Tensor products of idempotent semimodules. An algebraic approach

In the framework of idempotent mathematics, analogs of the classical kernel theorems of L. Schwartz and A. Grothendieck are studied. Idempotent versions of nuclear spaces (in the sense of A. Grothendieck) are discussed. The so-called algebraic approach is used. This means that the basic concepts and results (including those of "topological" nature) are simulated in purely algebraic terms.

This is an expository paper on tensor products where the standard approaches for constructing concrete instances of algebraic tensor products of linear spaces, via quotient spaces or via linear maps of bilinear maps, are reviewed by reducing them to different but isomorphic interpretations of an abstract notion, viz., the universal property, which is based on a pair of axioms.

In this note we describe conditions under which, in idempotent functional analysis, linear operators have integral representations in terms of idempotent integral of V. P. Maslov. We define the notion of nuclear idempotent semimodule and describe idempotent analogs of the classical kernel theorems of L. Schwartz and A. Grothendieck. Our results provide a general description of a class of subsemimodules of the semimodule of all bounded functions with values in the Max-Plus alg...

This paper suggests an algebraic version of the theorem on the existence of eigenvectors for linear operators in abstract idempotent spaces. Earlier, the theorem on the existence of eigenvectors was only known for the cases of a free finite-dimensional semimodule and for compact operators in semimodules of real continuous functions.

The use of a tensor product perspective has enriched functional analysis and other important areas of mathematics and physics. The context of operator spaces is clearly no exception. The aim of this manuscript is to kick off the development of a systematic theory of tensor products and tensor norms for operator spaces and its interplay with their associated mapping ideals. Based on the theory of tensor products in Banach spaces, we provide the corresponding natural definition...

These are the lecture notes for a short course on tensor categories. The coverage in these notes is relatively non-technical, focussing on the essential ideas. They are meant to be accessible for beginners, but it is hoped that also some of the experts will find something interesting in them. Once the basic definitions are given, the focus is mainly on k-linear categories with finite dimensional hom-spaces. Connections with quantum groups and low dimensional topology are po...

In this paper we consider Idempotent Functional Analysis, an `abstract' version of Idempotent Analysis developed by V. P. Maslov and his collaborators. We give a review of the basic ideas of Idempotent Analysis. The correspondence between concepts and theorems of the traditional Functional Analysis and its idempotent version is discussed; this correspondence is similar to N. Bohr's correspondence principle in quantum theory. We present an algebraical approach to Idempotent ...

We present the basic concepts of tensor products of vector spaces, emphasizing linear algebraic and combinatorial techniques as needed for applied areas of research. The topics include (1) Introduction; (2) Basic multilinear algebra; (3) Tensor products of vector spaces; (4) Tensor products of matries; (5) Inner products on tensor spaces; (6) Direct sums and tensor products; (7) Background concepts and notation.

In this paper, we present an algebraic approach to idempotent functional analysis, which is an abstract version of idempotent analysis. The basic concepts and results are expressed in purely algebraic terms. We consider idempotent versions of certain basic results of linear functional analysis, including the theorem on the general form of a linear functional and the Hahn-Banach and Riesz-Fischer theorems.

We give a survey on classical and recent results on dual spaces of topological tensor products as well as some examples where these are used.