Harmonic Analysis and Unitary Representations. Spring 2016

A. Yu. Pirkovskii

Syllabus

Lecture 1 (20.01.2016).

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 (updated 18.02.2016)

Lecture 2 (27.01.2016).

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.
References: [EdFour, FollandReal, RudinReal, Havin, Deitmar, Zhel, NaimReps].
Exercises for Lecture 2

Lecture 3 (03.02.2016).

Harmonic analysis on Z and T: the uniqueness theorems and the Fourier inversion formulas, the convolution on l^1(Z) and on L^1(T), the Fourier transforms on Z and T as algebra homomorphisms, the Fourier algebras A(Z) and A(T). Harmonic analysis on R: the Fourier transform, the Riemann-Lebesgue lemma, the convolution on L^1(R), the Fourier transform as an algebra homomorphism, the Fourier algebra A(R), the Fourier transform of the derivative and the derivative of the Fourier transform.
References: [EdFour, FollandReal, RudinReal, Havin, Deitmar, HelemFA, RudinFA, Hoerm1, Gurarii].

Lecture 4 (10.02.2016).

Harmonic analysis on R: the Fourier transform as an automorphism of the Schwartz space, the Parseval-Plancherel identities, the Fourier transform of tempered distributions, the uniqueness theorem, the Fourier inversion formula, the Plancherel theorem on R.
References: [HelemFA, RudinFA, FollandReal, RudinReal, Hoerm1, Havin, Gurarii, Deitmar].
Exercises for Lectures 3 and 4 (updated 05.05.2016)

Lecture 5 (17.02.2016).

Locally compact spaces, Urysohn's lemma (without proof), Radon measures, the Riesz-Markov-Kakutani theorem on positive functionals on C_c(X) (without proof). Locally compact groups, examples, 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.
References: [DeitEcht, HR, FD, FollandReal, Loomis, NaimRings, RudinReal, KnappLie, ZhShtern].
Exercises for Lecture 5

Lecture 6 (24.02.2016).

The Haar measure on some Lie groups. Invariant measures and invariant functionals. The existence of a Haar measure on a locally compact group.
References: [DeitEcht, Folland, HR, Zhel, BourInt68, FD, FollandReal, Loomis, NaimRings, WeilInt, ZhShtern].

Lecture 7 (09.03.2016).

The uniqueness of the Haar measure. The modular character. Continuous representations of topological groups. Some continuity criteria for representations.
References: [DeitEcht, Folland, HR, Zhel, BourInt68, FD, FollandReal, Loomis, NaimRings, WeilInt, NaimReps].
Exercises for Lectures 6 and 7

Lecture 8 (16.03.2016).

Unitary representations. The left and right regular representations of a locally compact group G on L^p(G). Irreducible representations. The unitary dual of a locally compact group. The Fourier transform.
References: [DeitEcht, Folland, Zhel, BourInt68, NaimReps, KirReps, KirHarm].
Exercises for Lecture 8

Lecture 9 (23.03.2016).

Banach algebras, Banach *-algebras, C*-algebras. Examples. Products of Radon measures. Convolution of functions on a locally compact group. The group Banach algebra L¹(G). Complex measures. The variation of a complex measure. The Jordan decomposition of real measures. Integration w.r.t. a complex measure. Complex Radon measures. The Riesz-Markov-Kakutani theorem on bounded functionals on C₀(X) (without proof). Products of complex Radon measures.
References: [DeitEcht, Folland, HR, Zhel, BourInt68, BourSpec, FD, FollandReal, Gurarii, HelemBA, HelemFA, Kaniuth, KirReps, Loomis, Murphy, NaimRings, Palmer, Pir, RudinFA].

Lecture 10 (30.03.2016).

The measure algebra M(G) of a locally compact group G. The embedding of L¹(G) into M(G). The structure of M(G). Approximate identities in Banach algebras. Examples. Dirac nets in L¹(G). Representations of Banach algebras.
References: [Folland, HR, BourInt68, FD, FollandReal, HelemBA, Kaniuth, KirReps, Palmer, RudinReal].

Lecture 11 (06.04.2016).

The Gelfand-Dunford-Pettis integral. Relations between representations of a locally compact group G and representations of the group algebras L¹(G) and M(G).
References: [DeitEcht, Folland, HR, Zhel, DiestUhl, KirReps, Loomis, NaimRings, Palmer].
Exercises for Lectures 9-11

Lecture 12 (13.04.2016).

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 unital commutative algebra. 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).
References: [DeitEcht, Folland, HR, Zhel, BourSpec, FD, Gurarii, HelemBA, HelemFA, Kaniuth, Loomis, Murphy, NaimRings, Palmer, Pir, RudinFA].

Lecture 13 (20.04.2016).

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)). 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.
References: [DeitEcht, Folland, HR, Zhel, BourSpec, FD, Gurarii, HelemBA, Kaniuth, Loomis, Murphy, NaimRings, Palmer, Pir, RudinFA].

Lecture 14 (27.04.2016).

Products and unitizations of C*-algebras. 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. The endomorphism algebra of an irreducible *-module (an infinite-dimensional version of Schur's lemma). A corollary: all irreducible unitary representations of a locally compact abelian group are 1-dimensional.
References: [DeitEcht, Folland, HR, Dixmier, Zhel, BourSpec, FD, Gurarii, HelemBA, Kaniuth, Loomis, Murphy, NaimRings, Palmer, Pir, RudinFA].
Exercises for Lectures 12-14

Lecture 15 (11.05.2016).

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 theorem for the L¹-Fourier transform.
References: [DeitEcht, Folland, HR, BourSpec, FD, Gurarii, Kaniuth, Loomis, NaimRings].

Lecture 16 (18.05.2016).

C*-envelopes of Banach *-algebras. An explicit construction of C*(A) in the case where A is commutative. The group C*-algebra of a locally compact group. The isomorphism between C*(G) and C₀(G^) for a LCA group G. Remarks on reduced group C*-algebras. The Plancherel theorem: construction of the Plancherel measure on G^.
References: [DeitEcht, Folland, HR, BourSpec, FD, Gurarii, Kaniuth, Loomis, NaimRings].

Lecture 17 (25.05.2016).

The Plancherel theorem on LCA groups (cont'd). The canonical map from G to G^^. A weak version of the Fourier inversion formula.
References: [DeitEcht, Folland, HR, BourSpec, FD, Gurarii, Kaniuth, Loomis, NaimRings].

Lecture 18 (01.06.2016).

The regularity of L¹(G). The Pontryagin duality. The Fourier algebra and the Fourier-Stieltjes algebra of a LCA group. The Fourier inversion formula.
References: [DeitEcht, Folland, HR, BourSpec, FD, Gurarii, Kaniuth, Loomis, NaimRings].
Exercises for Lectures 15-18

Lecture 19 (08.06.2016).

The uniform boundedness of continuous representations of compact groups. 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 are finite-dimensional and separate the points of G.
References: [DeitEcht, Folland, HR, FD, HofMorr, Robert, Zhel, EdComp, NaimRings, NaimReps].

Lecture 20 (15.06.2016).

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). The structure of R(G) and of L²(G). Decomposing the regular representations into irreducibles.
References: [DeitEcht, Folland, HR, FD, HofMorr, Robert, Zhel, EdComp, NaimRings, NaimReps].

Lecture 21 (27.06.2016).

The Fourier transform on compact groups. The uniqueness theorem. The Fourier transform of matrix elements of irreducible representations. The dual Fourier transform. The Plancherel theorem for compact groups. Coalgebras, bialgebras, Hopf algebras, Hopf *-algebras. Examples. The Hopf *-algebra structure on the algebra of representative functions. Comodules. An equivalence between G-modules and R(G)-comodules. The group X_*(A) of *-characters of a Hopf *-algebra A. The adjoint pair of functors G → R(G) and A → X_*(A). Integrals on Hopf algebras. The compactness of the group X_*(A) for a Hopf *-algebra A with a positive integral. The functors G → R(G) and G → Rep(G) are conservative. The Tannaka-Krein reconstruction theorem in terms of R(G). The Tannaka group of tensor-preserving self-conjugate transformations of the forgetful functor Rep(G) → Vect. The Tannaka-Krein reconstruction theorem in terms of Rep(G). The Fourier transform as an isomorphism between the two versions of the Tannaka-Krein theorem.
References: [Folland, EdComp, JS, Hoch, Abe].

References (in Russian)