Operator-stable quantum probability operators (Q1600618)

Let \(V(z)\), \(z\in {\mathbb R}^{2s}\), denote a projective unitary representation of \({\mathbb R}^{2s}\) satisfying the Weyl-Segal commutation relation. Given a probability operator (i.e. a positive operator with unit trace) \(T\) in the space \({\mathcal L}_1\) of nuclear operators on a separable infinite-dimensional Hilbert space \({\mathcal H}\), let \(\widehat{T}(z) = \operatorname {tr} TV(z)\), \(z\in {\mathbb R}^{2s}\). Define the convolution \(\star\) by setting \(\widehat{T_1\star T_2} = \widehat{T}_1 \cdot \widehat{T}_2\), and let \(T\circ \mu\) be defined for all probability operator \(T\) and probability measure \(\mu\) by \(\widehat{T\circ \mu} = \widehat{T}\cdot \widehat{\mu}\), where \(\widehat{\mu}\) denotes the classical Fourier transform, or characteristic function, of \(\mu\). Finally, given a positive diagonal matrix \(A\), let the linear transformation \(U_A\) of the space \({\mathcal L}_2\) of Hilbert-Schmidt operators on \({\mathcal H}\) be defined from \((\widehat{U_AT})(z) = \widehat{T}(A z)\), \(z\in {\mathbb R}^{2s}\). Let now \(\{ A_{kn}\), \(T_{kn}\}\) be a uniformly infinitesimal triangular array of admissible positive diagonal matrices and probability operators with \(A_{kn}=A_n\) and \(T_{kn}=T\), \(k=1,2,\ldots , n\), \(n\geq 1\). The author shows that if there exists a sequence \((c_n)_{n\geq 1}\) of vectors in \({\mathbb R}^{2s}\) such that the sequence \((\star_{k=1}^n U_{A_{kn}} T_{kn})\circ \delta_{c_n}\) converges to a probability operator \(S\) where \(\delta_{c_n}\) denotes the Dirac measure at \(c_n\), then \(S\) is Gaussian, i.e. \(\widehat{S}(z) = e^{-(1/2)\langle qz,z\rangle + i \langle z,z_0\rangle}\), where \(z_0\in {\mathbb R}^{2s}\) and \(q\) is a nonnegative selfadjoint operator on \({\mathbb R}^{2s}\). This yields a partial generalization of the non-commutative limit theorem of \textit{K. Urbanik} [Stud. Math. 78, 59--75 (1984; Zbl 0535.60014)] who considered diagonal matrices with constant coefficients.