Banach algebras: ideals, quotients, holomorphic functional calculus, weak* topology, Banach Alaoglu's theorem, Gelfand's transform.
C*-algebras: order, approximate units, ideals,

quotients, the characterization of commutative C*-algebras, continuous functional calculus, states and representations, the universal representation.

Operator topologies and approximation theorems: von Neumann's bicommutation theorem, Kaplansky's density theorem and Kadison's transitivity theorem.

The spectral theorem for bounded normal operators: the Borel functional calculus, commutative von Neumann algebras, the group algebra .