# Banach Alg. Lecture 3

Results/Exercises

- Prove that in a unital banach algebra the left invertible (reps. right invertible) elements form open sets.
- Let B be a unital banach algebra, be on the boundry of the invertible (meaning left and right invertible) elements. Prove is a topological zero divisor in , meaning with and both tend to zero.

Examples of spectrum

- If is cmpt and Hausdorff, set , then the spectrum of any is its range: .
- If is loc. compt and not compt, then set (cont. functions that vanish at infinity). Then for , .
- For a top. sp. set . Then
- If for a measure then essential range of = where ranges over all nonempty open sets containing .
- For the spectrum is .
- The spectrum of an element of ? Later ….

### I.5 Spectral Radius

Lemma I.5.1 Let be a banach algebra, . Then exists and is equal to .

proof: The case is trivial so assume and fix . For all use the Euclidean algorithm to write where . Then using the product inequality

Take the root and notice that as the terms on the rhs approach to and 1. It follows that

This holds for all m from which the following inequality follows

The opposite inequality is true which shows they are all in fact equalities.QED.

In view of the lemma, define the spectral radius of convergence of as .

Thm I.5.1 For have . Further, such that and .

proof: WLOG is unital. Set and fix . From it can be concluded that such that for

Recall the von neumann series associated to an element : . Recall in lecture 2 it was proved that locally in the resolvent set . It follows that converges aboslutely and uniformly for . This shows the desired inclusion.

Note if then and the second claim follows. Now assume . Proceeding by contradiction assume . The resolvent function is holomorhpic off the spectrum thus holomophic in an open set containing . Now choose such that is holomorphic in an openset containing . Choose a bnd. linear functional apply it to to get that

Converges in an openset containing ; in particular is bounded. By the principle of uniform boundedness is bounded say by then

implying that which is a contradiction.QED.

The principal of uniform boundedness says (according to wikipedia, a short proof is also on the wiki page):

**Theorem.** Let *X* be a Banach space and *Y* be a normed vector space. Suppose that *F* is a collection of continuous linear operators from *X* to *Y*. The uniform boundedness principle states that if for all *x* in *X* we have

then

.

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- August 31, 2009 / 10:23 am

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- Banach Algebras (Course)

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