Local completeness of operator algebras

Behncke, H.; Cuntz, J.

doi:10.1090/S0002-9939-1977-0428048-9

Local completeness of operator algebras
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by H. Behncke and J. Cuntz

Proc. Amer. Math. Soc. 62 (1977), 95-100

DOI: https://doi.org/10.1090/S0002-9939-1977-0428048-9

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Abstract:

A normed $\ast$-algebra $\mathcal {A}$ is called a local ${C^\ast }$-algebra, if all its maximal commutative $\ast$-subalgebras are ${C^\ast }$-algebras. It is shown that any local ${C^\ast }$-algebra dense in $\mathcal {K}(\mathcal {H})$, the algebra of compact operators on the Hilbert space $\mathcal {H}$ equals $\mathcal {K}(\mathcal {H})$. The same result holds also for local ${C^\ast }$-algebras dense in $A{W^\ast }$-algebras without a ${\text {II}_1}$ summand.

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