Some Remarks for Künneth Formula on Bounded Cohomology

Abstract

Abstract. Künneth formula is to compute (co)-homology of A⊗B for known (co)-homology of the complexes A and B. In the ordinary case, this is done by using elementary homological methods in an abelian category. However, when we consider the bounded cochain complex with values in R and its structure as a real Banach space, the techniques of homological algebra for constructing Künneth type formulas on it are not effective. The most notable facts are the image of a morphism of Banach spaces is not necessarily closed, and also the closed summand of a Banach space need not be a topological direct summand. The main goal of this paper is to construct the abstract theory of Künneth type formula on bounded cohomology with real coefficients in the suitable category of Banach spaces with some restricted conditions.