In this thesis، we introduce and study the Banach algebra of compact operator. Admittedly, we prove that if X is a Banach space, Then the collection of compact operators on X, is a Banach algebra. We also prove that if H is a separable Hilbert space, then K(H) is a C ?-algebra. The C ?-algebra KpHq contains the space F(H) of finite rank operators on H as a dense subset. It has an identity element if and only if H is finite dimentional.
