Integral decomposition of functionals on 𝐶*-algebras

Halpern, Herbert

doi:10.1090/S0002-9947-1972-0296710-7

Integral decomposition of functionals on $C^{\ast }$-algebras
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Trans. Amer. Math. Soc. 168 (1972), 371-385 Request permission

Abstract:

The spectrum of the center of the weak closure of a ${C^ \ast }$-algebra with identity on a Hilbert space is mapped into a set of quasi-equivalence classes of representations of the ${C^ \ast }$-algebra so that every positive $\sigma$-weakly continuous functional on the algebra can be written in a central decomposition as an integral over the spectrum of a field of states whose canonical representations are members of the respective quasi-equivalence classes except for a nowhere dense set. Various questions relating to disjointness of classes, factor classes, and uniformly continuous functionals are studied.

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