Journal Article

COMPLETENESS OF *-SYMMETRIC GELFAND–NAIMARK–SEGAL INNER PRODUCT SPACES

E. Chetcuti and J. Hamhalter

in The Quarterly Journal of Mathematics

Volume 63, issue 2, pages 367-373
Published in print June 2012 | ISSN: 0033-5606
Published online June 2012 | e-ISSN: 1464-3847 | DOI: https://dx.doi.org/10.1093/qmath/haq041
COMPLETENESS OF *-SYMMETRIC GELFAND–NAIMARK–SEGAL INNER PRODUCT SPACES

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Every state ϱ on a C*-algebra A induces a *-symmetric semi-inner product (x, y)↦ ϱ(y* x) + ϱ(xy*) (x, yA). The main scope of the paper is to characterize those states for which the induced *-symmetric Gelfand–Naimark–Segal inner product space is complete. It is shown that this happens precisely when ϱ is a finite convex combination of pure states. (It is well known that the same conclusion follows if one considers the non-symmetric semi-inner product (x, y) ↦ ϱ(y* x).) In so doing, we exhibit an interesting connection between convexity properties of states, the transitivity of irreducible representations and Banach space properties of the quotients of C*-algebra s by kernels of states.

Journal Article.  0 words. 

Subjects: Pure Mathematics

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