von Neumann Algebras
نویسنده
چکیده
For every selfadjoint operator T in the Hilbert space H, f(T) makes sense not only in the obvious case where / is a polynomial but also if / is just measurable, and if fn(x)-+f(x) for all x£R (with (/,) bounded) then fn(T)-+f(T) weakly, i.e. <fn(T)£9i)+<f(T)£9ti)V£9fiCH. Moreover the set {f(T)9 f measurable} is the set of all operators S in H invariant under all unitary transformations of H which fix T. More generally, if (Tf), i=\, ..., k, are operators in H then the weak closure of the set of polynomials in Ti9 T* is the space of all operators in H invariant under all the unitaries fixing the Ti9 as follows from the bicommutation theorem of von Neumann (1929):
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