nLab Israel Gelfand

Israel Moiseevich Gel’fand (Израиль Гельфанд, 1913–2009) played a preeminent role in Russian mathematics for many years. His legendary seminar in Moscow ran for 50 years, and set the pattern for many others, including Sullivan’s seminar at CUNY and Drinfeld’s at Chicago. His name is attached to many mathematical achievements, including

• the Gelfand representation? of a Banach algebra,

• an important special case, the Gelfand-Naimark theorem describing commutative C*-algebras,

• the Gelfand-Naimark-Segal construction describing general C-algebras,

• the Gelfand-Fuks cohomology of a foliation,

• the Gelfand-Kirillov dimension? of an associative algebra,

• the Bernstein-Gelfand-Gelfand resolution for representations of simple Lie groups.

• Gel'fand triple

• Gelfand-Retakh theory of quasideterminants

He is the author of a 6-volume treatise on distribution theory (various volumes in collaboration with Shilov, Vilenkin, Graev).

Selected writings

The characterization of states on group algebras and what came to be known as the Gelfand-Raikov theorem:

• И. М. Гельфанд, Д. А. Райков, Неприводимые унитарные представления локально бикомпактных групп, Матем. сб., 13(55):2–3 (1943) 301–316 [mathnet pdf]

  
  

  Israel Gelfand, Dmitri Raikov, Irreducible unitary representations of locally bicompact groups, Recueil Mathématique. N.S., 13(55) 2–3 (1943) 301–316 [mathnet:eng/sm6181]

Introducing the GNS construction:

• Israel Gelfand, Mark Naimark, On the imbedding of normed rings into the ring of operators on a Hilbert space, Matematicheskii Sbornik. 12 (2): 197–217 (1943)

  reprinted in:

  Robert Doran (ed.), $C^\ast$-Algebras: 1943–1993, Contemporary Mathematics 167, AMS 1994 (doi:10.1090/conm/167)

One of the first comprehensive series of texts on generalized functions (also known as a distributions):

• Israel M. Gelfand, Georgiy E. Shilov, Generalized Functions Vol. 1-6 (1958), Acad. Press (1966-1968), AMS (2016) [ISBN:978-1-4704-2885-3]

Introducing what came to be called BGG resolutions of Lie group representations by Verma modules built using regular differential operators:

• Joseph N. Bernstein, Israel M. Gelfand, Sergei I. Gelfand: Differential operators on the base affine space and a study of $\mathfrak{g}$-modules, in: I. M. Gelfand (ed.), Lie groups and their representations, Adam Hilger (1975) 21-64 [pdf, pdf]

On quasideterminants

• I. M. Gel'fand, V. S. Retakh, Determinants of matrices over noncommutative rings, Funct.Anal.Appl. 25 (1991), no.2, pp. 91–102. engl. transl. 21 (1991), pp. 51–58.

• I.M. Gel’fand, V.S. Retakh, A theory of noncommutative determinants and characteristic functions of graphs, Funct.Anal.Appl. 26 (1992), no.4, pp. 231–246.

• I.M. Gel’fand, V.S. Retakh, Quasideterminants I, Selecta Mathematica, N. S. 3 (1997) no.4, pp. 517–546; doi

• Israel Gelfand, Sergei Gelfand, Vladimir Retakh, Robert Lee Wilson, Quasideterminants, Advances in Mathematics 193 (2005) 56–141 doi

• Israel Gelfand, Daniel Krob, Alain Lascoux, Bernard Leclerc, Vladimir Retakh, J.-Y. Thibon, Noncommutative symmetric functions, Adv. in Math. 112 (2) (1995) 218–348