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In this essay we study the concepts of nuclearity in C-algebras, and amenability in Banach algebras. A proof on the "every amenable C-algebra is nuclear" is studied. A Banach algebra A is called amenable if every continous derivation D : A ! E , is inner i.e. there exists anif in this denition E is a dual Banach A-module and norm continuity replace by w-continuity then we have Connes-amenability; Connes-amenability was rst considered for von Neumann algebras in [J-K-R] (and thus should perhaps be called Johnson-Kadison-Ringrose amenability). The reason why this notion of amenability is usually associated with A. Connes are his papers [Conn 1] and [Conn 2] The name "Connes-amenability" seems... 

In this thesis, we consider the model of random interlacement on transient graghs, which was first introduced by Sznitman for the special case of Zd(d > 3) in 2010’s. in Sznitman’s case, it was shown that on Zd: for any intensity u > 0, the interlacement set is almost surely connected. The main result of this thesis says that for transient, transitive graphs, the above property holds if and only if the graph is amenable. In paticular, we show that in nonamenable transitive graphs, for small values of the intensity u the interlacement set has infinitely many infinite clusters. We also provide examples of nonamenable transitive graphs, for which the interlacement set becomes connected for...