory of local observables by H. J. Borchers and J. Yngvason Transplantation of local nets and geometric modular action on Robertson-Walker space-times by D. Buchholz, J. Mund, and S. J. Summers Classification of subsystems, local symmetry generators and intrinsic definition of local observables by S. Carpi and R. Conti From local perturbation theory to Hopf- and Lie-algebras of Feynman graphs by A. Connes and D. Kreimer The flat tube theorem for vector valued functions by C. D'Antoni and L. Zsido Point interactions by G. Dell'Antonio Perturbative algebraic field theory, and deformation quantization by M. Dutsch and K. Fredenhagen Sum rules for the free energy in the mean field spin glass model by F. Guerra Fractals in noncommutative geometry by D. Guido and T. Isola What I woud like to understand by R. Haag The Rohlin property for automorphisms of $C^*$-algebras by M. Izumi Environment induced localization and superselection rules in a gas of pyramidal molecules by G. Jana-Lasinio, C. Presilla, and C. Toninelli Connes-Moscovici-Kreimer Hopf algebras by D. Kastler The structure of the automorhpism group of an approximately finite dimensional factor by Y. Katayama and M. Takesaki Braiding and extensions of endomrophisms of subfactors by Y. Kawahigashi Bicategories of operator algebras and Poisson manifolds by N. P. Landsman Notes for a quantum index theorem introduction by R. Longo Conformal field theory and Doplicher-Roberts reconstruction by M. Muger On the distance between MASA's in type II$_1$ factors by S. Popa Recent results concerning E$_o$-semigroups of $\mathfrak{B}(\mathfrak{H})$ by R. T. Powers Locality and modular invariance in 2D conformal QFT by K.-H. Rehren Tensor products of Banach spaces and the Stone-Weierstrass problem of $C^*$-algebras by S. Sakai Perron-Frobenius theory for positive maps on trace ideals by R. Schrader Space- and time-like superselection rules in conformal quantum field theory by B. Schroer Finite quantum groupoids and inclusions of finite type by K. Szlachanyi On generalizations of the spectrum condition by R. Verch Algebraic orbifold conformal field theories by F. Xu.

A Banach space X is prime if every infinite-dimensional complemented subspace contains a further subspace which is isomorphic to X. A Banach space X is said to be primary if whenever X — Y © Z, X is isomorphic to either Y or Z. The classical examples of prime spaces are the spaces ipj 1 < p < oc. Many spaces derived from the £p-spaces in various ways are primary (see for example [AEO] and [CL]). The primarity of B(H) was shown by Blower [B] in 1990, and Arias [A] has recently developed further techniques which are used to prove the primarity of Cι, the space of trace class operators (this was first shown by Arazy [Arl, Ar2]). It has become clear that these techniques are not naturally confined to a Hubert space context; in the present paper we wish to extend the results to a variety of tensor products and operator spaces of ^-spaces (and in some cases £p-spaces). We also include some related results. Some of the intermediate propositions (on factoring operators through the identity) may actually be true for a wider class of Banach spaces (those with unconditional bases which have nontrivial lower and upper estimates). In fact, the combinatorial aspects of the factorization can be applied quite generally, and may have other applications. The proofs of primarity, however, rely on Pelczyήski's decomposition method which is not so readily extended. We have thus kept mainly to the case of injective and projective tensor products of tv spaces throughout. The results we obtain apply to the growing study of polynomials on Banach spaces since polynomials may be considered as symmetric multilinear operators with an equivalent norm (see [FJ], [M],