Subnormal operators and the Kaplansky density theorem (Q1971675)

It is shown that the entries in the matrix representation of the minimal normal extension of a pure subnormal operator \(S\) can be taken from \(W^*(S)\), and this applies to subnormal tuples of operators, too. Furthermore, if \(S= (S_1,\dots, S_n)\) is a subnormal \(n\)-tuple of operators in a von Neumann algebra \({\mathcal M}\) then each \(S_j\) is the SOT-limit of a sequence \(N_k(j)\) of normal operators from \({\mathcal M}\) with \(\|N_k(j)\|\leq\|S_j\|\) and such that the family \(\{N_k(j): j= 1,\dots, n\}\) is commuting for each \(k\).