Real and Complex Analysis (Rudin)

ELEMENTARY THEORY OF BANACH ALGEBRAS 363
If Xl - X2 E J, then cp(x l ) = CP(X2)· If Xl - X2 ¢ J, cp(x l ) 11 cp(x2) = 0. The set of
all cosets of J is denoted by AI J; it is a vector space if we define
cp(x) + cp(y) = cp(x + y), Acp(X) = cp(AX) (2)
for X and YEA 'and scalars A. Since J is a vector space, the operations (2) are well
defined; this means that if cp(x) = cp(x') and cp(y) = cp(y'), then
cp(x) + cp(y) = cp(x') + cp(y'), Acp(X) = Acp(X'). (3)
Also, cp is clearly a linear mapping of A onto AjJ; the zero element of AIJ is
cp(O) = J.
Suppose next that A is not merely a vector space but a commutative algebra
and that J is a proper ideal of A. If x' - X E J and y' - y E J, the identity
x'y' - xy = (x' - x)y' + x(y' - y) (4)
shows that x'y' - xy E J. Therefore multiplication can be defined in AIJ in a
consistent manner:
cp(X)cp(y) = cp(xy) (X and YEA). (5)
It is then easily verified that AIJ is an algebra, and cp is a homomorphism of A
onto AjJ whose kernel is J.
If A has a unit element e, then cp(e) is the unit of AIJ, and AIJ is afield if and
only if J is a maximal.feal.
To see this, suppose.x E A and X ¢ J, and put
I = {ax + y: a E A, Y E J}. (6)
Then I is an ideal in A which contains J properly, since x E I. If J is maximal,
I = A, hence ax + y = e for some a E A and y E J, hence cp(a)cp(x) = cp(e); and
this says that every nonzero element of AIJ is invertible, so that AIJ is a field. If J
is not maximal, we can choose x as above so that I "# A, hence e ¢ I, and then
cp(x) is not invertible in AjJ.
18.15 Quotient Norms Suppose A is a normed linear space, J is a closed subspace
of A, and cp(x) = x + J, as above. Define
IIcp(x)1I = inf {lix + yll: y E J}. (1)
Note that IIcp(x)1I is the greatest lower bound of the norms of those elements
which lie in the coset cp(x); this is the same as the distance from x to J. We call
the norm defined in AIJ by (1) the quotient norm of AIJ. It has the following
properties:
(a) AIJ is a normed linear space.
(b) If A is a Banach space, so is AIJ.
(c) If A is a commutative Banach algebra and J is a proper closed ideal, then AIJ
is a commutative Banach algebra.
These are easily verified: