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.