# First lecture

Disclaimer: I don’t know if I’ll keep this up… and other stuff

Recall (or learn) that a Banach space is a real or complex normed vector space such that is complete with respect to that norm. In this class probably only the complex version will be relevant. Also, for a T2 space K, the sup norm is defined

For a measurable function f, the essential supremum is the smallest number m such that the set where f is bigger than m is measure 0. For a given measure then define a space

Also consists of series indexed by the integers which are convergent when summed with the p-norm, i.e. summing pth powers of the terms (and if the answer is finite taking the pth root).

### I.1 Basic Definitions

A complex banach algebra is an algebra over which is a banach space s.t..

for all

A banach algebra need not be unital (example below). It if it unital, say with identity then there is an isomorphism of algebras so in the case of unital B the identity is suppressed in the notation to just 1. Note the product inquality implies . Here is some notation

- is the space of linear operators on a banach space X
- are the spaces of continuous, continuous that vanish at infinity, and continuous bounded functions. The operations are defined pointwise.
- = cont. funct. on [0,1] diff.’able n times.

Now some examples:

- If is a banach space, then any norm closed sub algebra of is a banach algebra.
- is a banach algebra.
- if is any banach space, can make it to a banach algebra by def mult: for all .
- If is a compact and T2 then (with the sup norm) is a unital banach algebra.
- If is locally compact, T2 but not compact (i.e. the real line) then $latexC_0(\mathscr{X})$ is a non unital banach algebra.
- For any top. sp , is a banach algebra.
- For a measure the space is a banach algebra. (note elements of this algebra are in general equivalence classes of functions, because…. )
- (Disk algebra) The algebra of continuous functions on that are holomorphic in .
- normed by
- with convolution as product.
- with convolution as product:

Prop. I.1.1 Let B be a banach algebra and is closed two sided ideal then is a banach algebra.

proof: Clear that is an algebra. A basic functional analysis result says if is banach and is closed then is banach (critch says to use a criterion that banach iff absolutely convergent implies convergent; see wiki absolute convergence; also use that the norm of is and that the inf is actually reazlied, these are enough pieces to show is banach. For the inquality let e>0 and take such that

(i.e. points that just a little bit away from the closest point in the coset to the identity) then multiplying:

and take lim as e goes to 0. QED.

Exercise: if is a normed alg/, then its completion is a banach algebra.

### I.2 Regular representations

Here is a unital banach alg. for any element a, there is a linear operator sending (Similarly there are operator ). Note because the operator achieves the bound when acting on 1.

There’s a natural map given by . (similarly for the right multiplication). This gives the left regular representation. Note L is injective and .

Lemma: iff T commutes with for all b.

proof: the (=>) direction is trivial. For the converse, say T commutes as in the lemma then (for )

QED.

Conclusion: LB is closed (commutting with everything above is a closed condition/intersection of closed things). Now the open mapping theorem (Banach-Schauder) in functional analysis says that a surjective cont. linear transformation of a banach space X to another Y is an open mapping. This is necessary to conclude L gives an iso from to . Thus replacing by have for sure that the identity is norm 1.

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- Published:
- August 26, 2009 / 1:39 pm

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- Banach Algebras (Course)

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