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Proceedings of the American Mathematical Society

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Characterizations of certain classes of hereditary $ C\sp *$-subalgebras


Author: Masaharu Kusuda
Journal: Proc. Amer. Math. Soc. 116 (1992), 999-1005
MSC: Primary 46L55
MathSciNet review: 1127142
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Abstract: This paper characterizes the class of full hereditary $ {C^ * }$-subalgebras and the class of hereditary $ {C^ * }$-subalgebras that generate essential ideals in a given $ {C^ * }$-algebra in terms of a certain projection of norm one from the enveloping von Neumann algebra of the $ {C^ * }$-algebra onto the enveloping von Neumann algebra of a hereditary $ {C^ * }$-subalgebra. For a $ {C^ * }$-dynamical system $ (A,G,\alpha )$, it is also shown that an $ \alpha $-invariant $ {C^ * }$-subalgebra $ B$ of $ A$ is a hereditary $ {C^ * }$-subalgebra belonging to either of the above classes if and only if the reduced $ {C^ * }$-crossed product $ B{ \times _{\alpha r}}G$ is a hereditary $ {C^ * }$-subalgebra, of the reduced $ {C^ * }$-crossed product $ A{ \times _{\alpha r}}G$, belonging to the same class as $ B$. Furthermore similar results for $ {C^ * }$-crossed products are also observed.


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DOI: https://doi.org/10.1090/S0002-9939-1992-1127142-5
Article copyright: © Copyright 1992 American Mathematical Society