Harmonic Analysis and Unitary Representations. Fall 2018

A. Yu. Pirkovskii

Syllabus

Lecture 1 (20.09.2018).

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.
References: [Havin, KirHarm, Mackey].
Exercises for Lecture 1

Lecture 2 (27.09.2018).

The Pontryagin duality for ℤ, 𝕋, and ℝ. The Stone-Weierstrass theorem (without proof). Harmonic analysis on ℤ and 𝕋: the Fourier transform, the Riemann-Lebesgue lemma, the Plancherel theorem, the uniqueness and density theorems, the Fourier inversion formulas. Convolution on 𝓁¹(ℤ) and L¹(𝕋), the Fourier transforms as algebra homomorphisms.
References: [EdFour, FollandReal, RudinReal, Havin, Deitmar, Zhel, NaimReps].
Exercises for Lecture 2

Lecture 3 (04.10.2018).

Harmonic analysis on ℝ (a survey): the Fourier transform, the Riemann-Lebesgue lemma, the Plancherel theorem, the uniqueness and density theorems, the Fourier inversion formula. Convolution on L¹(ℝ), the Fourier transform as an algebra homomorphism. Locally compact spaces, Radon measures, the Riesz-Markov-Kakutani theorem on positive functionals on C_c(X) (without proof).
References: [HelemFA, RudinFA, FollandReal, RudinReal, Hoerm1, Havin, Gurarii, Deitmar].
Exercises for Lecture 3

Lecture 4 (11.10.2018).

Locally compact groups. The Haar measure. An explicit construction of the Haar measure on a Lie group. Examples. Invariant measures and invariant functionals. The existence of a Haar measure on a locally compact group.
References: [DeitEcht, Folland, HR, Zhel, FD, FollandReal, Loomis, NaimRings, KnappLie, WeilInt, ZhShtern].
Exercises for Lecture 4

Lecture 5 (18.10.2018).

Uniformly continuous functions on topological groups. The uniqueness of the Haar measure. The modular character.
References: [DeitEcht, Folland, HR, Zhel, FD, Loomis, NaimRings, WeilInt].
Exercises for Lecture 5

Lecture 6 (26.10.2018).

The L¹-algebra of a locally compact group. Banach algebras, Banach *-algebras, C*-algebras. Examples. Approximate identities.
References: [DeitEcht, Folland, HR, Zhel, BourSpec, Dixmier, FD, Gurarii, HelemBA, HelemFA, Kaniuth, Loomis, Murphy, NaimRings, Palmer, Pir].
Exercises for Lecture 6

Lecture 7 (02.11.2018).

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, the spectral radius). Maximal ideals in commutative algebras. The maximal spectrum and the character space of a commutative unital algebra.
References: [DeitEcht, Folland, HR, Zhel, BourSpec, FD, Gurarii, HelemBA, HelemFA, Kaniuth, Loomis, Murphy, NaimRings, Palmer, Pir].

Lecture 8 (09.11.2018).

The closedness of maximal ideals of a unital Banach algebra. The 1-1 correspondence between the character space and the maximal spectrum of a unital commutative 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 (unital case). The maximal spectrum and the Gelfand transform for C(X). Functorial properties of the Gelfand transform (the adjoint functors X → C(X) and A → Max(A)).
References: [DeitEcht, Folland, HR, Zhel, BourSpec, FD, Gurarii, HelemBA, HelemFA, Kaniuth, Loomis, Murphy, NaimRings, Palmer, Pir].
Exercises for Lectures 7-8

Lecture 9 (16.11.2018).

Unitization. The unitization of C₀(X) and the one-point compactification of X. The spectrum of an element of a nonunital algebra. Modular ideals. The maximal spectrum and the Gelfand transform for nonunital Banach algebras. Products and unitizations of C*-algebras.
References: [DeitEcht, Folland, HR, Zhel, BourSpec, FD, Gurarii, HelemBA, Kaniuth, Loomis, Murphy, NaimRings, Palmer, Pir].
Exercises for Lecture 9

Lecture 10 (23.11.2018).

The spectral radius of a selfadjoint element. The uniqueness of a C*-algebra norm. The automatic continuity of *-homomorphisms to C*-algebras. 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. C*-envelopes of Banach *-algebras. An explicit construction of C*(A) in the case where A is commutative.
References: [DeitEcht, Folland, HR, Zhel, BourSpec, FD, Gurarii, HelemBA, Kaniuth, Loomis, Murphy, NaimRings, Palmer, Pir].
Exercises for Lecture 10

Lecture 11 (30.11.2018).

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.
References: [DeitEcht, Folland, HR, BourSpec, FD, Gurarii, Kaniuth, Loomis, NaimRings].

Lecture 12 (07.12.2018).

The group C*-algebra of a locally compact group. The isomorphism between C*(G) and C₀(Ĝ) for a LCA group G. The Plancherel theorem: construction of the Plancherel measure on Ĝ.
References: [DeitEcht, Folland, HR, BourSpec, FD, Gurarii, Kaniuth, Loomis, NaimRings].

Lecture 13 (14.12.2018).

The Plancherel theorem (end of proof). The Pontryagin duality.
References: [DeitEcht, Folland, HR, BourSpec, FD, Gurarii, Kaniuth, Loomis, NaimRings].
Exercises for Lectures 11-13

References (in Russian)