No abstract

No abstract

No abstract

No abstract

Introduction. The purpose of this paper is to study noncommutative C*-algebras as Banach spaces. The Gelfand representation of an abelian C*-algebra as the algebra of all continuous complex-valued functions on its spectrum has made it possible to apply the techniques of measure theory and the topological properties of compact Hausdorff spaces to the study of such algebras. No such structure theory of general C*-algebras is available at present. Many theorems about the Banach space structure of abelian C*-algebras are stated in terms of topological or measure-theoretic properties of their spectra; although much work has been done of late in studying an analogous dual object for general C*-algebras, the generalization is far from exact. For this reason we shall confine our study primarily to W*-algebras in which the lattice of self-adjoint projections will be used as a substitute for the Borel sets of the spectrum of an abelian C*-algebra. Using a theorem of Takeda [15] we shall be able to extend some of our results to general

A net {aα} of positive, norm one elements of a C*-algebra A excises a state f of A ifThis notion has been used explicitly by the second author [4, 5, 6] for pure states, but the present paper will explore it more fully. The name is motivated by the following example. Let K be the unit disk in the complex plane, A = C(K) and f(a) = a(0). Define an(reiθ) = ϕn(r), whereNote that the sets {t ∊ K:an(t) ∊ 0} form rings about 0 with radii tending to 0. In this sense the sequence {an} “cuts out” the state f and, in the limit, isolates it from all other states.

No abstract

No abstract