27 research outputs found
Approximation properties for dynamical W*-correspondences
Let be a locally compact quantum group, and von Neumann
algebras with an action by . We refer to these as
-dynamical W-algebras. We make a study of
-equivariant --correspondences, that is, Hilbert spaces
with an --bimodule structure by -preserving normal maps,
and equipped with a unitary representation of which is equivariant
with respect to the above bimodule structure. Such structures are a Hilbert
space version of the theory of -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
Weak Amenability of Locally Compact Quantum Groups and Approximation Properties of Extended Quantum SU(1, 1)
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