1. UNIFOR M ALGEBRAS: SOM E BASIC NOTIONS, RESULTS, AND EXAMPLES

We shall be mainly concerne d wit h uniform (o r su p normed) algebras : close d subalge -

bras A o f C(X), X compac t Hausdorff , whic h contain th e constants . Frequentl y w e will

assume A separate s th e point s of X, an d almos t alway s A wil l not b e close d unde r con

jugation.

The simples t an d bes t exampl e is the dis c algebra A = A(D) consistin g of all continu -

ous / o n th e close d dis c D i n C analyti c o n its interior D° . Fro m th e maximum prin -

ciple we know A an d A\T, T= dD, ar e isometrically isomorphi c objects , and w e have

our choice o f investigating A o n D, o r T, o r any se t between, so that ther e ar e man y

candidates fo r X. An d of course a particular X ma y enjo y specia l properties, as for exam

ple in

WERMER'S MAXIMALIT Y THEORE M

1.1A . = A(D)\T is a maximal (proper) closed

subalgebra of C(T).

Trivially thi s is false if T i s replaced b y an y large r subse t of D. Befor e proceeding ,

we will note a simple recent proof (du e to Lumer), since it provides a good simple exampl e

of th e us e of functional analyti c tools . W e only have t o know tha t (i ) the ope n bal l of

unit radiu s about th e identity i n a Banach algebr a consist s of invertibles (for th e proof ,

write th e obviou s power serie s for ( 1 - x

)- 1

) , an d (ii ) /* * e

ind

f(e

w)dd

= 0 fo r n 1

implies feA (fo r fGC(T)).

Suppose ACBC C(T), wit h B closed . I f e~

w

G £ the n of course B = C(T) by

Stone-Weierstrass. Thu s we can assume \\e~ w - B\\ = inf \\e~ w - b\\ 0, o r ||1 -V0Z||O

for al l beB. An d eve n if we had || 1 - e ie b\\ 1 fo r som e b the n e w beB~l (th e

set o f invertibles in B), so (e'

wb~1)'

b =

e~w

G B again . Thu s \\l-e

wb\\l

fo r

all b, o r || 1

-ewB\\=

1.

Now by th e Hahn-Banac h an d Ries z representatio n theorem s w e have a complex

measure JJL o f norm 1 o n T whic h is 1 a t 1 an d orthogona l to e

ldB

(vi a the fact

say, that th e dua l (C(T)/e

wB)*

i s a subspace of M(T)= C(7)*) . Bu t

M 0 )=

1 =

HMI I

implies / i i s a probability measur e an d sinc e

eind

G e

i6Bf

n l

^

ie

ind ^ g ) = 0 , H 1,

hence als o for n - 1, an d ix(dd) = dO/lir. Bu t no w

ix{eind b) = ii{ew e i( n-^d b)= 0 , n 1,

implies b E A, a s we know fro m (ii) .

In ou r exampl e of the dis c algebra, D i s the spectru m A = AA (o r maxima l idea l

1