**MTH 510: Operator Theory and Operator Algebras (4)**

**Pre-requisites**: *MTH 503 Functional Analysis *

*Course Contents*:

Banach Algebras, Ideals, Quotients, homomorphisms, Unitization

Invertible Elements, Spectrum, Gelfand-Mazur Theorem, Spectral Radius Formula

Commutative Banach Algebras, The Gelfand Transform, Applications to Fourier Transforms, Weiner's Theorem, Stone-Weierstrass Theorem

Compact and Fredholm Operators, Atkinson's Theorem, Index Theory

C* algebras, uniqueness of the norm, Commutative C* algebras, Gelfand-Naimark theorem, Spectral Mapping theorem

Functional Calculus, Positive Operators, Polar Decomposition

Weak and Strong Operator Topologies, Von Neumann Algebras, Double Commutant Theorem

Spectral measure, Spectral Theorem for Normal Operators, Borel Functional Calculus

Multiplicity Theory, Abelian Von Neumann Algebras, Classification of normal operators upto unitary equivalence

*Suggested Books*:

- G. J. Murphy,
*C* Algebras and Operator Theory*(Academic Press Inc, 1990) - J. B. Conway,
*A Course in Functional Analysis*(2^{nd}Ed) (Springer, 1990) - R. G. Douglas,
*Banach Algebra Techniques in Operator Theory*(2^{nd}Ed) (Springer, 1998) - K. R. Davidson,
*C* Algebras by Example*(Fields Institute Monograph, AMS 1996) - R. V. Kadison and J. R. Ringrose,
*Fundamentals of the Theory of Operator Algebras - Vol. I*(Academic Press Inc, 1983) - W. A. Arveson,
*A Short Course in Spectral Theory*(Springer 2002)

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