The K"unneth formula for nuclear $DF$-spaces and Hochschild cohomology

We consider complexes $(\X, d)$ of nuclear Fr'echet spaces and continuous boundary maps $d_n$ with closed ranges and prove that, up to topological isomorphism, $ (H_{n}(\X, d))^$ $\iso$ $H^{n}(\X^,d^),$ where $(H_{n}(\X,d))^$ is the strong dual space of the homology group of $(\X,d)$ and $ H^{n}(\X^,d^)$ is the cohomology group of the strong dual complex $(\X^,d^)$. We use this result to establish the existence of topological isomorphisms in the K"{u}nneth formula for the cohomology of complete nuclear $DF$-complexes and in the K"{u}nneth formula for continuous Hochschild cohomology of nuclear $\hat{\otimes}$-algebras which are Fr'echet spaces or $DF$-spaces for which all boundary maps of the standard homology complexes have closed ranges. We describe explicitly continuous Hochschild and cyclic cohomology groups of certain tensor products of $\hat{\otimes}$-algebras which are Fr'echet spaces or nuclear $DF$-spaces.

💡 Research Summary

The paper investigates the relationship between homology and cohomology for complexes built from nuclear Fréchet spaces and nuclear DF‑spaces, focusing on the case where all boundary operators have closed ranges. The authors first prove a duality theorem: for a complex ((\mathcal X,d)) whose terms are nuclear Fréchet spaces and whose differentials (d_n) are continuous with closed images, the strong dual of the homology group is topologically isomorphic to the cohomology of the strong dual complex, \