ACTA issues

On the bundle convergence of orthogonal series and SLLN in noncommutative $L_2$-spaces

Barthélemy Le Gac, Ferenc Móricz

Acta Sci. Math. (Szeged) 64:3-4(1998), 575-599

Abstract. We construct a new majorant for the consecutive partial sums of a finite sum $\sum ^n_{k=1}\xi_k$ whose terms are pairwise orthogonal vectors in a noncommutative $L_2(${\eufm A},$\phi )$ space. Here {\eufm A} is a $\sigma $-finite von Neumann algebra, and $\phi $ is a faithful and normal state defined on {\eufm A}. We extend a theorem of Tandori [15] on the convergence of the orthogonal series $\sum ^\infty_{k=1} \xi_k$ from the classical commutative case to the noncommutative one, in terms of bundle convergence. As it is known, hence almost sure convergence follows. The condition imposed on $\|\xi_k\|^2$ in our theorem is weaker than that in the noncommutative Rademacher-Menshov theorem proved by Hensz, Jajte and Paszkiewicz [5]. We also deduce an improved strong law of large numbers for an orthogonal sequence $(\xi_k)$ of vectors in $L_2(${\eufm A},$\phi )$ as well as new criteria for bundle convergence of a given subsequence of the partial sums of the series $\sum ^\infty_{k=1}\xi_k$. As a by-product, we improve a theorem of Hensz [3] by weakening the condition and strengthening the conclusion in it.

AMS Subject Classification (1991): 46L50, 60F15, 42C15

Keyword(s): von Neuman algebra, faithful and normal state, scalar product, prehilbert space, $L_2$-completion, Gelfand-Naimark-Segal representation theorem, cyclic vector, bundle convergence, almost sure convergence, orthogonal vectors in noncommutative setting, Rademacher-Menshov inequality and theorem, convergence of a given subsequence of partial sums, Cesàro average of partial sums, strong law of large numbers

Received September 25, 1997 and in revised form April 6, 1998. (Registered under 3310/2009.)