Essential Central Spectrum and Range in a W*-Algebra

Citation

Gray, Leonard Jeffrey (1973) Essential Central Spectrum and Range in a W*-Algebra. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/R9P2-KB57. https://resolver.caltech.edu/CaltechTHESIS:06282018-082103482

Abstract

Halpern has defined a center valued essential spectrum, Σ_I(A), and numerical range, Wʓ(A), for operators A in a von Neumann algebra ɸ. By restricting our attention to algebras ɸ which act on a separable Hilbert space, we can use a direct integral decomposition of ɸ to obtain simple characterizations of these qualities, and this in turn enables us to prove analogues of some classical results.

since the essential spectrum is defined relative to a central ideal, we first show that, under the separability assumption, every ideal, modulo the center, is an ideal generated by finite projections. This leads to the following decomposition theorem:

Theorem: Z = ʃ_Λ ⊕ c(λ)dµ ∈ Σ_I(A) if and only if c(λ) ∈ σ_e(A(λ)) µ-a.e., where A = ʃ_Λ ⊕ A(λ)dµ and σ_e is a suitable spectrum in the algebra ɸ(λ).

Using mainly measure-theoretic arguments, we obtain a similar decomposition result for the norm closure of the central numerical range:

Theorem: Z = ʃ_Λ ⊕ c(λ)dµ ∈ Wʓ(A) if and only if c(λ) ∈ W(A(λ)) µ-a.e.

By means of these theorems, questions about Σ_I(A) and W (A) in ɸ can be reduced to the factors ɸ(λ). As examples, we show that spectral mapping holds for Σ_I, namely f(Σ_I(A)) = Σ_I(f(A)), and that a generalization of the power inequality holds for Wʓ(A).

Dropping the separability assumption, we show that central ideals can be defined in purely algebraic terms, and that the following perturbation result holds:

Thereom: Σ_I(A + X) = Σ_I(A) for all A ∈ ɸ if and only if X ∈ I.

Thesis Files