About: Spectrum of a C*-algebra

Facets (new session)
Description
Metadata
Settings
- Rule:
- Inverse Functional Properties:
- "Same As":

About: Spectrum of a C*-algebra Goto Sponge NotDistinct Permalink

An Entity of Type : owl:Thing, within Data Space : dbpedia.demo.openlinksw.com associated with source document(s)
QRcode icon

http://dbpedia.demo.openlinksw.com/describe/?url=http%3A%2F%2Fdbpedia.org%2Fresource%2FSpectrum_of_a_C%2A-algebra

In mathematics, the spectrum of a C*-algebra or dual of a C*-algebra A, denoted Â, is the set of unitary equivalence classes of irreducible *-representations of A. A *-representation π of A on a Hilbert space H is irreducible if, and only if, there is no closed subspace K different from H and {0} which is invariant under all operators π(x) with x ∈ A. We implicitly assume that irreducible representation means non-null irreducible representation, thus excluding trivial (i.e. identically 0) representations on one-dimensional spaces. As explained below, the spectrum Â is also naturally a topological space; this is similar to the notion of the spectrum of a ring.

Attributes	Values
rdf:type	Thing
rdfs:label	Spectrum of a C*-algebra (en)
rdfs:comment	In mathematics, the spectrum of a C-algebra or dual of a C-algebra A, denoted Â, is the set of unitary equivalence classes of irreducible -representations of A. A -representation π of A on a Hilbert space H is irreducible if, and only if, there is no closed subspace K different from H and {0} which is invariant under all operators π(x) with x ∈ A. We implicitly assume that irreducible representation means non-null irreducible representation, thus excluding trivial (i.e. identically 0) representations on one-dimensional spaces. As explained below, the spectrum Â is also naturally a topological space; this is similar to the notion of the spectrum of a ring. (en)
rdfs:seeAlso	Primitive spectrum
foaf:depiction
dcterms:subject	Spectral theory C*-algebras
Wikipage page ID	918414 (xsd:integer)
Wikipage revision ID	976859698 (xsd:integer)
Link from a Wikipage to another Wikipage	Primitive ideal Tannaka–Krein duality Spectral theory Lie group -representation Compact space Natural transformation Separable space Connected space Nilpotent Annihilator (ring theory) Locally compact State (functional analysis) Space (mathematics) Spectrum of a ring Banach space Topological group Topology Hausdorff space Heisenberg group Irreducible representation Locally compact group Duality (mathematics) Fourier transform Ring (mathematics) Hilbert space James Glimm Surjective C-algebras Homeomorphism Discrete topology Dimension Borel set C*-algebra Polish space Pontryagin duality Group algebra of a locally compact group Pure state Idempotent If and only if Kuratowski closure axioms Semi-simple Unimodular group Unitary representation Zariski topology Plancherel theorem Topological space Simple module J. M. G. Fell G. Mackey GNS construction Gelfand transformation Compact topological group dbr:Weak_containment
sameAs	Spectrum of a C-algebra Spectrum of a C-algebra Spectrum of a C*-algebra
dbp:wikiPageUsesTemplate	dbt:Reflist dbt:See_also dbt:Functional_Analysis dbt:SpectralTheory
thumbnail	wiki-commons:Special:FilePath/3-dim_commut_algebra,_subalgebras,_ideals.svg?width=300
has abstract	In mathematics, the spectrum of a C-algebra or dual of a C-algebra A, denoted Â, is the set of unitary equivalence classes of irreducible -representations of A. A -representation π of A on a Hilbert space H is irreducible if, and only if, there is no closed subspace K different from H and {0} which is invariant under all operators π(x) with x ∈ A. We implicitly assume that irreducible representation means non-null irreducible representation, thus excluding trivial (i.e. identically 0) representations on one-dimensional spaces. As explained below, the spectrum Â is also naturally a topological space; this is similar to the notion of the spectrum of a ring. One of the most important applications of this concept is to provide a notion of dual object for any locally compact group. This dual object is suitable for formulating a Fourier transform and a Plancherel theorem for unimodular separable locally compact groups of type I and a decomposition theorem for arbitrary representations of separable locally compact groups of type I. The resulting duality theory for locally compact groups is however much weaker than the Tannaka–Krein duality theory for compact topological groups or Pontryagin duality for locally compact abelian groups, both of which are complete invariants. That the dual is not a complete invariant is easily seen as the dual of any finite-dimensional full matrix algebra Mn(C) consists of a single point. (en)
gold:hypernym	Set
prov:wasDerivedFrom	wikipedia-en:Spectrum_of_a_C*-algebra?oldid=976859698&ns=0
page length (characters) of wiki page	11918 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Spectrum_of_a_C*-algebra
is rdfs:seeAlso of	Gelfand representation
is Link from a Wikipage to another Wikipage of	List of functional analysis topics Multiplier algebra Primitive ideal Projectionless C-algebra Approximately finite-dimensional C-algebra Jacobson radical George Mackey Approximate identity Stone–von Neumann theorem Stone–Čech compactification Spectrum (disambiguation) Spectrum of a ring Banach space Ergodic flow Banach–Stone theorem Kazhdan's property (T) AW-algebra Hereditary C-subalgebra Differentiable manifold C*-algebra Unitary representation Noncommutative geometry

Faceted Search & Find service v1.17_git139 as of Feb 29 2024

Alternative Linked Data Documents: ODE Content Formats:

RDF

ODATA

Microdata

About

OpenLink Virtuoso version 08.03.3330 as of Mar 19 2024, on Linux (x86_64-generic-linux-glibc212), Single-Server Edition (378 GB total memory, 67 GB memory in use)
Data on this page belongs to its respective rights holders.
Virtuoso Faceted Browser Copyright © 2009-2024 OpenLink Software