905 research outputs found
The planar algebra of a semisimple and cosemisimple Hopf algebra
To a semisimple and cosemisimple Hopf algebra over an algebraically closed
field, we associate a planar algebra defined by generators and relations and
show that it is a connected, irreducible, spherical, non-degenerate planar
algebra with non-zero modulus and of depth two. This association is shown to
yield a bijection between (the isomorphism classes, on both sides, of) such
objects.Comment: 16 pages, 20 figures; content adde
Guionnet-Jones-Shlyakhtenko subfactors associated to finite-dimensional Kac algebras
We analyse the Guionnet-Jones-Shlyakhtenko construction for the planar
algebra associated to a finite-dimensional Kac algebra and identify the factors
that arise as finite interpolated free group factors.Comment: 18 pages, 21 figures, corrected typo
Stochastic integration in Fock space
In this paper, using purely Hubert space-theoretic methods, an analogue of the Itό integral is constructed in the symmetric Fock space of a direct integral § of Hilbert spaces over the real line. The classical Itό integral is the special case when §=L2[0, ∞). An explicit formula is obtained for the projection onto the space of 'non-anticipating functionals', which is then used to prove that simple non-anticipating functionals are dense in the space of all non-anticipating functionate. After defining the analogue of the Itό integral, its isometric nature is established. Finally, the range of this 'integral' is identified; this last result is essentially the Kunita-Watanabe theorem on square-integrable martingales
Unitary equivalence to integral operators
A bounded operator A on L2(X) is called an integral operator if there exists a measurable function k on X x X such that, for each f) in L 2(X), ∫\k(x,y)ƒ(y)\d μ (y) < ∞ a.e. and Aƒ(x)= ∫ k(x,y)ƒ(y)d μ (y) a.e. (Throughout this paper, (X, μ ) will denote a separable, sigma -finite measure space which is not purely atomic.) An integral operator is called a Carleman operator if the inducing kernel k satisfies the stronger requirement: ∫\k(x,y)\ 2d μ (y) < ∞ for almost every x in X
Hilbert von Neumann modules
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 -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).Comment: 20 page
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