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Proceedings of the American Mathematical Society

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

 
 

 

Local completeness of operator algebras


Authors: H. Behncke and J. Cuntz
Journal: Proc. Amer. Math. Soc. 62 (1977), 95-100
MSC: Primary 46L05
DOI: https://doi.org/10.1090/S0002-9939-1977-0428048-9
MathSciNet review: 0428048
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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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DOI: https://doi.org/10.1090/S0002-9939-1977-0428048-9
Article copyright: © Copyright 1977 American Mathematical Society

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