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On limit theorems in JW-algebras

Karimov A., Mukhamedov F.

Probability and Mathematical Statistics

2010

Vol. 30, Fasc. 1

153--165

In the present paper, we study bundle convergence in JW- algebra and prove certain ergodic theorems with respect to such convergence. Moreover, conditional expectations of reversible JW-algebras are considered. Using such expectations, the convergence of supermartingales is established.

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 in a von Neumann algebra and in a von Neumann subalgebra

Le Gac B., Moricz F.

Bulletin of the Polish Academy of Sciences. Mathematics

2004

Vol. 52, nr 3

283-295

Let H be a separable complex Hilbert space, A a von Neumann algebra in L(H), [phi] a faithful, normal state on A, and B a commutative von Neumann subalgebra of A. Given a sequence (X[n] : n is less than or equal to 1) of operators in B, we examine the relations between bundle convergence in B and bundle convergence in A.

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.