23 April 2021

Happy Birthday HM

Today, 50 years ago, Takahashi, a Colombian doctoral student at Tulane University, the son of a Japanese immigrant, brought Hilbert modules on non-commutative C*-algebras to the world of mathematics. These had been conceived by Kaplansky 18 years earlier but, due to his discouraging misgivings, childbirth was incomplete. After a very long period of latency, delivery was finally achieved with Takahashi as the skilled midwife.


33] K. H. Hofmann and J. R. Liukkonen (eds.), Recent Advances in the Representation

Theory of Rings and C

∗

-Algebras, Memoirs Amer. Math. Soc. 148 (1974).


TAKAHASHI-TesisAbril23-1971.pdf
FM-IMAGENES-ActaTesisTakahashi.png

ACTA DE NACIMIENTO

Firman el acta de la defensa de la tesis:

  1. Frank D. Quigley

  2. Jerome Goldstein

  3. Ronald J Knill

  4. Karl H. Hofmann

  5. John Dauns

  6. Frank Birtel

  7. Michael W Mislove

  8. John R Liukkonen

  9. Laszlo Fuchs

  10. Teishiro Saito

  11. John R Ringrose

  12. James T Rogers Jr


Nota aclaratoria de AT:

No alcanzo a descifrar la segunda firma, pero Jerry debió estar presente pues era uno de los jueces. No estoy seguro de Rogers, pero es probable pues mi profesor en Geometría Dif.

Creo que por allí andaban también Joachim Lambek y Alfred Goldie, pero no recuerdo bien.

FRANK-guiaHMH-PubPorAño.pdf
FRANK-guiaHM-31Mar-2017.pdf

La lista de Frank

Michael Frank han mantenido una lista de publicaciones relacionadas con los Módulos de Hilbert. La lista que publico es la actualizada al 31 de Marzo de 2017. Desde el artículo de Kaplanski en 1953 en que se originan hasta las 28 publicaciones sobre la cuestión publicadas en el primer trimestre del 2017. La gráfica de al lado muestra el número de publicaciones por año.

Campos donde los MH se usan

(Según la Guía de Frank)

  1. K-theory and KK-theory of operator algebras (G. G. Kasparov’s approach).

  2. Cuntz semigroups of C*-algebras and applications.

  3. Strong Morita equivalence of C*-algebras and Hilbert C*-modules, and its application to group representation theory and crossed product C*-algebras.

  4. Normal operator-valued weights (resp., conditional expectations) of finite index between C*-algebras / Correspondences of C*-algebras / Regular C*-valued weights.

  5. AW*-algebras and monotone complete C*-algebras.

  6. Completely positive mappings between C*-algebras and Hilbert C*-modules.

  7. Cuntz-(Krieger-Nica-)Pimsner algebras.

  8. (Higher-rank) Graph C*-algebras, Quiver Algebras.

  9. Mathematical and theoretical physics, quantum probability.

  10. Hilbert product systems.

  11. Unbounded operators, quantum groups and other applications.

  12. Vector bundles, (F)Hilbert bundles ↔ projective C*-modules, Hilbert C*-modules.

  13. Rotation C*-algebras, noncommutative (super-)tori and related structures.

  14. Other non-commutative geometry.

  15. Applications in algebraic and geometric topology.

  16. L2-methods in algebraic and geometric topology.

  17. Hilbert modules over pro-C*-algebras / locally C*-algebras.

  18. Operator spaces and operator algebras, and Hilbert C*-modules.

  19. Ternary rings of operators - a different setting for Hilbert C*-modules.

  20. Prediction theory of multivariate stochastic processes / Quantum stochastics.

  21. Wavelet theory, (modular) frames and Hilbert C*-modules.

Birth of Hilbert Modules

In […] the number of the article on Frank's list.

Conception

1953

I. Kaplansky. Modules over operator algebras. Amer. J. Math., 75, (1953). 839-858.[1012]. In this article, Kaplansky conceives of Hilbert modules and gives birth to modules on commutative algebras. He anticipates that the delivery of non-commutative can be very difficult. Takahashi 18 years later that Kaplansky was wrong. And to top it off, Paschke and Rieffel also demonstrate how to give birth to the non-commutative Hilbert modules one year after. The three midwives would be far from thinking that the concept was going to be so successful.

Múltiple Light up

1971-1972


  1. A. O. Takahashi (Marzo de 1937), . On Hilbert Modules, Dissertation, Tulane University, New Orleans, La., 1971, [1836]. Advisor: K.H. Hofmann.This dissertation studies the non-commutative Hilbert modules for the first time and proves the two fundamental results of the theory: the Representation Theorem and the Duality Theorem. Unfortunately, Takahashi did not publish his thesis until 1979: A duality between Hilbert modules and Fields of Hilbert spaces. Rev. Colomb. Mat.,13, (1979). 93-120 [1836]. And Hilbert modules and their representations. Rev. Colomb. Mat., 13, (1979). 1-38 [1838]. The Revista Colombiana de Matemáticas is an excellent journal, reviewed in the MR and ZB, but little known in the world academic circuit. Takahashi is descendant of Gauss vía Hilbert, then from Germany school.

  2. W. L. Paschke, Hilbert B-Modules and Completely Positive Maps, Dissertation, University of Oregon, 1972. Advisor: Paul Civin. Published as Inner Product Moduleso Ver B*-Algebras, Transactions of The American Mathematical Society, Volume 182, August, 1973, [1526, 1527]: one trailblazing paper in Hilbert C*-module theory. He cites Takahashi's dissertation as Fields of Hilbert modules, which he learns about from his director of thesis Hoffman's recension in K. H. Hofmann, see above 1, . Paschke is a descendant of Poisson, of the French school.

  3. M. A. Rieffel (22 de Diciembre de 1937) , Induced representations of C*-algebras (preprint).1972/74, [1644, 1645, 1646]: the other trailblazing papers, about Hilbert C*- modules and (strong) Morita equivalence of C*-algebras. Rieffel had done his dissertation with Kadison at Columbia in 1963. Rieffel had done his dissertation with Kadison at Columbia in 1963. A descendant of Poisson, of the French school. And he went back to her, Rieffel worked with Connes.


Annunciation

Año 1972


  1. K. H. Hofmann, Representation of algebras by continuous sections, Bull. Amer. Math. Soc. 78 (1972), 291-373. [835]. In this review article, Hofmann announces the theorems that his student Takahashi finds in his thesis that he quotes incorrectly. But he did not realize how important Takahashi's discoveries were. This is reflected in the fact that he was allowed to leave for Colombia very soon without publishing his thesis in a front-line magazine. Takahashi arrived in Tulane in August 1969 and returned to Colombia in August 1971. He had a scholarship from the Ford Foundation for four years. Hofmann in a personal letter of 2018 regrets not having maintained contact with his student.