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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenWed, 31 Jan 2024 19:09:11 +0000Essential Central Spectrum and Range in a W*-Algebra
https://resolver.caltech.edu/CaltechTHESIS:06282018-082103482
Authors: {'items': [{'id': 'Gray-Leonard-Jeffrey', 'name': {'family': 'Gray', 'given': 'Leonard Jeffrey'}, 'show_email': 'NO'}]}
Year: 1973
DOI: 10.7907/R9P2-KB57
<p>Halpern has defined a center valued essential spectrum, Σ<sub>I</sub>(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.</p>
<p>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:</p>
<p><u>Theorem</u>: Z = ʃ<sub>Λ</sub> ⊕ c(λ)dµ ∈ Σ<sub>I</sub>(A) if and only if c(λ) ∈ σ<sub>e</sub>(A(λ)) µ-a.e., where A = ʃ<sub>Λ</sub> ⊕ A(λ)dµ and σ<sub>e</sub> is a suitable spectrum in the algebra ɸ(λ).</p>
<p>Using mainly measure-theoretic arguments, we obtain a similar decomposition result for the norm closure of the central numerical range:</p>
<p><u>Theorem</u>: Z = ʃ<sub>Λ</sub> ⊕ c(λ)dµ ∈ Wʓ(A) if and only if c(λ) ∈ W(A(λ)) µ-a.e.</p>
<p>By means of these theorems, questions about Σ<sub>I</sub>(A) and W (A) in ɸ can be reduced to the factors ɸ(λ). As examples, we show that spectral mapping holds for Σ<sub>I</sub>, namely f(Σ<sub>I</sub>(A)) = Σ<sub>I</sub>(f(A)), and that a generalization of the power inequality holds for Wʓ(A).</p>
<p>Dropping the separability assumption, we show that central ideals can be defined in purely algebraic terms, and that the following perturbation result holds:</p>
<p><u>Thereom</u>: Σ<sub>I</sub>(A + X) = Σ<sub>I</sub>(A) for all A ∈ ɸ if and only if X ∈ I.</p>https://thesis.library.caltech.edu/id/eprint/11094