Banach algebras, Gelfand theory, $C^{*}$-algebras the GNS construction, spectral
theorem for normal operators, Fredholm operators. The L-infinity functional
calculus for normal operators.
Suggested books and references:
Conway, J.B., A Course in Functional Analysis, Springer, 1985.
Douglas, R. G., Banach Algebra Techniques in Operator Theory, Academic Press, 1972.