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MAT 1011HF (MAT495H1F) INTRODUCTION TO LINEAR OPERATORS G. A. Elliott MWF 2 in HU 1018, 215 Huron St. **Topics (and cross-listed)****: **The course will survey the branch of mathematics developed (in its abstract form) primarily in the twentieth century and referred to variously as functional analysis, linear operators in Hilbert space, and operator algebras, among other names (for instance, more recently, to reflect the rapidly increasing scope of the subject, the phrase non-commutative geometry has been introduced). The intention will be to discuss a number of the topics in Pedersen's textbook Analysis Now. Students will be encouraged to lecture on some of the material, and also to work through some of the exercises in the textbook (or in the suggested reference books). *Prerequisites: * Elementary analysis and linear algebra (including the spectral theorem for self-adjoint matrices). *Textbook: * Gert K. Pedersen, Analysis Now *Recommended references: * Paul R. Halmos, A Hilbert Space Problem Book Mikael Rordam, Flemming Larsen, and Niels J. Laustsen, An Introduction to K-Theory for C*-Algebras Bruce E. Blackadar, Operator Algebras: Theory of C*-Algebras and von Neumann Algebras *MAT1016HS TOPICS IN OPERATOR ALGEBRAS: K-THEORY AND C*-ALGEBRAS G. A. Elliott MWF 2 in Room 1018, 215 Huron St. The theory of operator algebras was begun by John von Neumann eighty years ago. In one of the most important innovations of this theory, von Neumann and Murray introduced a notion of equivalence of projections in a self-adjoint algebra (*-algebra) of Hilbert space operators that was compatible with addition of orthogonal projections (also in matrix algebras over the algebra), and so gave rise to an abelian semigroup, now referred to as the Murray-von Neumann semigroup. Later, Grothendieck in geometry, Atiyah and Hirzebruch in topology, and Serre in the setting of arbitrary rings (pertinent for instance for number theory), considered similar constructions. The enveloping group of the semigroup considered in each of these settings is now referred to as the K-group (Grothendieck's terminology), or as the Grothendieck group. Among the many indications of the depth of this construction was the discovery of Atiyah and Hirzebruch that Bott periodicity could be expressed in a simple way using the K-group. Also, Atiyah and Singer famously showed that K-theory was important in connection with the Fredholm index. Partly because of these developments, K-theory very soon became important again in the theory of operator algebras. (And in turn, this theory became increasingly important in other branches mathematics.) The purpose of this course is to give a general, elementary, introduction to the ideas of K-theory in the operator algebra context. (Very briefly, K-theory generalizes the notion of dimension of a vector space.) The course will begin with a description of the method (K-theoretical in spirit) used by Murray and von Neumann to give a rough initial classification of von Neumann algebras (into types I, II, and III). It will centre around the relatively recent use of K-theory to study Bratteli's approximately finite-dimensional C*-algebras---both to classify them (a result that can be formulated and proved purely algebraically), and to prove that the class of these C*-algebras---what Bratteli called AF algebras---is closed under passing to extensions (a result that uses the Bott periodicity feature of K-theory). Students (undergraduate students are welcome) will be encouraged to prepare oral or written reports on various subjects related to the course, including both basic theory and applications. *Prerequisites:* An attempt will be made to supply the necessary prerequisites when needed (rather few, beyond just elementary algebra and analysis). *Textbook: * Mikael Rordam, Flemming Larsen, and Niels J. Laustsen, An Introduction to K-Theory for C*-Algebras *Recommended References: * Edward G. Effros, Dimensions and C*-Algebras Bruce E. Blackadar, Operator Algebras: Theory of C*-Algebras and von Neumann Algebras

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