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

On trace zero matrices

In this note, we shall try to present an elementary proof of a couple of closely related results which have both proved quite useful, and also indicate possible generalisations