MAXIMAL IDEALS IN TOPOLOGICAL ALGEBRAS

By

E. BECKENSTEIN

in Brooklyn, N. Y., U.S.A.

L. NARICI

in Jamaica, N. Y., U.S.A.

AND

G. BACHMAN

in Brooklyn, N. Y., U.S.A.

1. In troduct ion

If M is a maximal ideal in a complex commutative Banach algebra with

identity, then M has codimension one and consists entirely of singular ele-

ments. Gleason [1] has shown the converse statement to hold as well. In

this paper, Gleason's result is extended to a certain class of topological al-

gebras. Throughout the paper X denotes a complex commutative Hausdorff

topological algebra with identity e and X ' denotes the space of continuous

linear functionals on X .

2. Pre l iminar ie s

We list a few facts about topological algebras here, referring to Michael

[2] for proofs.

X is called local ly m-convex if X is locally convex and there is a neighbor-

hood base at 0, {U}, such that for all such U , U U ~ U . In locally m-convex

algebras there is a 1-1 correspondence between the set Jl/c of all closed max-

imal ideals and the set q) of nontrivial continuous complex homomorphisms

on X ([2], p. 11). Accordingly we shall identify ~ and ./go.

X is called a Q-algebra if the invertible elements in X form an open set;

in Q-algebras all maximal ideals are closed so, letting d{ denote the set of all

159

160 E. BECKENSTEIN L. NARICI AND G. BACHMAN

maximal ideals, ..r =-~ 'c . If X is a locally m-convex barreled complete

algebra, then the following properties are euqivalent: (1) X is a Q-algebra;

(2) the spectrum of each x ~ X is compact; (3) ~/g = ~ ' c is compact in the

a(X', X) topology. ([2], p. 56).

3. C h a r a c t e r i z a t i o n of M a x i m a l Ideals

T h e o r e m . Let X be a locally m-convex barreled complete Q-algebra

and let M be a linear subspace of X of codimension one which contains

no invertible elements. Then M is a maximal ideal.

P r o o f . The hypothesis in conjunction with the results quoted in the pre-

ceding section show that J/r is a(X', X)-compact. Let C(dr denote the com-

mutative Banach algebra (with sup norm) of continuous complex valued

functions on ~ ' . For any x E X let ~ denote the function sending any h ~ ~e'

into h(x); let ~b denote the complex homomorphism x- - , ~ from X into

C(~t'). Note that @(X) -- .Y is a normed algebra with identity.

Since codlin M = 1, then the sets {pe + M} cover X as p runs through

the complex numbers C. Thus .~ = U, ~c(#~ + ~/) where 37/= ~k(M). If

~ )~7/, then there is some x ~ M such that ~ -- g , and it follows that h ( e - x ) = 0

for all h e ~/r Since M consists solely of singular elements, however, there

exists ([2], pp. 20 and 56) h o ~ . / / such that ho(x ) = 0 and it follows that

ho(e ) = 0 which cannot be. Therefore ~ ~ 37/and 37/is seen to be a subspace

of ~ of eodimension one.

Next we show that 37I is closed in ~ .

If 37/is not closed in ,(', then ~7/must be dense in ~ . Consequently there is

a sequence of points {~,} from 37I such that ~n - , ~. Thus for 1 > e > 0 and

sufficiently large n,

Since ~(h) = 1, it follows that s = h(x~) ~ 0 for any h ~ ~t' and sufficiently

large n. Since M consists only of singular elements, this cannot be and ~/

is seen to be closed.

Now consider ~ c i y c C(d[) . We contend that 37I is a subspace of

consisting entirely of singular elements.

MAXIMAL IDEALS IN TOPOLOGICAL ALGEBRAS 161

Certainly a;/ consists of singular elements of C ( J l ) - - a n d therefore of i f - -

because for any 2 e 37I there is an h e ~ such that g(h) = 0. Since .~ is

Banach algebra with identity, the set of singular elements is closed and it

follows that ~r contains singular elements only.

To show that aTI is of codimension 1 in ~ , consider g e )~, xn ~ g where

~ E )~and 2~ = # ~ +fin where fin ~ M. We show that {/~} is a Cauchy sequence

in order to show that {fin} is Cauchy. If this were done, then there would be

# e C a n d f ~ ~r such that ~t n ~ # and 33 n o f so that g = / ~ + f , thus proving

that ~r is of codimension one in f t . If {/In} were not Cauchy then for some

positive e' and any N , there must be no, m0 > N such that [/Z~o -/~mo I > e'.

Now, given ~ > 0, we choose N such that for all m, n > N

II II = II o ,n- + ran- ,:11 <

With no and m o as above it follows that

II ~ + o,.o - ~,.,o)-'(.Ono - .~,.o)II < ~*' / I V.o - ~,.,o I =< * .

Since (# ,o - 12mo)-l(Yno- Ymo) ~2~I' we have exhibited a singular element

in 2 as close to ~ as we please. Since this is contradictory, {~tn} must be a

Cauchy sequence and therefore ~ is of eodimension one.

By the result in [1] mentioned in the Introduction, ~ is an ideal in f t .

Hence, since A;/is a closed subspace of .~, it follows that ~r is an ideal in 2 .

Thus ~b-l(~ r) is an ideal in X. Now ~ - 1 ( ~ / ) ~ M and e r so

~-1(37/) = M and M is seen to be an ideal in X. Since codim M = 1, the

maximality of M is clear.

REFERENCES

1. A. Gleason, A characterization of maximal ideals, J. d'Analyse Math., 19 (1967), 171-2. 2. E. A. Michael, Locally multiplicatively-convex topological algebras, Memoirs Amer.

Math. Sot., 11 (1952).

POLYTECHNIC INSTITUTE OF BROOKLYN BROOKLYN, NEW YORK, U.S.A.

ST. JOHN'S UNIVERSITY JAMAICA, NEW YORK, U.S.A.

POLYTECHNIC INSTrrUTE OF BROOKLYN BROOKLYN, N~w YORK, U.S.A.

(Received June 15, 1970)