Ortho-independent states of the C. C. R. algebra
A simple characterisation of the normally distributed \({\mathbb{R}}^ n\)- valued random variables \((n>1)\) with covariance matrix \(\sigma^ 2I\) (\(\sigma\geq 0)\) due to \textit{M. Kac} [Am. J. Math. 61, 726-728 (1939; Zbl 0022.06102)] - namely stochastic independence of the components \((0X)_ i\) \(i=1,...,n\) for each orthogonal matrix O - is generalised to a result about regular states of the (smallest) \(C^*\)-algebra of the CCR's, \({\mathcal U}(V,\beta)\) `\textit{J. Manuceau, M. Sirugue, D. Testard} and \textit{A. Verbeure}, Commun. Math. Phys. 32, 231-243 (1973; Zbl 0284.46039)]. Here \(\beta\) is a (possibly degenerate) symplectic form on the real vector space V and \({\mathcal U}(V,\beta)\) is generated by unitaries \(W_ f\), \(f\in V\), satisfying \(W_ fW_ g=\exp \{-i\beta (f,g)\}W_{f+g}.\) When \(\beta\) is the imaginary part of the inner product of a complex pre- Hilbert space of dimension \(\leq 2\), the result is: For a regular state of \({\mathcal U}(V,Im<.,.>)\), the following are equivalent: (i) \({\hat \omega}\)(f\(+g)={\hat \omega}(f){\hat \omega}(g)\) whenever \(<f,g>=0\) (ii) \({\hat \omega}\)(f\(+g)={\hat \omega}(f){\hat \omega}(g)\) whenever \(Re<f,g>=0\) (iii) \({\hat \omega}\)(f)\(=\exp \{im(f)-\sigma^ 2\| f\|^ 2/2\}\) where \(\sigma\geq 1\), \(m\in V^*\)- the algebraic dual of. \(\omega\) is the 'generating function' \(f\to \omega (W_ f)\) and regularity means continuity in \({\mathbb{R}}\) of the maps \(t\to {\hat \omega}(f+tg)\) (f,g\(\in V).\) The states of the form (iii) with \(m=0\) are precisely the extremal elements in the convex set of 'universally invariant' states (\({\hat \omega}\)(Uf)\(={\hat \omega}(f)\forall f\in V\) and unitary U) [\textit{I. E. Segal}, Ill. J. Math. 6, 500-523 (1962; Zbl 0106.428)].
normally distributed \({\mathbb{R}}^ n\)-valued random variables
scientific article; zbMATH DE number 3967134