# Lecture 2

### I.3 Adjunction of a unit

If is a nonunital banach algebra. Make it unital as follows with norm

and multiplication

Clear that is a unit. Also use smart factoring and triangle inquality to show

so we get a bonafied banach algebra.

### I.4

Let B be a unital banach algebra, and . The spectrum of a, . If is nonunit set . The complement of the spectrum of a is the resolvent set. Define a function (in the nonunital case its a map to ) by

Thm: The spectrum is nonempty and . Further, is holmorphic in the resolvent set.

Note, has values in ; for such a function to be holomorphic means for each point in its domain there is an open nbd for which it is represented by a convergent power series. Another definition would be that after applying any functional , the resulting composition is holomorphic. The first definition implies the second, the opposite implication is a hard thm.

proof: WLOG is unital. (Step 1) let then formally

the latter series (as a function in z) converges locally uniformly in consequently the product of the series with can be taken term by term (it be regarded as the limit of the taking the product with all finite subsums); doing this shows indeed this series is the inverse. This gives the second claim of the thm. Note tends to 0 as z tends to infinity.

(Step 2) fix . Write

For

conv. abs. hence get local uniform convergence inside . Check again this series is actually an inverse by multiplying things out. The conclusion is that is holomorphic in the claimed region.

(Step 3) By step 2, the resolvent set is open and is holomorhphic in it. Its also a bnd function that tends to 0, so it the spectrum were empty it would be an entire function (using the Banach Louville thm; can reduce to ususal complex case by applying a functional to B) hence identically 0, this is an obvious contradiction. QED.

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You’re currently reading “Lecture 2,” an entry on Math Meandering

- Published:
- August 28, 2009 / 12:57 pm

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- Banach Algebras (Course)

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