Infinite tensor products of von Neumann algebras. I.

Abstract

Theorem 3.2. Assume that Ii\\Tt\\^ + oom I. // Tt G 3lt , then the following three conditions are equivalent : (0 (8)^(8)31,. (ii) For any CE.T and any (X)£c, (T t^ t)Gc or $§Ttxt = Q. (iii) (8) Tt is a strong limit of { Tj : / rte( 3U . in. // r.ea;, then (g) Tt e (®a,)'. Theorem 4.2. Let §Xt Z>g a finite von Neumann algebra with the coupling operator Ct for every c € /. (i) Let cpt be a normal trace on Slt for each t€.I such that 0 t(l)< + °°. // S^t((l~O )< + °°3 £^0w ^er^ is one and only one normal trace <p on (g)2lt such that