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Kernel representations of *-semigroups associated with infinitely divisible states. (English) Zbl 0787.46050

Accardi, L. (ed.), Quantum probability and related topics. Volume VII. Singapore: World Scientific,. QP-PQ. 31-50 (1992).
Summary: An indefinite space representation \(\lambda(b)= (e,\pi(b)e)\) of a conditionally positive function \(\lambda\) on a unital \(*\)-semigroup \({\mathcal B}\ni b\) is defined. An exponential \(*\)-representation \(\pi^ \otimes\) of \(\mathcal B\), associated with an infinitely divisible state \(\rho= e^ \lambda\) is reconstructed in a pseudo-Fock space \({\mathfrak F}\). The correspondent Araki-Woods construction is given by a \(*\)-projection on a pre-Hilbert subspace \({\mathcal F}\subseteq{\mathfrak F}\). It is proved that the projection of the kernel algebra \(C \pi^ \otimes({\mathcal B})\) defines the correspondence of the kernel calculus on \({\mathfrak F}\) with a quantum calculus on Fock space \({\mathcal F}\), associated with \(\rho\).
For the entire collection see [Zbl 0779.00006].

MSC:

46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras
47D07 Markov semigroups and applications to diffusion processes
81S25 Quantum stochastic calculus
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