The dual space of an operator algebra

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

Weakly compact operators and the strong* topology for a Banach space

“…Therefore, there exists λ 0 such that for each λ ≥ λ 0 , we have G i (x λ ) < δ, for all 1 ≤ i ≤ k, which shows, by (1), that x λ ∈ O, equivalently, U (x λ ) ≤ δ, for each λ ≥ λ 0 .…”

“…Given a JB*-triple E, a norm-one functional φ in E * and a norm-one element e ∈ E * * with φ(e) = 1, the mapping x → x φ = φ{x, x, e} 1 2 defines a prehilbertian seminorm on E which does not depend on the element e (compare [4, Proposition 1.2]). By the classical little Grothendieck inequality for JB*-triples we know that when E is a JB*-triple, then s * (E, E * ) coincides with the topology on E generated by all the seminorms of the form x ϕ , where ϕ and e are norm-one elements in E * and E * * , respectively, and satisfy ϕ(e) = 1 (see [17, §4]).…”

“…In Akemann's work, a key role is played by seminorms of the form x → φ(x * x + xx * ) 1 2 where φ ranges over the positive linear forms on the C*-algebra. While this looks like an specific C * -algebra situation, from the point of view of locally convex topologies it is not.…”

“…While this looks like an specific C * -algebra situation, from the point of view of locally convex topologies it is not. Indeed, as a consequence of the noncommutative generalisation of the little Grothendieck inequality, see [11], [19], the locally convex topology on a C * -algebra A generated by the seminorms x → φ(x * x + xx * ) 1 2 coincides with the one generated by all seminorms of the form x → Sx where it is required that S is a bounded linear map from A into a Hilbert space. (There is an analogous theory for JB * -triples; we give a brief discussion in §3 below.…”

Weakly almost periodic and uniformly continuous functionals on the Fourier algebra of any locally compact group

“…that for some sequence vn E K C PX(G) and all <i> G VN(G), 4>(vn) ~* *Poi<P)-Thus v" is weak Cauchy sequence in A(G). By a theorem of Sakai [14] (see also [1]) the predual of a lf*-algebra is weakly sequentially complete.…”