Real and Complex Analysis (Rudin)

364 REAL AND COMPLEX ANALYSIS
If X E J, II IP(X) II = O. If X ¢ J, the fact that J is closed implies that II IP(X) II > O.
It is clear that IIAIP(x)1I = I AI IIIP(x) II· If Xl and X2 E A and € > 0, there exist YI
and Y2 E J so that
Hence
(i = 1, 2). (2)
IIIP(XI + x2)11 ~ Ilxl + X2 + YI + Y211 < IIIP(XI)II + IIIP(X2)11 + 2£, (3)
which gives the triangle inequality and proves (a).
Suppose A is complete and {1P(x n )} is a Cauchy sequence in AIJ. There is a
subsequence for which
(i = 1, 2, 3, ... ), (4)
and there exist elements Z; so that Z; - X n' E J and liz; - z;+lll < 2-;. Thus {z;} is
a Cauchy sequence in A; and since A i~ complete, there exists z E A such that
liz; - zll---+ o. It foliows that lP(x"j) converges to lP(z) in AIJ. But if a Cauchy
sequence has a convergent subsequence, then the full sequence converges. Thus
AIJ is complete, and we have proved (b).
To prove (c), choose Xl and X2 E A and € > 0, and choose YI and Y2 E J so
that (2) holds. Note that (Xl + YI)(X2 + Y2) E XIX2 + J, so that
Now (2) implies
111P(XIX2)11 ~ II(xl + YI)(X2 + Y2)11 ~ Ilxl + Ylllllx2 + Y211. (5)
Finally, if e is the unit element of A, take Xl ¢ J and X2 = e in (6); this gives
111P(e)1I ~ 1. But e E q;(e), and the definition of the quotient norm shows that
II lP(e) II ~ Ilell = 1. So 111P(e)1I = 1, and the proof is complete.
18.16 Having dealt with these preliminaries, we are now in a position to derive
some of the key facts concerning commutative Banach algebras.
Suppose, as before, that A is a commutative complex Banach algebra with
unit element e. We associate with A the set ~ of all complex homomorphisms of
A; these are the homomorphisms of A onto the complex field, or, in different
terminology, the multiplicative linear functionals on A which are not identically O.
As before, u(x) denotes the spectrum of the element X E A, and p(x) is the spectral
radius of x.
Then the following relations hold:
18.17 Theorem
(a) Every maximal ideal M of A is the kernel of some h E ~.
(b) A E u(x) if and only if h(x) = Afor some h E ~.
(c) X is invertible in A if and only if h(x) :F Of or every h E ~.
(d) h(x) E u(x)for every X E A and h E ~.
(e) I h(x) I ~ p(x) ~ Ilxllfor every X E A and h E~.
(6)