Harmonic Analysis and Unitary Representations. Fall 2022

A. Yu. Pirkovskii

Syllabus

Lecture 1 (06.09.2022).

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.
Exercises for Lecture 1

Lecture 2 (13.09.2022).

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.

Lecture 3 (20.09.2022).

The Stone-Weierstrass theorem (without proof). Corollaries: the Weierstrass approximation theorems, the Riemann-Lebesgue lemma for the Fourier transform on T. 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.
Exercises for Lectures 2 and 3

Lecture 4 (27.09.2022).

The Fourier transform on R. The Riemann-Lebesgue lemma. Convolution on L¹(R), the Fourier transform as an algebra homomorphism. Basic results of harmonic analysis on R: the uniqueness and density theorems, the Plancherel theorem, the inversion formula. Ingredients of the proofs: the Schwartz space, the Fourier transform as a topological automorphism of the Schwartz space, the Fourier transform of tempered distributions. Locally compact topological spaces. Radon measures.
Exercises for Lecture 4

Lecture 5 (04.10.2022).

The Riesz-Markov-Kakutani theorem on positive functionals on C_c(X) (without proof). Locally compact groups. Uniformly continuous functions on topological groups. The uniform continuity of compactly supported continuous functions. The Haar measure. An explicit construction of the Haar measure on a Lie group. Examples.
Exercises for Lecture 5

Lecture 6 (11.10.2022).

The existence of a Haar measure on a locally compact group.

Lecture 7 (18.10.2022).

The uniqueness of the Haar measure. The modular character.
Exercises for Lectures 6 and 7

Lecture 8 (25.10.2022).

Continuous representations of topological groups. Some continuity criteria for representations. Unitary representations. The left and right regular representations of a locally compact group G on L^p(G).

Lecture 9 (01.11.2022).

Algebraically irreducible representations. Schur's lemma. Irreducible (=topologically irreducible) representations of topological groups. An infinite-dimensional version of Schur's lemma for unitary representations (statement only). A corollary: irreducible unitary representations of a topological abelian group are 1-dimensional. The invariance of the orthogonal complement of an invariant subspace for a unitary representation. A corollary: decomposing finite-dimensional unitary representations into irreducibles. The unitary dual of a locally compact group. The Fourier transform. The Gelfand-Dunford-Pettis integral for vector-valued functions. The existence and the basic properties of the integral. The uniform boundedness of continuous representations of compact groups.
Exercises for Lectures 8 and 9

Lecture 10 (08.11.2022).

The averaging procedure for compact groups. The averaging projection of B(E,F) onto Hom_G(E,F). The unitarizability of representations of compact groups on Hilbert spaces. The averaging projection preserves positivity and compactness of operators. Decomposing unitary representations of compact groups into finite-dimensional irreducibles (the Peter-Weyl theorem). A corollary: unirreps of a compact group G are finite-dimensional and separate the points of G.

Lecture 11 (15.11.2022).

Basic constructions of finite-dimensional G-modules. Matrix elements of representations. The fundamental map E⊗E* = End(E) → C(G). Characterizations of representative functions. The algebra R(G) of representative functions. The density of R(G) in C(G) for a compact group G (the Peter-Weyl approximation theorem).
Exercises for Lectures 10 and 11

Lecture 12 (22.11.2022).

The orthogonality relations for matrix elements of unirreps of a compact group. The structure of R(G) and of L²(G).

Lecture 13 (29.11.2022).

The Fourier transform on a compact group. The uniqueness theorem. Complex conjugate representations. An explicit formula for the Fourier transform for matrix elements of unirreps. A corollary: the Riemann-Lebesgue lemma for compact groups.
Exercises for Lectures 12 and 13

Lecture 14 (06.12.2022).

The Fourier cotransform and the Plancherel theorem for compact groups. Banach algebras, Banach *-algebras, C*-algebras: definitions and basic examples.

Lecture 15 (13.12.2022).

The L¹-algebra of a locally compact group. Approximate identities in Banach algebras. Dirac nets in L¹(G). A survey of Gelfand's theory of commutative Banach algebras (maximal ideals and characters, the Gelfand spectrum, the Gelfand transform, spectral properties of elements in C*-algebras, hermitian Banach *-algebras, the commutative Gelfand-Naimark theorem).
Exercises for Lectures 14 and 15

Lecture 16 (20.12.2022).

The dual of a locally compact abelian (LCA) group. The Fourier transform on LCA groups. Basic properties of the Fourier transform. L¹(G) is hermitian. The homeomorphism between the dual group and the Gelfand spectrum of L¹(G). The identification of the L¹-Fourier transform with the Gelfand transform of L¹(G). The uniqueness and density theorems for the L¹-Fourier transform. The Plancherel theorem and the Pontryagin duality (a survey).
Exercises for Lecture 16

References (under construction)