RTNCG before 1950

RTNCG before 1950: further reading

This page has a few references for the mini series of lectures given in September and October 2022. This is not at all meant to be exhaustive.

General references

G. Mackey, Harmonic analysis as the exploitation of symmetry: a historical survey; Bull. AMS (NS), Vol. 3, N. 1 (1980), pp. 543—698.
J. Dieudonné, History of functional analysis, North-Holland, 1981.
See in particular Chapter VII, "Spectral Theory after 1900"

Lecture 1: compact groups in the 1920s

Original papers (selected)

F. Peter and H. Weyl, Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuirlicher Gruppe, Math. Ann. 97 (1927), pp. 537–555.

English translation of the Peter--Weyl paper. (This was done rather quickly and my German is not good: use at your own risk!)

D. Hilbert, Grunzüge einer allgemeinen Theorie der Integralgleichungen, 1912.
Book edition of Hilbert's papers on integral equations. Hilbert wrote a preface with a summary of the results.

F. Riesz, Les systèmes d'équations linéaires à une infinité d'inconnues, 1913.
Riesz's presentation of Hilbert–Schmidt spectral theory.

F. Riesz, Über lineare Funktionalgleichungen, Acta Math. 41 (1916), pp.71—98.
The fundamental paper by F. Riesz on compact operators.

History books or articles

T. Hawkins, Emergence of the theory of Lie groups, An essay in the history of mathematics 1869–1926, Springer, 2000.
This is a magnificent book. Chapters 9 to 12 cover a very large portion of what I said in the first lecture, and much more.

A. Borel, Essays in the History of Lie Groups and Algebraic Groups, AMS/LMS History of Mathematics, vol. 21, 2001.
Chapter III, Hermann Weyl and Lie groups, is a very good source for Weyl's 1925 memoirs on semisimple Lie groups

C. Curtis, Pioneers of Representation Theory: Frobenius, Burnside, Schur and Brause , AMS/LMS History of Mathematics, vol.15, 1999
A good source for the work of Frobenius and Schur on finite groups

H. Weyl, David Hilbert and his mathematical work, Bull. Amer. Math. Soc. 50 (1944), pp. 612–654.
A mathematical obituary of Hilbert by Weyl. The article is reproduced at the end of the book by C. Reid cited below.

C. Reid, Hilbert, Springer, 1970.
A classic and lively biography of Hilbert, for all the nonmathematical aspects.

N. Bourbaki, Éléments d'histoire des mathématiques, Springer, 2007.
The Bourbaki books usually come with historical notes. Relevant to our story are: the note on Noncommutative algebra, that on Topological vector spaces, and that on Haar measure and convolution.

C. Chevalley and A. Weil, Hermann Weyl, Enseign. Math. 3 (3), pp. 157--187

Lecture 2: abelian locally compact groups in the 1930s

Original papers (selected)

A. Weil, L'intégration dans les groupes topologiques et ses applications, Paris (Hermann), 1940
Weil's monograph contains a synthesis of virtually all the results available at the time of writing, ca. 1936. It is still a very good read today.

A. Haar, Der Massbegriff in der Theorie der Kontinuirlichen Gruppen, Ann. of Math. (2) 34 (1933), no. 1, pp. 147–169
The paper which introduced Haar measure for second countable locally compact groups

L. Pontryagin, Les fonctions presque périodiques et l'analysis situs, C. R. Acad. Sci. Paris 196 (1933), pp. 1201–1203
This is Pontryagin's short note on almost periodic functions mentioned in my talk, building on work of Stepanoff and Tychonov (same CRAS volume, pp. 199–201)

L. Pontryagin, The theory of topological commutative groups, Ann. of Math. (2) 35 (1934), no. 4, pp. 361–388
This is the fundamental paper by Pontryagin on duality between compact and discrete groups, and structure of certain locally compact abelian groups.

E. van Kampen, Locally bicompact abelian groups and their character groups, Ann. of Math. (2) 36 (1935), pp. 448–463
This is van Kampen's generalization of Pontryagin duality to all (second countable) locally compact abelian groups

E. van Kampen, Almost periodic functions and compact groups, Ann. of Math. (2) 37 (1936), pp. 78–91
Contains van Kampen's results on Bohr compactifications (see also Weil's book above)

A. Weil, Sur les fonctions presque périodiques, C. R. Acad. Sci Paris 200 (1935), pp. 38–40
This is Weil's note introducing the Bohr compactification; see also his 1940 book, and his talk in the Séminaire Julia cited below.

N. Wiener, Tauberian theorems, Ann. of Math. (2) 33 (1932), pp. 1–100.
Wiener's famous paper on tauberian theorems, with the lemma about absolutely convergent Fourier series for 1/f as Lemma IIe.

R.E.A.C. Paley and N. Wiener, Analytic properties of the characters of abelian groups, Proc. ICM 1932 (Zurich), p. 95.

Séminaire de mathématiques dit Julia, 1934--1936, proceedings, , edited by Michèle Audin (link)
This is a kind of ancestor to the Bourbaki seminar, run by the young people who were just then founding the Bourbaki group. (Gaston Julia had agreed to sign his name so that they could book a classroom.) The talks were on mathematical actuality, and were written up. Several of them are very good expositions of some of the papers that are relevant to us (Henri Cartan on the Riesz paper on positive-definite functions, Claude Chevalley on Pontryagin duality, André Weil on almost periodic functions...).

History books or articles

J. Stewart, Positive-definite functions and generalizations, a historical survey, Rocky Mountain J. Math. 6 (1976), no. 3, pp. 409–434.
A good reference for the theory of positive-definite sequences and functions from 1910 to 1932.

G. Mackey, Harmonic analysis as the exploitation of symmetry: a historical survey, see general references above.

G. Mackey, Harmonic analysis and unitary group representations: the development from 1927 to 1950, Cahiers du séminaire d'histoire des mathématiques, 2è série, t. 2 (1992), pp. 13–42.
A good complement to the monograph-sized paper of Mackey mentioned above.

Lecture 3: Beginnings of operator algebras

Original papers (selected)

I. M. Gelfand, Collected works, 3 volumes, Springer, 1987
In Gelfand's collected works, papers have been sorted by theme.
The operator algebra papers are in Volume 1, and the papers on general representation theory are at the beginning of Volume 2.

J.von Neumann, Zur Algebra der Funktionaloperationen und Theorie der normalen Operatoren, Math. Ann. 102 (1929), pp.370–427
This is von Neumann's paper on the spectral theorem for normal operators, with detailed discussion of the weakly closed *-subalgebra generated by a set of operators

F. Murray and J. von Neumann, On rings of operators, Ann. of Math. (2) 37 (1936), pp.6–120
The first paper in the ``Rings of operators'' series

M. Stone, Linear transformations in Hilbert space and their applications to analysis, AMS, 1932
The book which ended up on the Chief Justice's desk

M. Stone, Applications of the theory of Boolean rings to general topology, Trans. AMS 41 (1937), pp.375–481
Contains the theorem about ideals in a Boolean algebra of functions on a locally compact space

S. Mazur, Sur les anneaux linéaires, C. R. Ac. Sci. Paris 207 (1938), pp.1025–1027
The original paper containing the Gelfand–Mazur theorem (but no proof)

I. Segal, The group algebra of a locally compact group, Trans. AMS 61 (1947), pp. 69–105
One of Segal's twin papers on locally compact groups and C*-algebras

I. Segal, Irreducible representations of operator algebras, Bull. AMS 53(2) (1947), pp. 73–88
Segal's other paper in the twin series on locally compact groups and C*-algebras

Lecture 4: Representations of noncompact groups

Original papers (selected)

E. Wigner, On unitary representations of the inhomogeneous Lorentz group, Ann. of Math. (2) 40 (1939), pp. 149–204

The Gelfand–Raikov paper on positive-definite functions, and the Gelfand–Naimark paper on SL(2,C),
are (translated) in Volume 2 of Gelfand's collected works (see above)

Harish-Chandra, Infinite unitary representations of the Lorentz group , Proc. Roy. Soc. London A 189 (1947), pp.372–401
This is more or less Harish-Chandra's thesis, back when he was a physicist.

V. Bargmann, Irreducible unitary representations of the Lorentz group, Ann. of Math. (2) 48 (1947), pp. 568–640

R. Godement, Sur les relations d'orthogonalité de Bargmann, C. R. Acad. Sci. Paris 225 (1947), pp. 657–659

History books or articles

G. Mackey, Harmonic analysis as the exploitation of symmetry: a historical survey, see general references above.

G. Mackey, Harmonic analysis and unitary group representations: the development from 1927 to 1950, Cahiers du séminaire d'histoire des mathématiques, 2è série, t. 2 (1992), pp. 13–42.