Real and Complex Analysis (Rudin)

ELEMENTARY THEORY OF BANACH ALGEBRAS 367
PROOF Let B be the closure in C = C(T) of the set of all functions of the form
The theorem asserts that B = C. Let us assume B :F C.
The set of all functions (3) (note that N is not fixed) is a complex algebra.
Its closure B is a Banach algebra which contains the function fo, where
fo(e i '1 = e ilJ • Our assumption that B :F C implies that lifo ~ B, for otherwise B
would contain f'O for all integers n, hence all trigonometric polynomials
would be in B; and since the trigonometric polynomials are dense in C
(Theorem 4.25) we should have B = C.
So fo is not invertible in B. By Theorem 18.17 there is a complex homomorphism
h of B such that h(fo) = O. Every homomorphism onto the
complex field satisfies h(l) = 1; and since h(fo) = 0, we also have
h(f'O) = [h(foW = 0 (n = 1, 2, 3, ... ). (4)
We know that h is a linear functional on B, of norm at most 1. The
Hahn-Banach theorem extends h to a linear functional on C (still denoted by
h) of the same norm. Since h(l) = 1 and IIhll ~ 1, the argument used in Sec.
5.22 shows that h is a positive linear functional on C. In particular, h(f) is
real for real f; hence h(j) = h(f). Since f 0" II is the complex conjugate of f'O, it
follows that (4) also holds for n = -1, -2, -3, .... Thus
I if n = 0,
{
h(f'O) = 0 if n of o.
(5)
Since the trigonometric polynomials are dense in C, there is only one
bounded linear functional on C which satisfies (5). Hence h is given by the
formula
(3)
(fe C). (6)
Now if n is a positive integer, gf'O e B; and since h is mUltiplicative on B,
(6) gives
g( - n) = ~ I" g(e i '1e illlJ dO = h(gf'O) = h(g)h(f'O) = 0, (7)
21t _"
by (5). This contradicts the hypothesis of the theorem.
We conclude with a theorem due to Wiener.
18.21 Theorem Suppose
IIII
00 00
f(e i '1 = L c II e'"IJ, L IcIII < 00, (1)
-00 -00