Local Cocycles Of Operator Algebras

The thesis consists of three chapters.In chapter 1, we prove that every local 1-cocycle from polynomial algebra C[x] into noncommutative polynomial algebra C[x]*C[y] is a 1-cocycle; In other words, every local derivation from C[x] into C[x]*C[y] is a derivation.In chapter 2, we study the local cocycle of a special kind of Banach algebra, generated linearly by its idempotents, under some conditions, we show that every local 2-cocycle of such a Banach algebra is still a 2-cocycle.In chapter 3, we introduce the notion of approximative local cocycle from Banach algebras into their Banach bimodules, and show that every approximative local k-cocycle of von Neumann algebra into itself is a k-cocycle,for k=2,3.