Citation
Gray, Leonard Jeffrey (1973) Essential Central Spectrum and Range in a W*Algebra. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/R9P2KB57. https://resolver.caltech.edu/CaltechTHESIS:06282018082103482
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)) µa.e., where A = ʃ_{Λ} ⊕ A(λ)dµ and σ_{e} is a suitable spectrum in the algebra ɸ(λ).
Using mainly measuretheoretic 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)) µa.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.
Item Type:  Thesis (Dissertation (Ph.D.))  

Subject Keywords:  Mathematics  
Degree Grantor:  California Institute of Technology  
Division:  Physics, Mathematics and Astronomy  
Major Option:  Mathematics  
Thesis Availability:  Public (worldwide access)  
Research Advisor(s): 
 
Thesis Committee: 
 
Defense Date:  2 April 1973  
Funders: 
 
Record Number:  CaltechTHESIS:06282018082103482  
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:06282018082103482  
DOI:  10.7907/R9P2KB57  
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  11094  
Collection:  CaltechTHESIS  
Deposited By:  Benjamin Perez  
Deposited On:  28 Jun 2018 18:28  
Last Modified:  14 Jun 2023 23:03 
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