Abstract
J. Glimm's Stone-Weierstrass theorem states that ifA is aC *-algebra,P(A) is the set of pure states ofA, andB is aC *-subalgebra which separates\(\overline {P(A)} \cup \left\{ 0 \right\}\), thenB=A. We show that ifB is aC *-subalgebra ofA andx an element ofA such that any two elements of\(\overline {P(A)} \cup \left\{ 0 \right\}\) which agree onB agree also onx, thenx∈B. Similar complements are given to other Stone-Weierstrass theorems. A theorem of F. Shultz states that ifx∈A **, the enveloping von Neumann algebra ofA, and ifx, x *, x, andxx * are uniformly continuous onP(A)∪{0}, then there is an element ofA which agrees withx onP(A). We show that the hypotheses onx *x andxx * can be dropped.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Akemann, C.A.: The general Stone-Weierstrass problem. J. Funct. Anal.4, 277–294 (1969)
Akemann, C.A., Anderson, J.: The Stone-Weierstrass problem forC *-algebras. In: Invariant subspaces and other topics, Operator Theory: Adv. Appl.6, 15–32 (1982)
Akemann, C.A., Shultz, F.W.: PerfectC *-algebras. Mem. Am. Math. Soc.326 (1985)
Archbold, R.J., Batty, C.J.K.: On factorial states of operator algebras. III. J. Op. Theory15, 53–81 (1986)
Batty, C.J.K., Archbold, R.J.: On factorial states of operator algebras. II. J. Op. Theory13, 131–142 (1985)
Brown, L.G.: Stable isomorphism of hereditary subalgebras ofC *-algebras. Pac. J. Math.71, 335–348 (1977)
Brown, L.G.: Semicontinuity and multipliers ofC *-algebras. Canad. J. Math.40, 865–988 (1988)
Brown, L.G., Green, P., Rieffel, M.A.: Stable isomorphism and strong Morita equivalence ofC *-algebras. Pac. J. Math.71, 349–363 (1977)
Busby, R.C.: Double centralizers and extensions ofC *-algebras. Trans. Am. Math. Soc.132, 79–99 (1968)
Dixmier, J.:C *-algebras. Amsterdam: North-Holland 1977
Effros, E.G.: Order ideals in aC *-algebra and its dual. Duke Math. J.30, 391–412 (1963)
Glimm, J.: A Stone-Weierstrass theorem forC *-algebras. Ann. Math.72, 216–244 (1960)
Kadison, R.V.: States and representations. Trans. Am. Math. Soc.103, 304–319 (1962)
Longo, R.: Solution of the factorial Stone-Weierstrass conjecture. An application of the theory of standard splitW *-inclusions. Invent. Math.76, 145–155 (1984)
Pedersen, G.K.:C *-algebras and their automorphism groups. London: Academic Press 1979
Popa, S.: Semi-regular maximal abelian *-subalgebras and the solution to the factor state Stone-Weierstrass problem. Invent. Math.76, 157–161 (1984)
Prosser, R.T.: On the ideal structure of operator algebras. Mem. Am. Math. Soc.45 (1963)
Sakai, S.: On the Stone-Weierstrass theorem ofC *-algebras. Tohoku Math. J.22, 191–199 (1970)
Shultz, F.W.: Pure states as a dual object forC *-algebras. Commun. Math. Phys.82, 497–509 (1982)
Author information
Authors and Affiliations
Additional information
Communicated by H. Araki
Rights and permissions
About this article
Cite this article
Brown, L.G. Complements to various Stone-Weierstrass theorems forC *-algebras and a theorem of Shultz. Commun.Math. Phys. 143, 405–413 (1992). https://doi.org/10.1007/BF02099015
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02099015