Banach *-algebra representation


Definition:

A representation of a Banach *-algebra π’œ is a *-homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath Ο€:π’œβŸΆβ„¬β’(H) of π’œ into the *-algebra of bounded operatorsMathworldPlanetmathPlanetmath on some Hilbert spaceMathworldPlanetmath H.

The set of all representations of π’œ on a Hilbert space H is denoted r⁒e⁒p⁒(π’œ,H).

Special kinds of representations:

  • β€’

    A subrepresentation of a representation Ο€βˆˆr⁒e⁒p⁒(π’œ,H) is a representation Ο€0∈r⁒e⁒p⁒(π’œ,H0) obtained from Ο€ by restricting to a closed π⁒(π’œ)-invariant subspace (http://planetmath.org/InvariantSubspace) 11by a π⁒(π’œ)- we a subspacePlanetmathPlanetmathPlanetmath which is invariantMathworldPlanetmath under every operator π⁒(a) with aβˆˆπ’œ H0βŠ†H.

  • β€’

    A representation Ο€βˆˆr⁒e⁒p⁒(π’œ,H) is said to be nondegenerate if one of the following equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath conditions hold:

    1. (a)

      π⁒(x)⁒ξ=0β€ƒβ€ƒβˆ€xβˆˆπ’œβŸΉΞΎ=0, where ξ∈H.

    2. (b)

      H is the closed linear span of the set of vectors π⁒(π’œ)⁒H:={π⁒(x)⁒ξ:xβˆˆπ’œ,ξ∈H}

  • β€’

    A representation Ο€βˆˆr⁒e⁒p⁒(π’œ,H) is said to be topologically irreducible (or just ) if the only closed π⁒(π’œ)-invariant of H are the trivial ones, {0} and H.

  • β€’

    A representation Ο€βˆˆr⁒e⁒p⁒(π’œ,H) is said to be algebrically irreducible if the only π⁒(π’œ)-invariant of H (not necessarily closed) are the trivial ones, {0} and H.

  • β€’

    Given two representations Ο€1∈r⁒e⁒p⁒(π’œ,H1) and Ο€2∈r⁒e⁒p⁒(π’œ,H2), the of Ο€1 and Ο€2 is the representation Ο€1βŠ•Ο€2∈r⁒e⁒p⁒(π’œ,H1βŠ•H2) given by Ο€1βŠ•Ο€2⁒(x):=Ο€1⁒(x)βŠ•Ο€2⁒(x),xβˆˆπ’œ.

    More generally, given a family {Ο€i}i∈I of representations, with Ο€i∈r⁒e⁒p⁒(π’œ,Hi), their is the representation βŠ•i∈IΟ€i∈r⁒e⁒p⁒(π’œ,βŠ•i∈IHi), in the direct sum of Hilbert spaces βŠ•i∈IHi, such that (βŠ•i∈IΟ€i)⁒(x):=βŠ•i∈IΟ€i⁒(x) is the direct sumMathworldPlanetmathPlanetmath of the family of bounded operators (http://planetmath.org/DirectSumOfBoundedOperatorsOnHilbertSpaces) {Ο€i⁒(x)}i∈I.

  • β€’

    Two representations Ο€1∈r⁒e⁒p⁒(π’œ,H1) and Ο€2∈r⁒e⁒p⁒(π’œ,H2) of a Banach *-algebra π’œ are said to be unitarily equivalent if there is a unitaryPlanetmathPlanetmath U:H1⟢H2 such that

    Ο€2⁒(a)=U⁒π1⁒(a)⁒U*β€ƒβ€ƒβˆ€aβˆˆπ’œ
  • β€’

    A representation Ο€βˆˆr⁒e⁒p⁒(π’œ,H) is said to be if there exists a vector ξ∈H such that the set

    π⁒(A)⁒ξ:={π⁒(a)⁒ξ:aβˆˆπ’œ}

    is dense (http://planetmath.org/Dense) in H. Such a vector is called a cyclic vectorMathworldPlanetmath for the representation Ο€.

Linked file: http://aux.planetmath.org/files/objects/9843/BanachAlgebraRepresentation.pdf

Title Banach *-algebra representation
Canonical name BanachalgebraRepresentation
Date of creation 2013-03-22 17:27:37
Last modified on 2013-03-22 17:27:37
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 23
Author asteroid (17536)
Entry type Definition
Classification msc 46H15
Classification msc 46K10
Defines subrepresentation
Defines cyclic representation
Defines cyclic vector
Defines nondegenerate representation
Defines topologically irreducible
Defines algebrically irreducible
Defines direct sum of representations
Defines unitarily equivalent