spectral invariance theorem (for $C^{*}$ -algebras)

The spectral permanence theorem ( entry) relates the spectrums $\sigma_{\mathcal{B}}(x)$ and $\sigma_{\mathcal{A}}(x)$ of an element $x\in\mathcal{B}\subseteq\mathcal{A}$ relatively to the Banach algebras $\mathcal{B}$ and $\mathcal{A}$ .

For $C^{*}$ -algebras (http://planetmath.org/CAlgebra) the situation is quite .

Spectral invariance theorem - Suppose $\mathcal{A}$ is a unital $C^{*}$ -algebra and $\mathcal{B}\subseteq\mathcal{A}$ a $C^{*}$ -subalgebra that contains the identity of $\mathcal{A}$ . Then for every $x\in\mathcal{B}$ one has

\sigma_{\mathcal{B}}(x)=\sigma_{\mathcal{A}}(x).

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

Theorem - Let $x\in\mathcal{B}\subset\mathcal{A}$ be as above. Then $x$ is invertible in $\mathcal{B}$ if and only if $x$ invertible in $\mathcal{A}$ .

Proof :

•

$(\Longrightarrow)$

If $x$ is invertible in $\mathcal{B}$ then it is clearly invertible in $\mathcal{A}$ .
•

$(\Longleftarrow)$

If $x$ is invertible in $\mathcal{A}$ , then so is $y=x^{*}x$ . Thus, $0\notin\sigma_{\mathcal{A}}(y)$ .

Since $y$ is self-adjoint (http://planetmath.org/InvolutaryRing), $\sigma_{\mathcal{A}}(y)\subseteq\mathbb{R}$ (see this entry (http://planetmath.org/SpecialElementsInACAlgebraAndTheirSpectralProperties)), and so $\mathbb{C}-\sigma_{\mathcal{A}}(y)$ has no bounded (http://planetmath.org/Bounded) connected components.

By the spectral permanence theorem (http://planetmath.org/SpectralPermanenceTheorem) we must have $\sigma_{\mathcal{B}}(y)=\sigma_{\mathcal{A}}(y)$ . Hence, $0\notin\sigma_{\mathcal{B}}(y)$ , i.e. $y$ is invertible in $\mathcal{B}$ .

It follows that $x^{-1}=(x^{*}x)^{-1}x^{*}=y^{-1}x^{*}\in\mathcal{B}$ , i.e. $x$ is invertible in $\mathcal{B}$ . $\square$

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

spectral invariance theorem (for C*-algebras)

spectral invariance theorem (for $C^{*}$ -algebras)