By Ralph Decker Bennett

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171 (2000), 278–307. [PP] M. Pimsner and S. Popa, Iterating the basic construction, Trans. Amer. Math. Soc. 310 (1988), 127–133. [R1] J. Renault, A Groupoid Approach to C∗ -Algebras, Lecture Notes in Math. 793, Springer-Verlag, Berlin–Heidelberg–New York 1980. [R2] J. Renault, The Fourier algebra of a measured groupoid and its multipliers, J. Funct. 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.

A unitary W from H βˆ ⊗α H onto µ µ µo H α ⊗β H will be called a pseudo-multiplicative unitary over the base N, with respect µo ˆ if to the representation α and the anti-representations β and β, (i) W intertwines α, β, βˆ in the following way: W (α(X) βˆ ⊗α 1) = (1 α ⊗β α(X))W ; No N ˆ ˆ W (1 βˆ ⊗α β(X)) = (1 α ⊗β β(X))W ; No N W (β(X) βˆ ⊗α 1) = (β(X) α ⊗β 1)W ; No N ˆ W (1 βˆ ⊗α β(X)) = (β(X) α ⊗β 1)W. No N (ii) The operator W satisfies: (1H α ⊗β W )(W βˆ ⊗α 1H ) No N = (W α ⊗β 1H )(σµo α ⊗β 1H )(1H α ⊗β W )σ2µ (1H βˆ ⊗α σµo )(1H βˆ ⊗α W ).

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