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Subalgebras of C*-algebras and von Neumann algebras

Published online by Cambridge University Press:  18 May 2009

Charles A. Akemann
Charles A. Akemann
Affiliation:
University of California, Santa Barbara
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Recent work [2, 6] on subalgebras of matrix algebras leads naturally to the following situation. Let A be a C*-subalgebra of the C*-algebra B andM be a weakly closed *-subalgebra of the von Neumann algebra N. Consider the following Conditions.

Condition 1. For every b≠ 0 in B there exists a ∈ A such that O≠ab ∈ A.

Condition 2. For every b∈B there exists a ≠ 0 in A such that ab ∈ A.

If we replace A by M and B by N in Conditions 1 and 2 we get von Neumann algebra versions which we shall call Condition 1'and Condition 2'. Clearly Condition 1 implies Condition 2, and both conditions suggest that A is some kind of weak ideal of B. This paper explores the extent to which this is true. The paper grew out of the author's attempts [1, 3] to generalize the Stone-Weierstrass theorem to C*-algebras.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 25 , Issue 1 , January 1984 , pp. 19 - 25
Copyright
Copyright © Glasgow Mathematical Journal Trust 1984

References

REFERENCES

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