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Israel Gelfand in nLab
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width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/132/#Item_2" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Selected writings</title></head> <body><div style="float:right;margin:0 20px 10px 20px;"><img width="250" src="http://www-history.mcs.st-andrews.ac.uk/BigPictures/Gelfand_4.jpeg" alt="[Portrait]" /></div> <p><strong>Israel Moiseevich Gel’fand</strong> (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</p> <ul> <li> <p>the <span class="newWikiWord">Gelfand representation<a href="/nlab/new/Gelfand+representation">?</a></span> of a <a class="existingWikiWord" href="/nlab/show/Banach+algebra">Banach algebra</a>,</p> </li> <li> <p>an important special case, the <a class="existingWikiWord" href="/nlab/show/Gelfand-Naimark+theorem">Gelfand-Naimark theorem</a> describing commutative <a class="existingWikiWord" href="/nlab/show/C%2A-algebra">C*-algebra</a>s,</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/Gelfand-Naimark-Segal+construction">Gelfand-Naimark-Segal construction</a> describing general C<em>-algebras,</em></p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/Gelfand-Fuks+cohomology">Gelfand-Fuks cohomology</a> of a <a class="existingWikiWord" href="/nlab/show/foliation">foliation</a>,</p> </li> <li> <p>the <span class="newWikiWord">Gelfand-Kirillov dimension<a href="/nlab/new/Gelfand-Kirillov+dimension">?</a></span> of an <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a>,</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/Bernstein-Gelfand-Gelfand+resolution">Bernstein-Gelfand-Gelfand resolution</a> for <a class="existingWikiWord" href="/nlab/show/representations">representations</a> of simple <a class="existingWikiWord" href="/nlab/show/Lie+groups">Lie groups</a>.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gel%27fand+triple">Gel'fand triple</a></p> </li> <li> <p>Gelfand-Retakh theory of <a class="existingWikiWord" href="/nlab/show/quasideterminant">quasideterminant</a>s</p> </li> </ul> <p>He is the author of a 6-volume treatise on <a class="existingWikiWord" href="/nlab/show/distribution">distribution</a> theory (various volumes in collaboration with Shilov, Vilenkin, Graev).</p> <h2 id="selected_writings">Selected writings</h2> <p>The characterization of <a class="existingWikiWord" href="/nlab/show/state+on+a+star-algebra">states</a> on <a class="existingWikiWord" href="/nlab/show/group+algebras">group algebras</a> and what came to be known as the <a class="existingWikiWord" href="/nlab/show/Gelfand-Raikov+theorem">Gelfand-Raikov theorem</a>:</p> <ul> <li id="GelfandRaikov43"> <p>И. М. Гельфанд, Д. А. Райков, <em>Неприводимые унитарные представления локально бикомпактных групп</em>, Матем. сб., 13(55):2–3 (1943) 301–316 [<a href="https://www.mathnet.ru/links/728d630fdf2220b1ea8bdb2a5f6a5488/sm6181.pdf">mathnet pdf</a>]</p> <p><br /></p> <p><a class="existingWikiWord" href="/nlab/show/Israel+Gelfand">Israel Gelfand</a>, <a class="existingWikiWord" href="/nlab/show/Dmitri+Raikov">Dmitri Raikov</a>, <em>Irreducible unitary representations of locally bicompact groups</em>, Recueil Mathématique. N.S., 13(55) 2–3 (1943) 301–316 [<a href="https://www.mathnet.ru/eng/sm6181">mathnet:eng/sm6181</a>]</p> </li> </ul> <p>Introducing the <a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a>:</p> <ul> <li id="GelfandNaimark43"> <p><a class="existingWikiWord" href="/nlab/show/Israel+Gelfand">Israel Gelfand</a>, <a class="existingWikiWord" href="/nlab/show/Mark+Naimark">Mark Naimark</a>, <em>On the imbedding of normed rings into the ring of operators on a Hilbert space</em>, Matematicheskii Sbornik. 12 (2): 197–217 (1943)</p> <p>reprinted in:</p> <p>Robert Doran (ed.), <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-Algebras: 1943–1993</em>, Contemporary Mathematics <strong>167</strong>, AMS 1994 (<a href="http://dx.doi.org/10.1090/conm/167">doi:10.1090/conm/167</a>)</p> </li> </ul> <p>One of the first comprehensive series of texts on <a class="existingWikiWord" href="/nlab/show/generalized+functions">generalized functions</a> (also known as a <a class="existingWikiWord" href="/nlab/show/distributions">distributions</a>):</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Israel+M.+Gelfand">Israel M. Gelfand</a>, <a class="existingWikiWord" href="/nlab/show/Georgiy+E.+Shilov">Georgiy E. Shilov</a>, <em>Generalized Functions</em> Vol. 1-6 (1958), Acad. Press (1966-1968), AMS (2016) [<a href="https://bookstore.ams.org/view?ProductCode=CHELGELFSET">ISBN:978-1-4704-2885-3</a>]</li> </ul> <p>Introducing what came to be called <em><a class="existingWikiWord" href="/nlab/show/BGG+resolutions">BGG resolutions</a></em> of <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> <a class="existingWikiWord" href="/nlab/show/representations">representations</a> by <a class="existingWikiWord" href="/nlab/show/Verma+modules">Verma modules</a> built using <a class="existingWikiWord" href="/nlab/show/regular+differential+operators">regular differential operators</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Joseph+N.+Bernstein">Joseph N. Bernstein</a>, <a class="existingWikiWord" href="/nlab/show/Israel+M.+Gelfand">Israel M. Gelfand</a>, <a class="existingWikiWord" href="/nlab/show/Sergei+I.+Gelfand">Sergei I. Gelfand</a>: <em>Differential operators on the base affine space and a study of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math>-modules</em>, in: I. M. Gelfand (ed.), <em>Lie groups and their representations</em>, Adam Hilger (1975) 21-64 [<a href="http://www.math.tau.ac.il/~bernstei/Publication_list/publication_texts/BGG-differ-operators.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/BGG.pdf" title="pdf">pdf</a>]</li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/quasideterminants">quasideterminants</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/I.+M.+Gel%27fand">I. M. Gel'fand</a>, <a class="existingWikiWord" href="/nlab/show/V.+Retakh">V. S. Retakh</a>, <em>Determinants of matrices over noncommutative rings</em>, Funct.Anal.Appl. <strong>25</strong> (1991), no.2, pp. 91–102. engl. transl. <strong>21</strong> (1991), pp. 51–58.</p> </li> <li> <p>I.M. Gel’fand, V.S. Retakh, A theory of noncommutative determinants and characteristic functions of graphs, Funct.Anal.Appl. <strong>26</strong> (1992), no.4, pp. 231–246.</p> </li> <li> <p>I.M. Gel’fand, V.S. Retakh, <em>Quasideterminants I</em>, Selecta Mathematica, N. S. 3 (1997) no.4, pp. 517–546; <a href="https://doi.org/10.1007/s000290050019">doi</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Israel+Gelfand">Israel Gelfand</a>, Sergei Gelfand, <a class="existingWikiWord" href="/nlab/show/Vladimir+Retakh">Vladimir Retakh</a>, Robert Lee Wilson, <em>Quasideterminants</em>, Advances in Mathematics 193 (2005) 56–141 <a href="https://doi.org/10.1016/j.aim.2004.03.018">doi</a></p> </li> <li> <p>Israel Gelfand, Daniel Krob, Alain Lascoux, <a class="existingWikiWord" href="/nlab/show/Bernard+Leclerc">Bernard Leclerc</a>, <a class="existingWikiWord" href="/nlab/show/Vladimir+Retakh">Vladimir Retakh</a>, J.-Y. Thibon, <em>Noncommutative symmetric functions</em>, Adv. in Math. <strong>112</strong> (2) (1995) 218–348</p> </li> </ul> <div class="property">category: <a class="category_link" href="/nlab/all_pages/people">people</a></div></body></html> </div> <div class="revisedby"> <p> Last revised on July 24, 2024 at 17:25:29. 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