THE KÜNNETH ISOMORPHISM IN BOUNDED COHOMOLOGY PRESERVING THE NORMS

Abstract

In this paper, for discrete groups G and K, we show that the cohomology of the complex of projective tensor product B∗(G)⊗̂B∗(K) is isomorphic to the bounded cohomology Ĥ∗(G×K) of G×K, which is the cohomology of B∗(G×K) as topological vector spaces, where B∗(G) is a complex of bounded cochains of G with real coefficients R. In fact, we construct an isomorphism between these two cohomology groups that carries the canonical seminorm in Ĥ∗(G × K) to the seminorm in the cohomology of B∗(G)⊗̂B∗(K).