Real and Complex Analysis (Rudin)

370 REAL AND COMPLEX ANALYSIS
14 Suppose A is a commutative Banach algebra with unit, and let .1. be the set of all complex homomorphisms
of A, as in Sec. 18.16. Associate with each x e A a function x on .1. by the formula
x(h) = h(x) (h e .1.).
x is called the Gelfand transform of x.
Prove that the mapping x ..... x is a homomorphism of A onto an algebra A of complex functions
on .1., with pointwise multiplication. Under what condition 01} A is this homomorphism an isomorphism?
(See Exercise 12.)
Prove that the spectral radius p(x) is equal to
IIxIL", = sup {i x(h) I: h e .1.}.
Prove that the range of the function x is exactly the spectrum u(x).
15 If A is a commutative Banach algebra without unit, let A I be the algebra of all ordered pairs (x, A),
with x e A and A a complex number; addition and multiplication are defined in the" obvious" way,
and lI(x, A) II = IIxll + IAI. Prove that Al is a commutative Banach algebra with unit and that the
mapping x ..... (x, 0) is an isometric isomorphism of A onto a maximal ideal of AI' This is a standard
embedding of an algebra without unit in one with unit.
16 Show that HaJ is a commutative Banach algebra with unit, relative to the supremum norm and
pointwise addition and mUltiplication. The mappingf ..... f(lX) is a complex homomorphism of HaJ,
whenever IIX I < 1. Prove that there must be others.
17 Show that the set of all functions (z - 1)2!, where fe H aJ, is an ideal in HaJ which is not closed.
Hint:
1(1- z)2(1 + £ - Z)-I - (1- z)1 < £ if I z I < 1, £ > O.
18 Suppose rp is an inner function. Prove that {rpf:fe HaJ} is a closed ideal in HaJ. In other words,
prove that if {In} is a sequence in H aJ such that rpfn ..... guniformly in U, then g/rp e HaJ.