Structure of CT ∗-Algebras

Abstract

Chapter 4 is devoted to structure of CT∗-algebras. In Sect. 4.1 it is proved that every CT∗-algebra \({\mathfrak A}\) is decomposed into \({\mathfrak A}=P({\mathfrak A})+Z({\mathfrak A})\) by the nondegenerate part \(P({\mathfrak A})\) and the nulifier \(Z({\mathfrak A})\) of \({\mathfrak A}\) using the corresponding projections \(P_{\mathfrak A}\) and \(N_{\mathfrak A}\), and more decompositions of \({\mathfrak A}\) are obtained decomposing the projections \(P_{\mathfrak A}\) and \(N_{\mathfrak A}\). Furthermore, the important equalities: \(P_{\mathfrak A}=I-N_{{\mathfrak A}^\tau }\) and \(P_{{\mathfrak A}^\tau }=I-N_{\mathfrak A}\) are verified. In Sect. 4.2 the notions of regular, semifinite, nondegenerate, singular and nilpotent CT∗-algebras are defined and their characterizations are investigated. In Sect. 4.3 it is shown that every commutative semisimple CT∗-algebra is isomorphic to the Banach ∗-algebra C0( Ω) ∩ L2( Ω, μ) consisting of the commutative C∗-algebra C0( Ω) of continuous functions on a locally compact space Ω vanishing at infinity on Ω and the L2-space L2( Ω, μ) defined by a regular Borel measure μ on Ω. The following subjects are discussed: Sect. 4.4.1 definition of regular, (semisimple, nondegenerate, singular and nilpotent) T∗-algebras. Sect. 4.4.2 construction of \(\lambda (\tilde {{\mathfrak A}})\) from \(\lambda ({\mathfrak A})\). Sect. 4.4.3 construction of semisimple CT∗-algebras from the C∗-algebra obtained by the uniform closure of \(\pi ({\mathfrak A})\). Sect. 4.4.4 the semisimplicity and the singularity of T∗-algebras. 3.4.6 Construction of a natural weight on the positive cone \((\pi ({\mathfrak A})^{\prime \prime })_+\) of the von Neumann algebra \(\pi ({\mathfrak A})^{\prime \prime }\). Sect. 4.4.6 the density of a ∗-subalgebra of a T∗-algebra under the strong∗ topology and under the uniform topology.