Isospectral deformation of discrete random Laplacians

Let Ti,T2,...,Td be commuting automorphisms of the probability space (X,n). We call the Zd action (X,T,//) a dynamical system and write Tnx = T^T22...T%d(x) for n G Zd. Denote by X the crossed product of the von Neumann algebra A = L°°(X,M(N,C)) with the dynamical system (X,T,n). The group Zd acts on A by automorphisms / i-> f(Tn) where f(Tn)(x) = f(Tnx) and the algebra X is obtained by completing the algebra of all polynomials in the variables T\,...,Td with coefficients in A K = Y, K»rn> ( * L) » = E KiLm(Tl)rn n e F C % d l + m = n with respect to the norm \\\K\\ \ = | ||A'(x)| | |oo, where K(x) is the bounded linear operator on l2(Zd) defined by (K(x)u)n = ^2mKm(x)un+m and where | | • | | is the operator norm on B(l2(Zd)) and | • 1 ^ the essential supremum norm. With the involution on X defined by (E*»o * = E*-(T"B)T"B n n X becomes a von Neumann algebra. It has the trace tv(K) = f Ti(L0(x)) dfi(x),Jx where Tr denotes the usual trace on the matrix algebra M(N, C). We call the elements in X discrete random operators. In X lies the set of discrete random Laplacians d C = {L = ar + (ar) * + b = ^ a ^ + (a^) * + 6}, On Three Levels, Edited by M. Fannes et al.