The purpose of this course is to introduce the notions and basic results of linear operators in Banach and Hilbert spaces, and to give important theorems for operators, representations and positive linear functionals in C*-algebras.

Linear operators of Banach spaces, the spectrum of an element of Banach algebras, the Gelfand representation of commutative Banach algebras, C*-algebras, positive elements of C*-algebras, Hilbert spaces, operators of Hilbert spaces, ideals in C*-algebras, approximate identities of C*-algebras, positive linear functionals of C*-algebras, representations of C*-algebras, the Gelfand-Naimark Representation, irreducible representations of C*-algebras, Kadison transitivity theorem.