Abstract
An abelian C ∗-algebra is known to be isomorphic to the algebra of all complex continuous functions vanishing at infinity on its maximal ideal space. In an earlier paper the author defined a structure on the closed left ideals of an arbitrary C ∗-algebra which was analogous to the structure topology on the maximal ideals which exists for the abelian case. This paper carries that work forward. The first section expands the technical base of the theory by showing that more of the desirable properties of a compact Hausdorff space carry over to the new structure. The following section contains applications of various sorts. For example, Theorem I.1 gives a new characterization of AW ∗-algebras. The last section characterizes the maximal and minimal (closed) left ideals of any C ∗-algebra.
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