Lectures on C*-algebras

Abstract: The following topics are presented in these notes: Elements of Banach algebras, Banach algebras of the form $L^1(G)$, where $G$ is a locally compact group, spectrum of elements of Banach algebras, the spectral theory of compact operators on Banach spaces, the holomorphic functional calculus in Banach algebras, the Gelfand transform on commutative Banach algebras and C*-algebras, the continuous functional calculus, the Gelfand duality between commutative C*-algebras and locally compact and Hausdorff topological spaces, positivity in C*-algebras, approximate units, ideals of C*-algebras, hereditary C*-subalgebras, multiplier algebras, Hilbert spaces, the C*-algebra $B(H)$ of bounded operators on a Hilbert space $H$, examples of concrete C*-algebras, the reduced group C*-algebra of a locally compact group $G$, locally convex topologies on the C*-algebra $B(H)$, the Borel functional calculus in $B(H)$, projections in $B(H)$ and the polar decomposition of elements of $B(H)$, C*-algebras of compact operators and the bicommutant theorem.