Random Walks on Graphs and Approximation of L2-Invariants

Abstract

Right multiplication operators $R_w: l_2G \rightarrow l_2G$, $w \in \C[G]$, are interpreted as random-walk operators on labelled graphs that are analogous to Cayley graphs. Applying a generalization of the graph convergence defined by R. Grigorchuk and A. \.{Z}uk \cite{Grigorchuk_Zuk_1} gives a new proof and interpretation of a special case of W. L\"{u}ck's famous Theorem on the Approximation of $l_2$-Betti numbers for countable residually finite groups. In particular, using this interpretation, the proof follows quickly from standard theorems about the weak convergence of probability measures that are characterized by their moments.