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Subalgebras of C*-algebras and von Neumann algebras
Glasgow mathematical journal, Tome 25 (1984) no. 1, pp. 19-25

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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. 
Akemann, Charles A. Subalgebras of C*-algebras and von Neumann algebras. Glasgow mathematical journal, Tome 25 (1984) no. 1, pp. 19-25. doi: 10.1017/S001708950000536X
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