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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Approximating maps and a Stone-Weierstrass theorem for $ C\sp{\ast} $-algebras


Author: John W. Bunce
Journal: Proc. Amer. Math. Soc. 79 (1980), 559-563
MSC: Primary 46L30
DOI: https://doi.org/10.1090/S0002-9939-1980-0572301-5
MathSciNet review: 572301
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Abstract: Let A be a $ {C^ \ast }$-algebra with identity and B a $ {C^ \ast }$-subalgebra of A which separates the pure states of A. We give an easy proof of the fact that, assuming there is a sequence of norm one linear maps $ {L_n}:A \to B$ such that $ {L_n}(b)$ converges weakly to b for each b in B, B must equal A. As corollaries we prove that if B separates the pure states of A, then $ B = A$ if B is nuclear, or if $ B = C_r^ \ast ({F_2})$ and $ A \subseteq VN({F_2})$, where $ {F_2}$ is the free group on two generators.


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DOI: https://doi.org/10.1090/S0002-9939-1980-0572301-5
Keywords: Stone-Weierstrass theorem, approximating maps, nuclear $ {C^ \ast }$-algebra, left regular representation of the free group on two generators
Article copyright: © Copyright 1980 American Mathematical Society