Approximating maps and a Stone-Weierstrass theorem for 𝐶*-algebras

Bunce, John W.

doi:10.1090/S0002-9939-1980-0572301-5

Approximating maps and a Stone-Weierstrass theorem for $C^{\ast }$-algebras
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by John W. Bunce

Proc. Amer. Math. Soc. 79 (1980), 559-563

DOI: https://doi.org/10.1090/S0002-9939-1980-0572301-5

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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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