Ilijas Farah's teaching.

**Functional Analysis II (Math 6462), winter 2022. **

**Course Information**

Lectures: MW 11:30--1:00pm (EST, aka `Toronto time')

The classes will be meeting online for the time being; hopefully we will move to the in-person instruction, the situation permitting. Stay tuned. If you are interested in auditing the course, please email me for a Zoom link.

Text: William Arveson, A short course on spectral theory, Springer Graduate Texts in Mathematics 209.

The plan is to cover most of the book, time permitting. More precisely, we will cover the following topics (the *italicized topics* are optional). The following plan may be too ambitious; if needed, we will slow down.

The spectrum of an operator. Banach algebras. Examples of Banach algebras. The regular representation of a Banach algebra. The general linear group of a Banach algebra. The spectrum of an element of a Banach algebra. Spectral radius. Ideals and quotients of Banach algebras. Commutative Banach algebras (Gelfand duality). *C(X) and the Wiener algebra*. Spectral permanence theorem. *Analytic functional calculus. *

Operators on Hilbert space. C*-algebras. Commutative C*-algebras (Gelfand-Naimark duality). Continuous functional calculus. The Spectral Theorem. Representations of Banach *-algebras. Borel functional calculus. Spectral measures. Compact operators (Hilbert-Schmidt operators, the trace). Unitizations of C*-algebras. Ideals and quotients of C*-algebras.

Asymptotics: Compact perturbations, Fredholm theory. The Calkin algebra. Riesz theory and compact operators (Fredholm alternative, spectral theory). Fredholm operators. The Fredholm index.

* Methods and applications. Maximal abelian von Neumann algebras. Toeplitz matrices and Toeplitz operators. The Toeplitz C*-algebra. Index theorem for continuous symbols. Some function theory in the Hardy space H^2. Spectra of Toeplitz operators with continuous symbol. States. The GNS construction. The Gelfand-Naimark(-Segal) theorem. *

**Prerequisites for this course** include, in addition to basic results of functional analysis (core results from Math 6461), the following. (All of the advanced necessary results will be briefly covered in lectures. `Advanced' is defined as `results that the students are not familiar with'.)

**Basic measure theory** (Borel sets. Borel measures. Radon measures - finite, sigma-finite, positive, signed, complex. Outer measures. Caratheodory Extension Theorem. Haar measures on locally compact groups. Absolute continuity of measures. Measure equivalence. Riesz representation theorem for functionals on C(X). Dominated Convergence Theorem. We may occasionally need other results, such as the Radon-Nikodym Theorem.)

**Basic complex analysis** (Analytic/holomorphic functions, winding number, Liouville's Theorem, possibly the Cauchy Integral Theorem.)

**Very basic algebra** (groups and such).

##### Homeworks

The first homework, due January 31

The first homework, corrected. (There was a typo in question 4c; my apologies for the inconvenience.)

Homework 1 solutions

The second homework, due February 13 (updated February 7, twice; thanks Daniel for pointing out to a few typos).

In Question 3 of Homework 2, assume that the homomorphism is surjective. For bonus points, figure out whether this assumption is necessary.

Homework 4 (corrected-there was a typo in Question 4), due March 28

##### Class recordings

Class 8, February 2, recording

Class 9, February 7, recording.

Class 10, February 9, recording.

Class 11, February 14, recording.

Class 12, February 16, recording.

Class 13, February 28, recording.

** There will be no class on March 9. The makeup class will take place on Friday, March 11, at 11:30-13:00 at Ross N627.**

** The life is back to normal. All classes from March 11 on will be taking place in person and they will not be broadcast or recorded.**

**Evaluation:**

Four graded assignments, dates to be announced, 20% each.

Short in-class presentation of a topic of choice (a list of suggested topics will be available soon), 20%.