Superalgebras and closed idealsAbstract. Let A be a Banach algebra (complex, commutative, unital) which is equipped

ROCZNIK.I POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXVIII (1988)

Hâ k a n He d e n m a l m

(Uppsala, Sweden)

Superalgebras and closed ideals

Abstract. Let A be a Banach algebra (complex, commutative, unital) which is equipped with a colletion of closed ideals whose intersection is JO}. For Banach superalgebras В containing A as a dense subalgebra, we define what it should mean that В inherits J from A.

The main result is that there exists a pseudo-Banach superalgebra s 4 { J ) of A such that В inherits j f from A if and only if the injection mapping A -* s é ( J ) extends to a bounded monomorphism В - > -?/(,/).

Introduction. All algebras we will consider are assumed complex, com­

mutative, and unital. For a Banach algebra, an ideal theory is a characteriza­

tion of its closed ideals and the corresponding quotient algebras. In Heden­

malm [2], [3], the ideal theories of closely related Banach algebras were compared in some typical situations. Here, we will explore the same problem from a different angle, namely when one of the algebras is a dense subalgebra of the other.

I. Notation and basic concepts. An epimorphism is a surjective homo­

morphism, and a monomorphism is an injective homomorphism.

Let A be a Banach algebra. We will denote by Jt{A ) the space of complex homomorphisms on A , endowed with the weak * topology induced by the (topological) dual space A*\ this is the Gelfand space or maximal ideal space of A. Recall that a complex homomorphism is a nonzero homomorphism A -* C, where C denotes the complex field.

We will denote by EUN(.4) the set of all equivalent submultiplicative unital norms, that is, those equivalent norms p which satisfy

p(x-y) ^ p(x)p (j), x, y e A, and p ( l ) = l . It is well known that this set is never empty.

Bornological algebras will appear in this paper. Good references are Allan, Dales, and McClure [1] and Waelbroeck [4], [5].

The so-called pseudo-Banach algebras (Allan, Dales, McClure [1]) con­

stitute a particularly interesting subclass — they are roughly speaking unions

°f Banach algebras that are directed with respect to inclusion, endowed

with the natural inductive limit bornology.

A linear mapping between two bornological algebras is called bounded if it maps bounded sets onto bounded sets.

A subalgebra A of a pseudo-Banach algebra В = [j Ba (where Ba is a

a s /

Banach algebra for every a in the index set I) is said to be a Banach subalgebra if it is equipped with a nornj that makes A a Banach algebra and the injection mapping A —> В is bounded. By the way the bornology on В is defined, a Banach subalgebra A of В must be contained in one of the Banach algebras Ba, and by the closed graph theorem, its norm is determined within equivalence. We speak of В as a pseudo-Banach superalgebra of A, or in case В is a Banach algebra, it is a Banach superalgebra of A. В is a minimal Banach superalgebra of A if it is a Banach superalgebra of A and A is dense in B.

Let f be any family of ideals in an algebra A. For ease of notation, we will use the convention of writing

r a d (/) = Ç] I.

Ief

2. Preliminaries. Let A be an arbitrary Banach algebra. The following lemma will prove useful.

Le m m a

2.1. Suppose px, p2eEUN(X). Then there exists a peEUN(A) such that p ^ min(pl5 p2).

P roof. Put

#1 = Pi 1 ([0, 1]) and = p2 1 ([0, 1]),

the respective closed unit balls. Since p1 and p2 are equivalent norms, there exists a such that

X 1 c 3 2 <=■ X&x.

Let $} be the closed convex hull of which is a subset of X$^

containing u âS2. It is easily checked that is a convex balanced neighborhood of 0 such that

(

2

1

Let p be the Minkowski functional of Д which is an equivalent norm on A.

Then & = p_ 1([0, 1]), and, by (2.1), p is submultiplicative. Hence p(l) ^ 1, but since \ n & 2, p(l) must equal 1. We conclude that

EUN(A)', that p ^ min (pi, p2) is obvious.

3. The problem and its solution. From now on, A is a fixed arbitrarily chosen Banach algebra and # is a family of closed Л-ideals such that

r a d (/) = П I = {0}.

/

/

Let В be a minimal Banach superalgebra of А. Фв denotes the closure operation in B. For every / е / , Фв (1) is a (closed) J3-deal, since A was dense in B. We write У (В) for the image of у under the mapping Фв .

D

3.1. We say that В inherits у from A if

(a) Фв (1) r\A = I for all I e y , which makes Фв a bijection у —► У {В), (b) rad ( y (В)) = П J = { 0}, and

(c) the quotient algebras A /l and В/Фв {1) are canonically isomorphic for all l e y .

R em ark 3.2. It should be observed that it follows from (c) of Defini­

tion 3.1 that A + J = В for every J e У (В), and since (A + J)/J is canonically isomorphic to A/{J n A ), it follows that Фв (1) n A = I for all l e y . Hence (a) is a consequence of (c).

The object o f this paper is to characterize those algebras В which inherit У from A. Our main result, Theorem 3.8, states that there exists a pseudo- Banach superalgebras .<&(У) of A such that В inherits J from A if and only if the canonical monomorphism A -* { / ) extends to a (unique) bounded monomorphism B -> sé (У).

It is now our intention to introduce a family {,я/р}ре^ of minimal Banach superalgebras of A such that sép inherits y from A. The main reason for doing so is that we will be able to show that every minimal Banach superalgebra В of A which inherits у from A is a (dense) Banach subalgebra of some У p. The pseudo-Banach algebra s é { y ) will be the union of all d p.

For all 1 e У , we will consider the norms in EUN(A//) as seminorms on A. Let JP = £Р{У) be the family of all mappings p: A x у -*• [0, oo) such that p (-, I)e E U N (A /I) and

p ( - ,I ) < C - \\- \\A/I for all l e y ,

for some constant C independent of 1. One should observe that SP is nonempty. By our condition rad (У) = {0}, the expression

N ip = supp(x, 1)

is an algebra norm on A for every pe2P. A p denotes the completion of A in the || • ||p-norm, which is a (minimal) Banach superalgebra of A. Put

sJp = A PIrad ( / (A p)), p e 0>.

If we can show that

rad ( y (Ap)) n A = [0J ,

the composition of the injection mapping A ~+Ap and the canonical epi-

morphism Ap -+ ,<tfp will be a bounded ( = continuous) monomorphism

A - * r fp. By the definition of the norm in A p, the canonical epimorphism A -+ A /I has a unique bounded extension L t : A p ->A/I, which is also an epimorphism, for every / е / . Clearly, ker Lj =з ФАр(1), and since Lj is ca­

nonical on A, ker Lj n I. The assertion follows, and hence we may regard .<Vp as a minimal Banach superalgebra of A.

P

3.3. A minimal Banach superalgebra В o f A inherits

from A if and only if the injection mapping A extends boundedly to a mono­

morphism В -+ s / p far some p e 0>.

Proof. We may assume without loss of generality that the norms of A and В are chosen in Е1Ж(Л) and EUN(B), respectively.

Let us deal with the “only if’ part of the assertion first. So, assume В inherits / from A. Since the norm of В belongs to EUN(B), the induced norm on В/Фв (1) belongs to EUN (В/Фв (/)), and by (c) of Definition 3.1, its restriction to A is in Е1Ж(Л//) for every l e / . Put

p(x, I) = ||

+ Ф

(/)||

/

в(/), x e A , l e / . In order to show that peâ? it only remains to check that

P (‘, 1) < C\\-\\A/l for all l e / for some constant C independent of 1. But evidently,

IMI

< C||x|Li,

for some constant C, since A is a Banach subalgebra of B, and consequently, p(x, I) = \\х + Фв (1)\\в/Фв(1) ^ C-\\x + I\\A/I, x e A , l e / .

Hence р е / and since p(x, I) ^ ||x||B,

IWI

^IWI

.

so the injection mapping A ~*Ap extends to a (unique) bounded homo­

morphism j: В —>Ap. Our next step is to show that j is a monomorphism.

Let y e k e r j be arbitrary. Then there exists a sequence {y„}® in A converging to у in the norm of B. Since yeker;, ||yj|p -*0 as n -*■ oo, and consequently

\\Уп

+ Д в и ^ 0 as n-^oo for all J e / ( B ) . Hence у erad(< /(£))= {0}, and the assertion follows.

We will now show that the composition /: В -> л/р of j and the canonical epimorphism A p -*■ p is a monomorphism, too. Let us for simplic­

ity regard В as a subalgebra of A p. By the definition of the norm in A p, the canonical epimorphism В —> B/J extends to a (unique) bounded epimorphism A j\ Ap -* B /J for every J e / ( B ) . Clearly, кегЛ, => ФА (J), and since A j is canonical on B, ker A j n B = J. The assertion follows, since

ker / = r a d ( / И „ ) ) п В = П <PA U )r> B = f W = 0.

Jef(B) J « / ( «

Let us turn to the “i f ’ part of the assertion. So, assume В is a Banach subalgebra of sé p for some p e .é . We have mentioned before that the canonical epimorphism A -> A /l has a unique bounded linear extension L}: sép -* A /I, an epimorphism which is canonical on A. Denote by the restriction to В of L,, which is bounded since В is a Banach subalgebra of sé p. We now intend to show that ker SFi = Фв(/) for all I e j . Obviously, Фв{1) crkerii'V Choose a В-Cauchy sequence c- A converging to an arbitrary x e k e r J ^ . Then ||x„ — уп\\л -* 0 as n-+ oo for some sequence

{уя}® c: / since \\xn + I\\A/I -> 0 as n -> oo, and it follows that Iy„]o° is another B-Cauchy sequence converging to x. We conclude that ker S£t = Фв {1)- Since S£i is canonical on A, ker F£l n A = I, and A /I and B/ker £Fl are canonically isomorphic. This shows that conditions (a) and (c) of Definition 3.1 are met.

(b) follows trivially, since rad ( / ( ^ p)) = |0j . The proof of the proposition is complete.

Rem ark 3.4. A consequence of Proposition 3.3 is the following. Let ) be the set of all closed T-ideals which contain an ideal in j . Then a minimal Banach super algebra of A inherits / from A if and only if it inherits j from A.

Putting В = sép, Proposition 3.3 has the following consequence.

C

3.5. For every pe0*, sé p inherits # from A.

There is a natural order relation on the set \sép}pe^: for p, q e write ség ^ sép if for some constant C,

\\x\\P < C||x||e, x e A .

Clearly, {sép}pe:? is partially ordered by “ =^”. We have the following result.

P

3.6. For any two plf p2e there is a pe0> such that

•^

and séPl

Proof. For every l e f , Lemma 2.1 tells us that there exists a P( *, 7)eEUN(y4//) such that

p { - ,I ) < min (pi (-, /), p2(-, I)).

Clearly this p will do.

The following proposition tells us that we may regard séq as a (dense Banach) subalgebra of sép if séq ^ sép, and therefore the order relation is just ordinary inclusion.

P

3.7. I f séq ^ s é p for two p, q e& , the injection mapping A -> sép has a (unique) bounded extension séq -* sép, which is a monomorphism.

Proof. By the assumptions on p and q, the injection mapping A -* Ap

extends uniquely to a bounded homomorphism /: Aq -» A p. Clearly, this will

define a bounded monomorphism s0q -* s0 p which extends the canonical monomorphism A —> .90p if we can show that

(3.1) and

(3.2)

for all / е / , since then

l( r a d ( f (Aq)j) c: rad ( f (Ap)) and Г 1 (rad ( j ( A p)) n l{Aq)) c rad ( f (A„)).

For / е / , denote by L I p the bounded epimorphism Ap ~* A /I which extends the canonical epimorphism A -* A /I, and let L l q\ Aq —*• A /I be defined analogously. It is easy to see that L I q = L I<pol; just check on A and remember that A is dense in Aq. Thus

ker L/)9 = Г 1 (ker L Up n l (Aq)),

and if we can show that kerL/p = ФЛр(7) and ker LIq = ФА (I), (3.1)— (3.2) will follow easily from this relation because / о /-1 is the identity mapping on the set of subsets of l(Aq). It suffices to verify the assertion for Lt p only, since the proof for L I q would be identical. We will employ the same type of argument as we used in the proof of Proposition 3.3. Obviously, ФАр{1) <=kerL/>p. Choose an Tp-Cauchy sequence {.x,,}® c A converging to an arbitrary x£k erL />p. Then \\

„ + I\\

^ ^ as n->

and hence there is a sequence {yn}^ <= / such that ||x„ — y„\\A -* 0 as n->oo. It follows that {у„}^

is another /l p-Cauchy sequence converging to x, and we conclude that kerL/ p = ФАр{1). The proof of the proposition is complete.

We will regard s0q as a subalgebra of $0p if s0q ^ $0p (p , q e &). Let А ( Я = U

peHf)

which is a pseudo-Banach algebra when endowed with its inductive limit bornology. We are now ready to formulate our main result.

T

3.8. A minimal Banach superalgebra В o f A inherits f from A if and only if the injection mapping A —> s0 { / ) extends boundedly to a monomorphism В —> я0 (,/).

P roof. The “only i f ’ part is clear by Proposition 3.3. On the other hand, if В is a Banach subalgebra of s 0 { # \ then В must be contained in some s0p, pG0>, by the way the bornology on s0 {J!) is defined, and therefore Proposition 3.3 proves the other direction, too.

E

3.9. (a) Let A be the disc algebra A(D), which consists of those holomorphic functions on D = {z e C : \z\ < 1 } that extend continuous­

ly to the boundary dD, and let # consist of the ideals zn • A (£>), n ^ 0, where

z is the coordinate function z(£) = £, { e D . Then ^ ( / ) = C [[z]], the

algebra of formal power series at the origin, and the sets O O

{ £ aHz n€ C [[zJi: \a„| < M„},

n — 0

where \M„)

ranges over all positive sequences, form a base of the born- ology on •</(,/).

(b) Assume A is semisimple, and let / = M { A \ the set of maximal ideals. Regard A as a subalgebra of Then is the uniform closure of A.

4

. Acknowledgements. I should like to thank professor Yngve Domar, who aroused my interest in this type of questions. I should also like to thank the Sweden-America Foundation for financial support.

References

[1] G. R. A lla n , H. G. D a le s , and J. P. M c C lu r e , Pseudo-Banach algebras, Studia Math. 15 (1971), 55-69.

[2] H. H e d e n m a lm , A comparison between the closed modular ideals in ll (w) and Ll (w), Math.

Scand. 58 (1986), 275-300.

[3] —, Bounded analytic functions and closed ideals, J. Analyse Math. 48 (1987), 142-165.

[4] L. W a e lb r o e c k , The holomorphic functional calculus and non-Banach algebras, Algebras in analysis, Academic Press (1975), 187-251.

[5] —, The holomorphic functional calculus as an operational calculus, Banach Center Publica­

tions 8, Warsaw 1982, 513-552.

d e p a r t m e n t o f m a t h e m a t ic s Up p s a l a u n i v e r is t y, Sw e d e n