Harmonic Analysis and Banach Algebras. Fall 2024
Syllabus
Lectures
Playlist
- Lecture 1 (06.09.2024) (video)
- Historical remarks on harmonic analysis and unitary
representations. The Pontryagin duality for finite abelian groups.
Harmonic analysis on a finite abelian group: the Fourier transform,
the Plancherel theorem, the Fourier inversion formula, the convolution, the Fourier transform as
an algebra homomorphism.
- Lecture 2 (13.09.2024) (video)
- The compact-open topology on the dual groups of Z, T, and R.
The isomorphisms Ẑ ≅ T, T̂ ≅ Z,
R̂ ≅ R. The Pontryagin duality for Z, T, and R.
The Fourier transform on Z and T. The Stone-Weierstrass theorem (without proof). Corollaries: the Weierstrass approximation
theorems, the Riemann-Lebesgue lemma for the Fourier transform on T.
- Lecture 3 (20.09.2024) (video)
- Convolution on 𝓁1(Z) and L1(T).
The Fourier transforms as algebra homomorphisms. The Plancherel theorem for the Fourier transforms on Z and T.
The uniqueness and density theorems, the inversion formulas, the Fourier algebras on Z and T.
The Fourier transform on R. The Riemann-Lebesgue lemma. Convolution on L1(R),
the Fourier transform as an algebra homomorphism.
Statements of the basic results of harmonic analysis on R: the uniqueness and density theorems, the Plancherel theorem,
the inversion formula.
- Lecture 4 (27.09.2024) (video)
(lecture notes)
- Harmonic analysis on R: the Schwartz space, the Fourier transform as a topological automorphism
of the Schwartz space, the Fourier transform of tempered distributions. Proofs of the uniqueness theorem, the density theorem,
and the Plancherel theorem for the Fourier transform on R. Locally compact topological spaces. Radon measures.
The Riesz-Markov-Kakutani theorem on positive functionals on Cc(X) (without proof).
Locally compact groups. Uniformly continuous functions on topological groups.
- Lecture 5 (04.10.2024) (video)
(lecture notes)
- The uniform continuity of compactly supported continuous functions on a locally compact group. The Haar measure.
An explicit construction of the Haar measure on a Lie group. Examples.
- Lecture 6 (11.10.2024) (video)
(lecture notes)
- The existence of a Haar measure on a locally compact group. The uniqueness of a Haar measure.
- Lecture 7 (18.10.2024) (video)
(lecture notes)
- The modular character and related integration formulas. Unimodular locally compact groups. Banach algebras, Banach *-algebras,
C*-algebras: definitions and basic examples.
- Lecture 8 (25.10.2024) (video)
- The L1-algebra of a locally compact group. Complex measures. The Riesz-Markov-Kakutani theorem on
bounded functionals on C0(X) (without proof). The measure algebra of a locally compact group.
An embedding of L1(G) into M(G).
- Lecture 9 (01.11.2024) (video)
(lecture notes)
- Approximate identities in normed algebras. Examples. Dirac nets in L1(G).
A survey of spectral theory in Banach algebras (the spectrum
of an algebra element, algebraic properties of the spectrum, properties of the multiplicative group of a Banach
algebra, the continuity of characters, the compactness and the nonemptiness of the spectrum,
the Gelfand-Mazur theorem).
- Lecture 10 (08.11.2024) (video)
(lecture notes)
- The spectral radius formula. The maximal spectrum and the character space of a commutative unital algebra.
The closedness of maximal ideals of a commutative unital Banach algebra.
The 1-1 correspondence between the character space and the maximal spectrum of a commutative unital Banach algebra.
Some properties of the weak* topology.
The Gelfand topology on the maximal spectrum. The compactness of the maximal spectrum (the unital case).
The Gelfand transform of a commutative unital Banach algebra. Remarks on the Jacobson radical. The maximal spectrum
and the Gelfand transform for subalgebras of C(X). Functorial properties of the Gelfand transform
(the adjoint functors X → C(X) and A → Max(A)).
- Lecture 11 (15.11.2024) (video)
(lecture notes)
- Unitization. The unitization of C0(X) and the one-point compactification of X. The nonunital spectrum
of an algebra element. Relations between the unital and nonunital spectra. Modular ideals. The maximal spectrum
and the Gelfand transform for nonunital Banach algebras. Products and unitizations of C*-algebras.
- Lecture 12 (22.11.2024) (video)
(lecture notes)
- Spectral properties of C*-algebras: the spectral radius of a normal element; the automatic continuity of *-homomorphisms;
the spectrum of a selfadjoint element. Hermitian *-algebras. A characterization of hermitian commutative Banach *-algebras.
The 1st (commutative) Gelfand-Naimark theorem. A category-theoretic interpretation of the Gelfand-Naimark theorem.
The dual of a locally compact abelian (LCA) group. The Pontryagin topology and the continuity of multiplication
on the dual group. The Fourier transform of complex Radon measures and of integrable functions
on LCA groups.
- Lecture 13 (29.11.2024) (video)
(lecture notes)
- Basic properties of the Fourier transform on LCA groups. A homeomorphism between the dual group
and the Gelfand spectrum of L1(G). Corollaries: L1(G) is hermitian; the dual group is locally compact.
The identification of the L1-Fourier transform with the Gelfand transform of L1(G).
The density theorem for the L1-Fourier transform.
- Lecture 14 (06.12.2024) (video)
- The uniqueness theorem for the L1-Fourier transform. A corollary: unitary characters separate the points.
Positive functionals on *-algebras. Positive definite functions on groups. Examples. Positive definite functions as positive functionals
on the group algebra. A characterization of positive definite functions on finite abelian groups. Functions of positive type
on locally compact groups as positive functionals on L1-group algebras. A continuous bounded function is
of positive type iff it is positive definite.
- Lecture 15 (13.12.2024) (video)
- The canonical map G → G^^. The dual Fourier transform of Radon measures. The injectivity of the dual Fourier transform.
Bochner's theorems for hermitian Banach algebras and for LCA groups. The Fourier and Fourier-Stieltjes algebras of an LCA group.
The Fourier inversion formula (statement only). Examples of dual (Plancherel) measures.
- Lecture 16 (20.12.2024) (video)
- The Fourier inversion formula and the Plancherel theorem for LCA groups. The Pontryagin duality (a sketch).
References