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 $\mathcal{A}$.

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 :

Title spectral invariance theorem (for $C^{*}$-algebras) SpectralInvarianceTheoremforCalgebras 2013-03-22 17:29:53 2013-03-22 17:29:53 asteroid (17536) asteroid (17536) 7 asteroid (17536) Theorem msc 46H10 msc 46L05 spectral invariance theorem invariance of the spectrum of $C^{*}$-subalgebras invertibility in $C^{*}$-subalgebras