Continuous analogues of Fock space. II. The spectral \(C^*\)-algebra
This is the second part of a study of a semigroup \(\{\alpha_ t\); \(t\geq 0\}\) of normal *-endomorphisms of the algebra B(H) of all bounded operators on a Hilbert space H such that \(\alpha_ t(1)=1\) for all \(t\geq 0\). In the first part [Mem. Am. Math. Soc. 80, No.409 (1989; Zbl 0697.46035)], the problem of classifying such semigroups up to outer conjugacy was reduced to the problem of classifying certain structure associated with them, called product systems. In this second part, the spectral \(C^*\) algebra associated with a product structure E is introduced as its topological spectral invariant. The author first introduces (Definition 2.12) a separable \(C^*\)-algebra \(C^*(E)\) whose non-degenerate *-representations corresponds to semigroups whose associated product systems are isomorphics to E. (Theorem 2.16 and Corollary 2 to Theorem 3.4). This \(C^*\)-algebra is nuclear (Corollary to Theorem 4.1). It is a continuous time analogue of the Cuntz algebra \({\mathcal O}_{\infty}.\) The author then defines its regular representations and, as its image there, the reduced \(C^*\)-algebra \(C^*_ r(E)\). It is shown that \(C^*_ r(E)\) have a nontrivial gauge-invariant closed two-sided ideals.
semigroup of normal *-endomorphisms of the algebra B(H)
spectral \(C^ *\) algebra associated with a product structure
continuous time analogue of the Cuntz algebra
scientific article; zbMATH DE number 4177905
scientific article