Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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



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
MathSciNet review: 503530
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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?)

  • [1] Heinz Bauer, Probability theory and elements of measure theory, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1981. Second edition of the translation by R. B. Burckel from the third German edition; Probability and Mathematical Statistics. MR 636091
  • [2] Claude Dellacherie, Un complément au théorème de Weierstrass-Stone, Séminaire de Probabilités (Univ. Strasbourg, Strasbourg, 1966/67) Springer, Berlin, 1967, pp. 52–53 (French). MR 0229005
  • [3] Jacques Dixmier, Les 𝐶*-algèbres et leurs représentations, Deuxième édition. Cahiers Scientifiques, Fasc. XXIX, Gauthier-Villars Éditeur, Paris, 1969 (French). MR 0246136
  • [4] Richard V. Kadison, A generalized Schwarz inequality and algebraic invariants for operator algebras, Ann. of Math. (2) 56 (1952), 494–503. MR 0051442,

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46L05, 46A40, 46K05

Retrieve articles in all journals with MSC: 46L05, 46A40, 46K05

Additional Information

Keywords: $ {C^\ast}$-subalgebra, Stone lattice, positive square root
Article copyright: © Copyright 1978 American Mathematical Society