Kolmogorov's factorization theorem for von Neumann algebras (Q1941354)

The Kolmogorov factorization theorem for positive definite functions states that, if \(K\) is a continuous positive definite function from \(\mathbb{Z} \times \mathbb{Z}\) to \(\mathbb{C}\), there is a continuous mapping \(x\) from \(\mathbb{Z}\) into a Hilbert space \(\mathcal{H}\) such that \(K(s,t)=x^*(s)x(t)\). This result was extended independently by \textit{G. D. Allen, F. J. Narcowich} and \textit{J. P. Williams} [Pac. J. Math. 61, 305--312 (1975; Zbl 0324.46041)] and \textit{R. A. Kunze} [Functional Analysis, Proc. Conf. Univ. California, Irvine 1966, 235--247 (1967; Zbl 0226.43011)] as follows: Let \(G\) be a separable Hausdorff space and \(\mathcal{H}\) be a Hilbert space. If \(K\) is a continuous positive definite function from \(G\times G\) into \(B(\mathcal{H})\), then there exists a separable Hilbert space \(\mathcal{K}\) and a continuous function \(X\) from \(G\) into \(B(\mathcal{H},\mathcal{K})\) such that \(X^*(s)X(t)=K(s,t)\). In the paper under review, the case where \(K\) takes values in a von Neumann algebra is considered. The main result is the following: Let \(\mathcal{M}\) be a von Neumann algebra acting on a Hilbert space \(\mathcal{H}\). Put \(\mathcal{K}=\sum_{i=1}^{\infty}\mathcal{H}\) and let \(\mathcal{M}_{1 \infty}\) be the set of all matrices \(T=[T_i]\) of \(B(\mathcal{H}, \mathcal{K})\) which satisfy \(T_i \in \mathcal{M}\). Let \(G\) be a separable Hausdorff space and \(K\) be a continuous positive definite function from \(G\times G\) into \(\mathcal{M}\). Then there exists a continuous map \(X\) from \(G\) into \(\mathcal{M}_{1 \infty}\) such that \(X^*(s)X(t)=K(s,t)\).