Proposal for Advanced Instructional School. ___________________________________________ Period : .February 1.. to .February 21.., 2015. Title : Advanced Instructional School on “Operator Theory/Algebras”. Venue : Indian Institute of Mathematical Sciences (IMSc) C.I.T. Campus, 4th Cross Street Taramani, Chennai Tamil Nadu – 600 113 ___________________________________________ Names of Organisers (with full address, email, tel. No. Etc.) : V. S. Sunder Indian Institute of Mathematical Sciences (IMSc) C.I.T. Campus, 4th Cross Street Taramani, Chennai Tamil Nadu – 600 113 ....................................... Email : ..sunder@imsc.res.in... Tel. No. ..91-44-22543210.. Vijay Kodiyalam Indian Institute of Mathematical Sciences (IMSc) C.I.T. Campus, 4th Cross Street Taramani, Chennai Tamil Nadu – 600 113 ...................................... Email : ..vijay@imsc.res.in.. Tel. No. ..91-44-22543212.. ___________________________________________ Email address to which applications can be sent :..sunder@imsc.res.in.. Common email address, if any : ................................................... Number of Participants : at most 20 from outside Chennai ___________________________________________ Brief description about the school for publicity : This school is intended for people in the second year of graduate school (M.Sc. or Ph.D.) who have had a first course in functional analysis (and know such things as Hahn-banach Theorem, the Priinciple of Uniform Boundedness and the Open Mapping Theorem). The goals of this school are manifold: (i) to teach some rudimentary theory of operators on Hilbert space, culminating in a formulation of the Spectral Theorem for bounded self-adjoint operators (and for commuting families of normal operators, if time permits) in terms of the existence and uniqueness of appropriate continuous and measurable functional calculii; (ii) to give a quick crash course on the basics of C*-algebras; (iii) to give a quick crash course on the basics of von Neumann algebras; (iv) to introduce students to the fundamentals of Hilbert C*-modules; (v) to introduce students to the fundamentals of free probability, the combinatorics of non-crossing partitions, and the connections with random matrix theory; and (vi) to give a brief introduction to the K-theory of C^*-algebras. For the uninitiated, already at the level of (i) above, one sees the importance of the topological and probabilistic viewpoints; the `non-commutative' counterparts of these two aspects are C*-algebras and von Neumann algebras. While (ii) and (iii) may be regarded as dealing with the fundamentals of `non-commutarive' general topology and meausre theory respectively, (iv) and (vi) may be viewed as the beginnings of non-commutative algebraic topology, and (v) as non-commutative probability theory. Interested people should send an email to sunder2imsc.res.in with a short description of their level of preparedness to benefit from this programme, including what chapters of what books on functional analysis/Hilbert space theory they have read. ___________________________________________ Speakers(affiliations) - no. of lecture hours - detailed syllabus V.S. Sunder (IMSc)) - 6 x 1.5 : sp. thm. for s.a operators on Hilbert spaces, bounded operators, self-adjoint operators, continuous and measurable functional calculii, relation to conventional formulation in terms of spectral measures, extension to normal operators (if time permits). Ref: V.S. Sunder, Operators on Hilbert space. Kunal Krishna Mukherjee (IIT-M): 6 x 1.5 hrs C* algebras, spectral radius formula, spectrum, Gelfand-Naimark for commutative C* algebras Spectrum of self-adjoints, unitaries, continuous functions of self-adjoints and unitaries, *-homomophisms, automatic continuity, independence of spectrum relative to subalgebras. Order structure, square roots, ideals, quotients. States, faithfulness, decomposition of functionals, pure states etc. Representations, GNS construction. Examples - Compact operators, Calkin algebra, Group C* algebras, tensor products, crossed products (if time permits) References - Functional Analysis, Sunder Operator Algebras I, Kadison-Ringrose C* algebras and Finite Dimensional Approximations, Brown-Ozawa R. Srinivasan (CMI) von Neumann algebras. 1. weak, strong topologies, double commutant theorem 2. and 3. projections and types of von Neumann algebras, (type I is B(H), finiteness is equivalent to existence of finite normal trace, semi-finite von Neumann algebras) 4, 5, 6 standard form and Tomita-Takesaki theorem and modulr theory. Ref: Dixmier, Jacques, von Neumann algebras Serban Stratila and Laszlo Zsido, Lectures on Von Neumann algebras. Takesaki, M., Tomita's theory of modular Hilbert algebras and its applications, Lecture Notes Math. Bratteli, O.; Robinson, D.W., Operator Algebras and Quantum Statistical Mechanics Partha Sarathi Chakraborty (IMSc) - 6x1.5 - Hilbert $C^*$-modules 1) Hilbert C* modules 2) Adjointable linear maps 3) Multiplier Algebras 4) Kasparov Stabilization 5) KSGNS theorem 6) Induced representations for C*-algebras. If time permits then I may discuss regular operators. Text book: Lance/ Raeburn-Williams/ Jensen-Thomsen. Vijay Kodiyalam (IMSc): Free probability The emphasis of this course will be on the combinatorics of free probability theory. We will begin with non-commutative probability spaces and distributions and the concepts of free independence and free product of these. Next we will discuss free cumulants and the combinatorics of non-crossing partitions. We conclude with the connection between ,free probability theory and random matrices. Ref: Nica, Speicher: Lectures on the combinatorics of free probability. S. Sundar (CMI): K-theory of $C^*$-algebras Brief discussion of the Serre-Swan theorem. Definition of K_{0} and K_{1}, homotopy invariance, short-exactness of K-theory, Bott periodicity, six term exact sequence. Ref: 1. K-theory for operator algebras -- Bruce Blackadar 2. An introduction to the K-theory for C*-algebras -- Rordam, Larsen and Laustsen. 3. K-theory and C*-algebras (a friendly approach) - Wegge Olsen.