Lectures on C*-algebras

TL;DR

This comprehensive lecture notes on C*-algebras covers foundational concepts, spectral theory, functional calculus, duality, and key examples, providing a thorough introduction to the structure and applications of C*-algebras.

Contribution

It offers an extensive, unified presentation of core topics in C*-algebra theory, integrating classical results with modern perspectives and examples.

Findings

Gelfand duality between commutative C*-algebras and topological spaces

Spectral theory of compact operators on Banach spaces

Structure and properties of multiplier and hereditary subalgebras

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.