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

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

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

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

$${\sigma }_{ℬ}(x)={\sigma }_{𝒜}(x).$$

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

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

Proof :

• $(⟹)$

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

• $(⟸)$

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

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

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

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