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

• Autorzy: Le Gac B. , Moricz F.
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Borel-Cantelli lemmas; bundle convergence; faithful and normal state; Gelfand-Naimark-Segal representation; independence in a von Neumann algebra; von Neumann algebra; weighted sum of operators

PL

algebra von Neumanna; lemat Borela-Cantelliego; niezależność w algebrze von Neumanna; reprezentacja Gelfanda-Naimarka-Segala; stan wierny i normalny

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2001
• Tom: Vol. 49, no 4
• Strony: 327--336
• Opis fizyczny: Bibliogr. 4 poz.

Twórcy

autor

Le Gac B.

• Université de Provence, Centre de Mathématiques et Informatique, 30 Rue Joliot-Curie, 13453 Marseille Cedex 13, France

autor

Moricz F.

• University of Szeged, Bolyai Institute, Aradi Vertandk Tere 1 6720 Szeged, Hungary

Bibliografia

• [1] E. Hensz, R.. Jajte, A. Paszkiewicz, The bundle convergence in von Neumann algebras and their L2-spaces, Studia Math., 120 (1996) 23-46.
• [2] R. Jajte, Strong limit theorems in noncommutative L2-spaces, Lecture Notes in Math., 1477, Springer, Berlin-Heidelberg-New York 1991.
• [3] B. Le Gac, F. Móricz, On the bundle convergence of orthogonal series and SLLN in noncommutative L2-spaces, Acta Sci. Math. (Szeged), 64 (1998) 575-599.
• [4] Y. Seo, Independence in noncommutative probability, Math. Japon., 46 (1997) 445-450.