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.)
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