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An operator-valued free Poincarè inequality

Ito Hyuga

Probability and Mathematical Statistics

2023

Vol. 43, Fasc. 2

241--246

The purpose of this short note is to give an operator-valued free Poincarè inequality, which provides a simple proof to (an improvement of) a lemma of Voiculescu (2000) asserting that the kernel of the free difference quotient is exactly the coefficients.

On the ergodic Hilbert transform in L2 over a von Neumann algebra

Kielanowicz K.

Probability and Mathematical Statistics

2007

Vol. 27, Fasc. 2

223--234

In this note a noncommutative version of Jajte’s theorem on the existence of the ergodic Hilbert transform is given. As a noncommutative counterpart of the classical almost everywhere convergence the bundle convergence of operators in a von Neumann algebra and its L2-space is used.

Bundle convergence of weiteighted sums of operators in noncommutative [L_2]-spaces

Le Gac B., Moricz F.

Bulletin of the Polish Academy of Sciences. Mathematics

2001

Vol. 49, no 4

327-336

The notion of bundle convergence for sequences of operators in a von Neumann algebra A equipped with a faithful and normal state phi as well as for sequences of vectors in their [L_2]-spaces were introduced by Hensz, Jajte and Paszkiewicz in 1996 as an appropriate substitute for almost everywhere convergence in the commutative setting. First, we prove that if (B_k : k = l, 2,...) is a sequence in A such that [sum{phi(B^*_k B_k)} < infinity] and if (a_k : k = l, 2,...) is a sequence of complex numbers such that [sum|a_k|^2 < infinity], then the *-homomorph image of the sequence [(sum_{k = 1}^n {a_k B_k : n = 1, 2,...})] in the [L_2] space given by the Gelfand-Naimark-Segal representation theorem is bundle convergent. Second, we prove a noncommutative version of the second Borel-Cantelli lemma in terms of bundle convergence. Such a version of the first Borel-Cantelli lemma cannot exist in this noncommutative setting. On closing, we raise two problems.