About: Gelfand–Mazur theorem     Goto   Sponge   NotDistinct   Permalink

An Entity of Type : yago:TheoremsInFunctionalAnalysis, within Data Space : dbpedia-live.openlinksw.com associated with source document(s)
QRcode icon
http://dbpedia-live.openlinksw.com/describe/?url=http%3A%2F%2Fdbpedia.org%2Fresource%2FGelfand%E2%80%93Mazur_theorem

In operator theory, the Gelfand–Mazur theorem is a theorem named after Israel Gelfand and Stanisław Mazur which states:A complex Banach algebra, with unit 1, in which every nonzero element is invertible, is isometrically isomorphic to the complex numbers.In other words, the only complex Banach algebra that is a division algebra is the complex numbers C.

AttributesValues
rdf:type
sameAs
foaf:isPrimaryTopicOf
rdfs:comment
  • In operator theory, the Gelfand–Mazur theorem is a theorem named after Israel Gelfand and Stanisław Mazur which states:A complex Banach algebra, with unit 1, in which every nonzero element is invertible, is isometrically isomorphic to the complex numbers.In other words, the only complex Banach algebra that is a division algebra is the complex numbers C.
rdfs:label
  • Gelfand–Mazur theorem
has abstract
  • In operator theory, the Gelfand–Mazur theorem is a theorem named after Israel Gelfand and Stanisław Mazur which states:A complex Banach algebra, with unit 1, in which every nonzero element is invertible, is isometrically isomorphic to the complex numbers.In other words, the only complex Banach algebra that is a division algebra is the complex numbers C. This follows from the fact that, if A is a complex Banach algebra, the spectrum of an element a ∈ A is nonempty (which in turn is a consequence of the complex-analycity of the resolvent function). For every a ∈ A, there is some complex number λ such that λ1 − a is not invertible. By assumption, λ1 − a = 0. So a = λ · 1. This gives an isomorphism from A to C.Actually, a stronger and harder theorem was proved first by Stanisław Mazur alone, but it was published in France without a proof, when the author refused the editor's request to shorten his already short proof. Mazur's theorem states that there are (up to isomorphism) exactly three real Banach division algebras: the fields of reals R, of complex numbers C, and the division algebra of quaternions H. Gelfand proved (independently) the easier, special, complex version a few years later, after Mazur. However, it was Gelfand's work which influenced the further progress in the area.
Link to the Wikipage edit URL
extraction datetime
Link to the Wikipage history URL
Wikipage page ID
page length (characters) of wiki page
Wikipage modification datetime
Wiki page out degree
Wikipage revision ID
Link to the Wikipage revision URL
dbp:wikiPageUsesTemplate
dct:subject
is Wikipage redirect of
Faceted Search & Find service v1.17_git39 as of Aug 10 2019


Alternative Linked Data Documents: iSPARQL | ODE     Content Formats:       RDF       ODATA       Microdata      About   
This material is Open Knowledge   W3C Semantic Web Technology [RDF Data] Valid XHTML + RDFa
OpenLink Virtuoso version 08.03.3319 as of Sep 1 2020, on Linux (x86_64-generic-linux-glibc25), Single-Server Edition (61 GB total memory)
Data on this page belongs to its respective rights holders.
Virtuoso Faceted Browser Copyright © 2009-2021 OpenLink Software