By Leonid Vainerman
Vainerman (mathematics and mechanics, U. de Caen, France) offers seven papers from the February 2002 assembly which introduced jointly physicists and mathematicians to debate such issues of quantum teams as multiplicative partial isometries and finite quantum groupoids; multiplier Hopf*-algebras with optimistic integrals; Galois activities via finite quantum groupoids, and quantum groupoids and pseudo-multiplicative unitaries"
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Additional info for Locally compact quantum groups and groupoids: proceedings of the meeting of theoretical physicists and mathematicians, Strasbourg, February 21-23, 2002
2 for the definitions). Let us define now the following representations of A on the Hilbert space H = L2 (A) ⊗ L2 (M): β(x) = x ⊗ 1, ˆ β(x) = (JA ⊗ JM )α(x ∗ )(JA ⊗ JM ). 1) to L2 (α(A) ). 7), we get that α(A) is isomorphic to L(L2 (M)) ⊗ A . Therefore, the relative tensor product H βˆ ⊗α H is isomorphic to L2 (A) ⊗ L2 (M) ⊗ L2 (M). 5) that the relative tensor product H α ⊗β H µ is also isomorphic to L2 (A) ⊗ L2 (M) ⊗ L2 (M). It is then possible to verify that 1L2 (A) ⊗ W is a pseudo-multiplicative unitary over the base A, with respect to α, β, ˆ and β.
With the above mentioned hypothesis, let us suppose that the base N is a sum of type I factors. , Cw (W ) = α(N ) . (ii) Kα,µ ⊂ Cn (W ). , Cn (W ) = Kα,µ . Theorem. With the above mentioned hypothesis, let us suppose that the base N is abelian. , Cw (W ) = α(N ) . (ii) Kα,µ ⊂ Cn (W ). , Cn (W ) = Kα,µ . 42 Michel Enock 5 Examples In this section, we give examples of pseudo-multiplicative unitaries of compact type. , proper) groupoids. 2), we recover the inclusions which are equipped, on some level of the Jones’ tower, with a conditional expectation.
Therefore, ωξ β is faithful, and does not depend on the choice of the fixed binormalized vector ξ . Let W be a pseudo-multiplicative unitary over the base N, with respect to the ˆ we shall say that W is of representation α and the anti-representations β and β; “compact type” if there exists a vector ξ fixed and binormalized by W , with respect to the normal faithful positive form µ = ωξ α = ωξ βˆ = ωξ β on N. We shall call µ the canonical normal positive form on N. We shall say that W is of “discrete type” if W = σµo W ∗ σµ is of compact type.