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 V, || - || such that V 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

||f||_\infty = sup_{x \in K} |f(x)|

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 \mu then define a space 

L^\infty(\mu) = \{ f | \mbox{ess sup} |f| < \infty\}

Also l^p(\mathbb{Z}) 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 B over \mathbb{C} which is a banach space B, | - |  s.t..

|xy| \le |x| \cdot |y| for all x,y, \in B

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

  • \mathscr{L}(X) is the space of linear operators on a banach space X
  • C(\mathscr{X}), C_0(\mathscr{X}), C_b(\mathscr{X}) are the spaces of continuous, continuous that vanish at infinity, and continuous bounded functions.  The +, \cdot operations are defined pointwise.
  • \mathbb{D} = \{z \in \mathbb{C} | z\bar z = 1\}
  • C^n[0,1] = cont. funct. on [0,1] diff.’able n times.

Now some examples:

  1. If X is a banach space, then any norm closed sub algebra of \mathscr{L}(X) is a banach algebra.
  2. \mathbb{C} is a banach algebra.  
  3. if X is any banach space, can make it to a banach algebra by def mult: x \cdot y = 0 for all x,y \in X.
  4. If \mathscr{X} is a compact and T2 then C(\mathscr{X}) (with the sup norm) is a unital banach algebra.
  5. If \mathscr{X} is locally compact, T2 but not compact (i.e. the real line) then $latexC_0(\mathscr{X})$ is a non unital banach algebra.
  6. For any top. sp \mathscr{X}, C_b(\mathscr{X} is a banach algebra.
  7. For a measure \mu the space L^\infty(\mu) is a banach algebra. (note elements of this algebra are in general equivalence classes of functions, because…. )
  8. (Disk algebra) The algebra of continuous functions on \overline{\mathbb{D}} that are holomorphic in \mathbb{D}.
  9. C^n[0,1] normed by || f|| = \sum_{k = 0}^n \frac{1}{k!} || f^{(k)}||_{\infty}
  10. L^1(\mathbb{R}) with convolution as product.
  11. l^1(\mathbb{Z}) with convolution as product: x*y = \sum_{k = -\infty}^\infty x(n-k)y(k)

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

proof: Clear that B/J is an algebra.  A basic functional analysis result says if B is banach and J is closed then B/J 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 x + J is \inf_j d(x,j) and that the inf is actually reazlied, these are enough pieces to show B/J is banach.  For the inquality let e>0 and take x,y such that

|x| < |x + J| + e, |y| < |y+J| + e

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

|(x+J)||(y+J)| = |xy + J| \le |xy| \le (|x+J| + e)(|y+J| + e)

and take lim as e goes to 0. QED

Exercise: if B is a normed alg/\mathbb{C}, then its completion is a banach algebra.

I.2 Regular representations

Here B is a unital banach alg. for any element a, there is a linear operator L_a \colon B \to B sending x \mapsto ax  (Similarly there are operator R_a).  Note |L_a| \le |a| because the operator achieves the bound when acting on 1.

There’s a natural map L \colon B \to \mathscr{L}(B) given by L(a) = L_a.  (similarly for the right multiplication).  This gives the left regular representation.  Note L is injective and |L| \le 1.  

Lemma: T \in \mathscr{L}(B) \in LB iff T commutes with R_b for all b.

proof: the (=>) direction is trivial.  For the converse, say T commutes as in the lemma then (for a = T1)

Tx = T(1\cdot x) = T(R_x 1) = R_xT1 = ax = L_ax

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 B to LB.  Thus replacing a by L_a have for sure that the identity is norm 1.

 

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