The strong law of large numbers for d-dimensional arrays in von Neumann algebras

Let A be a von Neumann algebra with faithful normal tracial state r; let (A) over tilde be the *-algebra of measurable operators in Segal's sense. The aim of this paper is to show that if an array (x((n) over bar), (n) over bar is an element of N-d) of self-adjoint pairwise independent identically distributed elements in (A) over tilde satisfies the condition r(\x((1) over bar\(log(+)\x((1) over bar)\)(d-1)) <infinity, then [GRAPHICS] converges to r(x((1) over bar)) bilaterally almost uniformly (where (n) over bar = (n(1),...,n(d)), (k) over bar = (k(1),...,k(d)), (1) over bar =(1,...,1)is an element of N-d).