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Index computation for amalgamated products of product systems. (English) Zbl 1223.46061

In [B. V. R. Bhat and M. Mukherjee, “Inclusion systems and amalgamated products of product systems.” Infin. Dimens. Anal. Quantum Probab. Relat. Top., 13, No. 1, 1–26 (2010; Zbl 1198.46050)], Bhat and the author introduced a way to amalgamate two Arveson systems (that is, product systems of Hilbert spaces) over a contraction morphism between the two Arveson systems. If the contraction morphism consists of rank-one operators of norm 1 (so that the Arveson systems are necessarily spatial), then the amalgamated system coincides with the product of spatial product systems introduced by the reviewer in [M. Skeide, “The index of (white) noises and their product systems.” Infin. Dimens. Anal. Quantum Probab. Relat. Top., 9, No. 4, 617–655 (2006; Zbl 1119.46051)]. In that case, the index of the product is the sum of the indices of the factors. Already Bhat and Mukherjee [loc. cit.] showed that the index increases by 1 as soon as the rank-one morphism is no longer normalized.

In the present paper, the author calculates the index of an arbitrary amalgamated product. As the index of an Arveson system is the dimension of the multiplicity space of its maximal type I subsystem (consisting of Fock spaces), this problem amounts to computing the index of all amalgamated products of type I systems. Using the explicit parametrization of all contraction morphisms from [B. V. R. Bhat, Cocycles of CCR-flows. Mem. Am. Math. Soc. 709 (2001; Zbl 0976.46050)], the author completely solves that problem. He also computes a number of distinguished classes of examples.

46L55Noncommutative dynamical systems
46L53Noncommutative probability and statistics
60J25Continuous-time Markov processes on general state spaces