spectral invariance theorem (for C*-algebras)

The spectral permanence theorem ( entry) relates the spectrums σ(x) and σ𝒜(x) of an element x𝒜 relatively to the Banach algebrasMathworldPlanetmath and 𝒜.

For C*-algebrasMathworldPlanetmath (http://planetmath.org/CAlgebra) the situation is quite .

Spectral invariance theorem - Suppose 𝒜 is a unital C*-algebra and 𝒜 a C*-subalgebraPlanetmathPlanetmath that contains the identityPlanetmathPlanetmathPlanetmathPlanetmath of 𝒜. Then for every x one has


The spectral invariance theorem is a straightforward corollary of the next more general theorem about invertible elements in C*-subalgebras.

Theorem - Let x𝒜 be as above. Then x is invertible in if and only if x invertible in 𝒜.

Proof :

  • ()

    If x is invertible in then it is clearly invertible in 𝒜.

  • ()

    If x is invertible in 𝒜, then so is y=x*x. Thus, 0σ𝒜(y).

    Since y is self-adjoint (http://planetmath.org/InvolutaryRing), σ𝒜(y) (see this entry (http://planetmath.org/SpecialElementsInACAlgebraAndTheirSpectralProperties)), and so -σ𝒜(y) has no boundedPlanetmathPlanetmath (http://planetmath.org/Bounded) connected componentsMathworldPlanetmathPlanetmath.

    By the spectral permanence theorem (http://planetmath.org/SpectralPermanenceTheorem) we must have σ(y)=σ𝒜(y). Hence, 0σ(y), i.e. y is invertible in .

    It follows that x-1=(x*x)-1x*=y-1x*, i.e. x is invertible in .

Title spectral invariance theorem (for C*-algebras)
Canonical name SpectralInvarianceTheoremforCalgebras
Date of creation 2013-03-22 17:29:53
Last modified on 2013-03-22 17:29:53
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 7
Author asteroid (17536)
Entry type Theorem
Classification msc 46H10
Classification msc 46L05
Synonym spectral invariance theorem
Synonym invariance of the spectrum of C*-subalgebras
Defines invertibility in C*-subalgebras