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Proceedings of the American Mathematical Society

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

 

 

Stable isomorphism of hereditary $ C\sp *$-subalgebras and stable equivalence of open projections


Author: Shuang Zhang
Journal: Proc. Amer. Math. Soc. 105 (1989), 677-682
MSC: Primary 46L05; Secondary 46M20
DOI: https://doi.org/10.1090/S0002-9939-1989-0961418-7
MathSciNet review: 961418
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Abstract: We relate the stable isomorphism of two hereditary $ {C^*}$-subalgebras to the stable equivalence of the corresponding open projections. We prove that if $ A$ is completely $ \sigma $-unital, then $ {\operatorname{her}}(p)$ and $ {\operatorname{her(}}q)$ generate the same closed ideal of $ A$ iff $ p \otimes 1 \sim q \otimes 1$ in $ {(A \otimes K)^{**}}$ iff the central supports of $ p$ and $ q$ in $ {A^{**}}$ are the same. If, in addition, $ p \bot q$, then the above three equivalent conditions are equivalent to the condition: $ p \otimes 1$ and $ q \otimes 1$ are in the same path component of open projections in $ {(A \otimes K)^{**}}$.


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DOI: https://doi.org/10.1090/S0002-9939-1989-0961418-7
Keywords: Hereditary $ {C^*}$-subalgebras, open projections, stable isomorphism, path component of open projections
Article copyright: © Copyright 1989 American Mathematical Society