Graduate Blog

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