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Indian Institute of Science Education and Research Bhopal
MTH 615: 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 (2nd Ed) (Springer, 1990)
- R. G. Douglas, Banach Algebra Techniques in Operator Theory (2nd 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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