# 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 .

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) 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