The theory of Banach coalgebras

Abstract

This paper is a thorough study of the general theory of Banach coalgebras. For any Banach spaces X and Y, let X [tensor product with circumflex] Y denote the completion of the algebraic tensor product X [tensor product] Y of X and Y with respect to the greatest cross-norm, and X [tensor product with two circumflexes] Y the completion of X [tensor product] Y with respect to the least cross-norm (whose associate is also a cross-norm). By the universal property of X [tensor product with circumflex] X, a Banach algebra can be defined as an associative continuous linear map X [tensor product with circumflex] X to X. A Banach coalgebra is defined to be a coassociative continuous linear map X to X [tensor product with two circumflexes] X. The most important example of a Banach coalgebra is the Banach space C(G) of all continuous complex-valued functions on a compact group with supremum norm and comultiplication C(G) to C(G) [tensor product with two circumflexes] C(G) congruent to C(G x G) induced by the group multiplication map G x G to G. Chapter I is a summary of the relevant properties of tensor products of Banach spaces. In Chapter II we develop the basic theory of Banach coalgebras. We give several concrete examples and employ general construction methods to create new Banach coalgebras from old ones. We conclude this chapter with an investigation of approximate coidentities, a concept corresponding to the notion of an approximate identity for Banach algebras. In Chapter III we establish the adjointness between the category of Banach coalgebras and the category of Banach algebras. A continuous linear map p: X [tensor product with circumflex] X to X induces a dual map p*: X* to (X [tensor product with two circumflexes] X)* (superset or equal to X* [tensor product with two circumflexes] X*) on the dual space. Defining X^0 to be the largest subspace of X* such that p*(X^0) subset or equal to X^0 [tensor product with two circumflexes] X^0 (which exists by Zorn's lemma), we obtain a contravariant functor (.)^0 from the category of Banach algebras to the category of Banach coalgebras which is adjoint to the dualization functor (.)*. As a corollary we obtain a bijection of the continuous algebra endomorphisms of the measure algebra M(G) of a compact group G with the continuous coalgebra maps of C(G) into M(G)^0. We conclude our work with an investigation of the relationship between the cofinite ideals in a Banach algebra X and the finite subcoalgebras of X^0.