Approximation properties for dynamical W*-correspondences

Let $\mathbb{G}$ be a locally compact quantum group, and $A,B$ von Neumann algebras with an action by $\mathbb{G}$. We refer to these as $\mathbb{G}$-dynamical W$^*$-algebras. We make a study of $\mathbb{G}$-equivariant $A$-$B$-correspondences, that is, Hilbert spaces $\mathcal{H}$ with an $A$-$B$-bimodule structure by $*$-preserving normal maps, and equipped with a unitary representation of $\mathbb{G}$ which is equivariant with respect to the above bimodule structure. Such structures are a Hilbert space version of the theory of $\mathbb{G}$-equivariant Hilbert C$^*$-bimodules. We show that there is a well-defined Fell topology on equivariant correspondences, and use this to formulate approximation properties for them. Within this formalism, we then characterize amenability of the action of a locally compact group on a von Neumann algebra, using recent results due to Bearden and Crann. We further consider natural operations on equivariant correspondences such as taking opposites, composites and crossed products, and examine the continuity of these operations with respect to the Fell topology.Comment: 44 pages. Comments are welcome! Apart from small stylistic changes, we have added some results on non-strong equivariant amenability properties in Section 6, and have reorganized some of the material in Section 5.5. We have added a table at the end of Section 5 to summarize some of the result

The Poisson Lie algebra, Rumin's complex and base change

Results from the forthcoming papers [BDK4, D3] are announced. We introduce a singular current construction, or base change, for pseudoalgebras which may be used to obtain a primitive Lie pseudoalgebra of type H from a suitable one of type K. When applied to representations, it derives the pseudo de Rham complex of type H from that of type K — which is related to Rumin’s construction from [Ru] — both with standard coefficients and with nontrivial Galois coefficients. In the latter case, the construction yields exact complexes of modules for the Poisson linearly compact Lie algebra P_2N exhibiting a nontrivial central action