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A characterization of $ C\sp{\ast} $-subalgebras


Author: Jan A. van Casteren
Journal: Proc. Amer. Math. Soc. 72 (1978), 54-56
MSC: Primary 46L05; Secondary 46A40, 46K05
DOI: https://doi.org/10.1090/S0002-9939-1978-0503530-5
MathSciNet review: 503530
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Abstract: Let A be a closed linear subspace of a $ {C^\ast}$-algebra B. Adjoin, if necessary, the identity 1 to B. Then A is a $ {C^\ast}$-subalgebra if and only if, for each x in A, the elements $ {x^\ast}$ and $ \vert x\vert + 1 - \vert\vert x\vert - 1\vert$ are in A. If 1 is in A, then A is a $ {C^\ast}$-subalgebra if and only if $ \vert x\vert$ is in A for each x in A. Here $ \vert x\vert$ denotes the unique positive square root of $ {x^\ast}x$ in B.


References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1978-0503530-5
Keywords: $ {C^\ast}$-subalgebra, Stone lattice, positive square root
Article copyright: © Copyright 1978 American Mathematical Society