PlanetMath: spectral invariance theorem (for $C^*$-algebras)

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (230)
Orphanage
Unclass'd
Unproven (519)
Corrections (16)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

spectral invariance theorem (for $C^*$

-algebras)

(Theorem)

The spectral permanence theorem (parent 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 -algebras the situation is quite simple.

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

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

The spectral invariance theorem is a straightforward corollary of the next more general theorem about invertible elements in -subalgebras.

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

Proof :

$(\Longrightarrow)$
If is invertible in $\mathcal{B}$ then it is clearly invertible in $\mathcal{A}$ .
$(\Longleftarrow)$
If is invertible in $\mathcal{A}$ , then so is . Thus, $0 \notin \sigma_{\mathcal{A}}(y)$ .

Since is self-adjoint, $\sigma_{\mathcal{A}}(y) \subseteq \mathbb{R}$ (see this entry), and so $\mathbb{C} - \sigma_{\mathcal{A}}(y)$ has no bounded connected components.

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

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

Anyone with an account can edit this entry. Please help improve it!

"spectral invariance theorem (for $C^*$

-algebras)" is owned by asteroid.

(view preamble)

Other names:

spectral invariance theorem, invariance of the spectrum of $C^*$

-subalgebras

Also defines:

invertibility in $C^*$

-subalgebras

This object's parent.

Log in to rate this entry.
(view current ratings)

Cross-references: connected components, proof, invertible, identity, contains, unital, Banach algebras, spectrums, spectral permanence theorem
There are 3 references to this entry.

This is version 4 of spectral invariance theorem (for $C^*$

-algebras), born on 2007-08-23, modified 2007-08-24.
Object id is 9886, canonical name is SpectralInvarianceTheoremForCAlgebras.
Accessed 1012 times total.

Classification:

AMS MSC:	46L05 (Functional analysis :: Selfadjoint operator algebras :: General theory of $C^*$-algebras)
	46H10 (Functional analysis :: Topological algebras, normed rings and algebras, Banach algebras :: Ideals and subalgebras)

Pending Errata and Addenda

None.

Discussion

forum policy

No messages.

Interact