The lecture titles below approximate what might happen in each class. I will adjust them as time goes on.
Tuesday 13 January
Hilbert spaces
Definition of a Hilbert space, examples, results from Fourier theory and Sturm-Liouville theory, orthonormal sets, orthonormal bases. Closed convex sets, orthogonal complements, basis expansions.
Background reading: Terry Tao's notes on Hilbert spaces, or lectures 14 and 15 here, or the first five sections in Chapter 3 of Melrose's notes, or Chapter 7 of Shapiro's notes, or ...
Thursday 15 January
Projection operators and introduction to the spectral theorem
Projection operators . Integral operators and Sturm-Liouville theory.
Background reading: Sections 6-11 of Chapter 3 in Melrose's notes.
Tuesday 20 January
The spectrum of a bounded operator
Compact operators, Hilbert’s spectral theorem.
Background reading: Section 3.18 of Melrose's notes.
First homework assigned. Please see these instructions, along with this set of sample solutions along with the TeX file used to create it (and an image file that you'll need to run TeX on the solutions file without generating an error).
Thursday 22 January
Spectrum for bounded operators and elements in Banach algebras
Spectrum of a bounded operator. Basic properties of the spectrum. Examples: compact operators, unitaries, isometries. Banach algebras.
Background reading: John Roe’s notes, Lecture 2. Ian Putnam’s notes, Chapter 1.
Tuesday 27 January
The spectral radius formula
First homework due. Second homework assigned.
Background reading: See this note on the spectral radius formula.
Thursday 29 January
The Gelfand-Naimark theory for commutative Banach algebras
Background reading: John Roe’s notes, Lecture 4. Ian Putnam’s notes, Chapter 1.
Tuesday 03 February
The Gelfand-Naimark theory for commutative Banach algebras, continued
Second homework due. Third homework assigned.
Background reading: John Roe’s notes, Lecture 5. Ian Putnam’s notes, Chapter 1.
Thursday 05 February
A look at the spectral theorem
Background reading: Jacob Lurie(!) has a nice discussion of spectral measures, although in a more general context than we discussed. This is part of a whole course on von Neumann algebras. See these notes of mine, or these other notes, for more on the spectral theory of Sturm-Liouville operators. We shall return to this topic soon, when we come to consider unbounded self-adjoint operators.
Tuesday 10 February
Step one of the GNS construction: positive elements
Third homework due. Fourth homework assigned.
Background reading:
Thursday 12 February
More on positive elements, including approximate units
Background reading: Section 1.6 of Ian Putnam’s notes and/or Lecture 7 of John Roe’s notes and/or Section 2 of Dana William’s notes
Tuesday 17 February
The GNS construction, part two
Fourth homework due. Fifth homework assigned.
Background reading: Section 4 of Dana William’s notes and/or Section 1.12 of Ian Putnam’s notes.
Thursday 19 February
The C-algebra of compact operators
*Background reading: My lecture notes on the compact operators and the Toeplitz algebra
Tuesday 24 February
The Toeplitz algebra
Fifth homework due. Sixth homework assigned.
Background reading: My lecture notes on the compact operators and the Toeplitz algebra
Thursday 26 February
Representations of ideals and quotients
Background reading: My lecture notes on the compact operators and the Toeplitz algebra
Tuesday 03 March
The Toeplitz algebra, concluded
Sixth homework due.
Background reading: My lecture notes on the compact operators and the Toeplitz algebra
Thursday 05 March
No class today
Tuesday 10 March
Spring Break
Thursday 12 March
Spring Break
Tuesday 17 March
Sturm-Liouville operators, after Weyl, part one
Thursday 19 March
Sturm-Liouville operators, after Weyl, part two
Tuesday 24 March
The Stone-von Neumann theorem, part one
Seventh homework assigned.
Thursday 26 March
The Stone-von Neumann theorem, part two
Tuesday 31 March
Recap
Seventh homework due. Eighth homework assigned.
Thursday 02 April
The canonical anti-commutation relations
Tuesday 07 April
The CAR algebra, part one
Eighth homework due. Ninth homework assigned.
Thursday 09 April
The CAR algebra, part two
Tuesday 14 April
The reduced group C*-algebra, part one
Ninth homework due. Tenth homework assigned.
Thursday 16 April
The reduced group C*-algebra, part two
Tuesday 21 April
Powers' theorem
Tenth homework due. Eleventh homework assigned.
Thursday 23 April
The irrational rotation algebra, part one
Tuesday 28 April
The irrational rotation algebra, part two
Eleventh homework due.
Thursday 30 April
Recap