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Hilbert von Neumann modules | | Panchugopal Bikram
; Kunal Mukherjee
; R. Srinivasan
; V.S. Sunder
; | | Date: |
23 Feb 2011 | | Abstract: | We introduce a way of regarding Hilbert von Neumann modules as spaces of
operators between Hilbert space, not unlike [Skei], but in an apparently much
simpler manner and involving far less machinery. We verify that our definition
is equivalent to that of [Skei], by verifying the ’Riesz lemma’ or what is
called ’self-duality’ in [Skei]. An advantage with our approach is that we can
totally side-step the need to go through $C^*$-modules and avoid the two stages
of completion - first in norm, then in the strong operator topology - involved
in the former approach.
We establish the analogue of the Stinespring dilation theorem for Hilbert von
Neumann bimodules, and we develop our version of ’internal tensor products’
which we refer to as Connes fusion for obvious reasons.
In our discussion of examples, we examine the bimodules arising from
automorphisms of von Neumann algebras, verify that fusion of bimodules
corresponds to composition of automorphisms in this case, and that the
isomorphism class of such a bimodule depends only on the inner conjugacy class
of the automorphism. We also relate Jones’ basic construction to the
Stinespring dilation associated to the conditional expectation onto a
finite-index inclusion (by invoking the uniqueness assertion regarding the
latter). | | Source: | arXiv, 1102.4663 | | Services: | Forum | Review | PDF | Favorites | |
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