THE KÜNNETH ISOMORPHISM IN BOUNDED COHOMOLOGY PRESERVING THE NORMS

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).