On the Blum-Hanson theorem for quantum quadratic processes

In this paper an analog of the Blum-Hanson theorem for quantum quadratic processes on the von Neumann algebra is proved, i.e., it is established that the following conditions are equivalent: i) P^(t)x is weakly convergent to x₀; ii) for any sequence {a_n} of nonnegative integrable functions on [1, ∞] such that ∫₁^∞ a_n(t) d(t) = 1 for any n and lim_n→∞ ||a_n||_∞ = 0, the integral ∫₁^∞ a_n(t)P^(t)x dt is strongly convergent to x₀ in L²(M, φ), where x ∈ M, p^(t) is a quantum quadratic process, M is a von Neumann algebra, and φ is an exact normal state on M.