By Leonid Vainerman
Vainerman (mathematics and mechanics, U. de Caen, France) provides 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 confident integrals; Galois activities by means of finite quantum groupoids, and quantum groupoids and pseudo-multiplicative unitaries"
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Extra resources for Locally compact quantum groups and groupoids: proceedings of the meeting of theoretical physicists and mathematicians, Strasbourg, February 21-23, 2002
When no confusion is possible about the representation and the anti-representation, we shall write H ⊗ψ K instead of H β ⊗γ K, and ξ ⊗ψ η instead of ξ β ⊗γ η. ψ β,γ For any ξ in D(Hβ , ψ o ), we define the bounded linear application λξ β,γ ψ from K to H β ⊗γ K by, for all η in K, λξ (η) = ξ β ⊗γ η. We shall write λξ if no confusion ψ ψ is possible. 10): β,γ λξ o = R β,ψ (ξ ) ⊗ψ 1K , where we use the canonical identification (as left N-modules) of L2 (N ) ⊗ψ K with K. We have: β,γ β,γ (λξ )∗ λξ = γ ( ξ, ξ β,ψ o ).
Anal. 145 (1997), 455–490. -L. Sauvageot, Produit tensoriel de Z-modules et applications, in Operator Algebras and their Connections with Topology and Ergodic Theory (H. ), Lecture Notes in Math. 1132, Springer-Verlag, Berlin 1985, 468–485. -L. Sauvageot, Sur le produit tensoriel relatif d’espaces de Hilbert, J. Operator Theory 9 (1983), 237–352. [St] S. ¸ Str˘atil˘a, Modular theory in operator algebras, Editura Academiei/Abacus Press, Bucuresti/Tunbridge Wells, Kent, 1981. Quantum groupoids and pseudo-multiplicative unitaries 47 [Sz] K.
We have therefore, for all ξ1 and η2 in H, ((id ∗ωξ2 ,η1 )(σµo W )ξ1 |η2 ) = (W (ξ1 βˆ ⊗α ξ2 )|η1 α ⊗β η2 ). µ µo 36 Michel Enock We can verify easily that this operator can be written also as (ωξ2 ,η1 ∗ id)(W σµ ). Using the intertwining property of W with respect to α, we get that (id ∗ωξ2 ,η1 )(σµo W ) belongs to α(N) . If ξ belongs to D(α H, µ), we shall write (id ∗ωξ )(σµo W ) instead of (id ∗ωξ,ξ )(σµo W ). , weak) closure of the linear span of these operators; we have Cn (W ) ⊂ Cw (W ) ⊂ α(N ) .