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Indian Institute of Science Education and Research Berhampur

Mathematics

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

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