## Doctoral Dissertations

Fall 1988

Dissertation

Mathematics

#### Degree Name

Doctor of Philosophy

Eric A Nordgren

#### Abstract

Given a norm closed unital algebra ${\cal A}$ of operators on Hilbert space, a sublattice of Lat$\sb{1/2}{\cal A}$ is identified, which we denote Lat$\sb{\rm cb}{\cal A}$. It is proved that Lat$\sb{\rm cb}{\cal A}$ is lattice isomorphic to Lat$\sb{1/2}{\cal A}\otimes{\cal K}$ (where ${\cal K}$ denotes the ideal of compact operators on an infinite dimensional separable Hilbert space). This isomorphism is used to prove theorems describing Lat$\sb{\rm cb}{\cal A}$ by carrying over know results concerning Lat$\sb{1/2}{\cal A}\otimes{\cal K}$. It is then shown that Lat$\sb{\rm cb}{\cal A}$ can always be written as the set of ranges of operators that intertwine the algebra with a complete contraction.

A characterization of Lat$\sb{\rm cb}{\cal A}$ is obtained when ${\cal A}$ is a certain type of abelian strictly cyclic algebra. The characterization applies to all strictly cyclic weighted shift commutants, provided the weighted shift has a monotonically decreased weight sequence.

Conditions under which the range of a diagonal operator is in Lat$\sb{1/2}{\cal A}$ are obtained and compared with the conditions needed for it to be in Lat$\sb{1/2}{\cal A}$*, when ${\cal A}$ is the commutant of a strictly cyclic weighed shift. A complete characterization of the invariant ranges of diagonal operators is obtained for the commutant of a strongly strictly cyclic weighted shift.

The invariant operator range question for C*-algebras and the question "is Lat$\sb{\rm cb}{\cal A} = {\rm Lat}\sb{1/2}{\cal A}$" when ${\cal A}$ is an abelian strictly cyclic algebra are related to the question "must the set of ranges of operators from a weakly closed linear manifold contain a maximal element"? It is proved that an affirmative answer to the latter question implies affirmative answers to the other questions.

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