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137 (1996), 466–543. [ES] M. -M. Schwartz, Kac Algebras and Duality of Locally Compact Groups, Springer-Verlag, Berlin 1992. [EV] M. -M. Vallin, Inclusions of von Neumann algebras and quantum groupoids, J. Funct. Anal. 172 (2000), 249–300. [GHJ] F. M. Goodman, P. de la Harpe and V. F. R. Jones, Coxeter Graphs and Towers of Algebras, Publ. Res. Inst. Math. Sci. 14, Springer-Verlag, New York 1989. [KV] J. Kustermans and S. Vaes, Locally compact quantum groups, Ann. Sci. École Norm. Sup. 33 (2000), 837–934.

Moreover, we get that ςN (M1 β ∗γ M2 ) = M2 γ ∗β M1 . In particular, we have No N (M1 ∩ β(N ) ) β ⊗γ (M2 ∩ γ (N) ) ⊂ M1 β ∗γ M2 N N and M1 β ∗γ γ (N ) = (M1 ∩ β(N) ) β ⊗γ 1. N N 26 Michel Enock More generally, if β is a non-degenerate normal involutive anti-homomorphism from N into a von Neumann algebra M1 , and γ a non-degenerate normal involutive homomorphism from N into a von Neumann algebra M2 , it is possible to define, without any reference to a specific Hilbert space, a von Neumann algebra M1 β ∗γ M2 .

Q ; q)n (1) a a ; q, z = (b; q)∞ 1 ϕ1 ; q, z . See [4] for an b b extensive treatment on q-hypergeometric functions. The analysis of the dual of quantum SU(1, 1) depends heavily on Al-Salam & Chihara polynomials and little q-Jacobi functions.

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