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 :

•
$(\u27f9)$
If $x$ is invertible in $\mathcal{B}$ then it is clearly invertible in $\mathcal{A}$.

•
$(\u27f8)$
If $x$ is invertible in $\mathcal{A}$, then so is $y={x}^{*}x$. Thus, $0\notin {\sigma}_{\mathcal{A}}(y)$.
Since $y$ is selfadjoint (http://planetmath.org/InvolutaryRing), ${\sigma}_{\mathcal{A}}(y)\subseteq \mathbb{R}$ (see this entry (http://planetmath.org/SpecialElementsInACAlgebraAndTheirSpectralProperties)), and so $\u2102{\sigma}_{\mathcal{A}}(y)$ 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 $\mathcal{B}$. $\mathrm{\square}$
Title  spectral invariance theorem (for ${C}^{*}$algebras) 

Canonical name  SpectralInvarianceTheoremforCalgebras 
Date of creation  20130322 17:29:53 
Last modified on  20130322 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 