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It is shown that for any C(*)-algebra A of operators on a separable Hilbert space, there is, for each self-adjoint operator x in the strong closure of A, a sequence {x(n)} of self-adjoint operators, each of which is the strong limit of a monotone increasing sequence of self-adjoint operators from A, such that {x(n)} is monotone decreasing and strongly convergent to x.

The “Up-Down” Problem For Operator Algebras