MATHEMATICS

A. Ya. HELEMSKII

DESCRIPTION OF ANNIHILATOR COMMUTATIVE BANACH ALGEBRAS

(Presented by Academician P. S. Novikov, 14 II 1964)

In the present paper we consider a special case of a very general problem: to describe, up to isomorphism, all commutative Banach algebras (c.b.a.) \(\mathfrak A\) with prescribed radical \(\mathfrak R\) and quotient algebra \(A=\mathfrak A/\mathfrak R\).

We shall call \(\mathfrak A\) algebraically decomposable if there exists a subalgebra \(B\) in \(\mathfrak A\) such that \(\mathfrak A=B\oplus\mathfrak R\), and, following \((^1)\), strongly decomposable if there exists such a closed subalgebra. In what follows, by an annihilator c.b.a. we shall mean a c.b.a. \(\mathfrak A\) with one-dimensional radical \(\mathfrak R\) such that, for any \(x\in\mathfrak A\), \(r\in\mathfrak R\), \(xr=0\).

Let \(\mathfrak B\) be an arbitrary c.b.a. with one-dimensional radical, and let \(\mathfrak B'\) be the c.b.a. obtained from \(\mathfrak B\) by adjoining an identity (if \(\mathfrak B\) itself has an identity, \(\mathfrak B'=\mathfrak B\)). It is shown in \((^2)\) that in \(\mathfrak B'\) there exists a maximal ideal which is an annihilator c.b.a.

Thus the case of a c.b.a. with one-dimensional radical is reduced to the case of annihilator c.b.a.’s. Below we give a complete description of algebraically decomposable annihilator c.b.a.’s, proceeding from the properties of the quotient algebra \(A=\mathfrak A/\mathfrak R\).

Let \(A\) be a semisimple c.b.a. Introduce the following notation: \(S(A)\) is the unit ball of the Banach space \(A\); \(T\) is the convex hull of the set \(S(A)\cdot S(A)=\{xy:x,y\in S(A)\}\); \(L(T)\) is the linear span of \(T\); finally, \(S=S(A)\cap L(T)\). Thus in \(L(T)\), along with the given norm (which will hereafter be called the \(S\)-norm), another norm is introduced, specified by the convex centrally symmetric absorbing set \(T\) (we shall call it the \(T\)-norm). The \(T\)-norm is, obviously, no weaker than the \(S\)-norm.

Lemma 1. In order that a linear functional \(f\) on \(L(T)\) be continuous in the \(T\)-norm, it is necessary and sufficient that
\[ |f(xy)|\le C_1\|x\|\|y\| \]
for some constant \(C_1\) and all \(x,y\in A\).

By \(L^*(S)\) and \(L^*(T)\) we denote the Banach spaces conjugate to \(L(T)\), taken respectively with the \(S\)-norm and the \(T\)-norm.

Let \(f\in L^*(T)\), and let \(F\) be its (arbitrary) linear extension to all of \(A\). Introduce in \(A\) a new norm \(\|\cdot\|_F\), setting, for \(x\in A\),
\[ \|x\|_F=\max\{\|x\|;\ |F(x)|\}. \]

With the aid of Lemma 1 one proves

Lemma 2. The norm \(\|\cdot\|_F\) satisfies the multiplicative condition (i.e.
\[ \|xy\|_F\le C_2\|x\|_F\|y\|_F \]
for a constant \(C_2\) and all \(x,y\in A\)).

Consequently, completing \(A\) in the norm \(\|\cdot\|_F\), we obtain a certain c.b.a.

The following holds.

Theorem 1. Let the c.b.a. \(\mathfrak A\) be the completion of \(A\) in the norm \(\|\cdot\|_F\). Then \(\mathfrak A\) is semisimple if and only if \(F\) is continuous on \(A\), and in this case \(\mathfrak A\) is isomorphic to \(A\). In the opposite case \(\mathfrak A\) is an algebraically decomposable annihilator c.b.a. with quotient algebra \(\mathfrak A/\mathfrak R\) isomorphic to \(A\).

In order to obtain, by means of an arbitrary \(F\), an annihilator c.B.a., we agree, in the case when \(F\) is continuous on \(A\), to adjoin directly to \(A\) an annihilating one-dimensional radical \(\mathfrak R\).

Thus, for any \(f \in L^*(T)\), taking its extension \(F\) to all of \(A\), we obtain, by the method described, an algebraically decomposable annihilator c.B.a. \(\mathfrak A\) such that \(\mathfrak A/\mathfrak R\) is isomorphic to \(A\). We shall call it an \(f\)-inflation of the algebra \(A\) and denote it by \(\mathfrak A(A,f)\). The correctness of this definition is shown by Lemma 3.

Lemma 3. Let \(f \in L^*(T)\); let \(F_1\) and \(F_2\) be two of its extensions to all of \(A\); and let \(\mathfrak A'\) and \(\mathfrak A''\) be two \(f\)-inflations obtained from \(A\) with the aid of the functionals \(F_1\) and \(F_2\). Then \(\mathfrak A'\) and \(\mathfrak A''\) are isomorphic.

We proceed directly to the classification of algebraically decomposable annihilator c.B.a.’s by means of \(f\)-inflations. The first step is Lemma 4.

Lemma 4. Let \(\mathfrak A\) be an arbitrary algebraically decomposable annihilator c.B.a. such that \(\mathfrak A/\mathfrak R\) is isomorphic to \(A\). Then \(\mathfrak A\) is isomorphic to an \(f\)-inflation \(\mathfrak A_0=\mathfrak A_0(A,f)\) for some \(f \in L^*(T)\).

Introduce in the set \(L^*(T)\) two equivalence relations:

\(1^\circ.\ \sim:\ f \sim g\) \((f,g \in L^*(T))\), if, for some constants \(C_3\) and \(C_4\),
\(|f(x)| \leq C_3 \max\{\|x\|, |g(x)|\}\) and \(|g(x)| \leq C_4 \max\{\|x\|, |f(x)|\}\) for all \(x \in L(T)\).

\(2^\circ.\ \simeq:\ f \simeq g\), if there exists an automorphism \(\omega\) of the c.B.a. \(A\) such that for all \(x \in L(T)\)
\(f(\omega x)=g(x)\).

Denote by \(\cong\) the composition of the equivalence relations \(\sim\) and \(\simeq\) (for the general definition see \((^3)\)). The basis of everything that follows is

Lemma 5. Let \(\mathfrak A_f=\mathfrak A_f(A,f)\) and \(\mathfrak A_g=\mathfrak A_g(A,g)\). Then \(\mathfrak A_f\) and \(\mathfrak A_g\) are isomorphic if and only if \(f \cong g\).

Lemma 5, together with Lemma 4, makes it possible to prove the theorem describing the required class of c.B.a.’s.

Theorem 2. Let \(A\) be a semisimple c.B.a. There exists a one-to-one correspondence between the classes of isomorphic algebraically decomposable annihilator c.B.a.’s \(\mathfrak A\) with \(\mathfrak A/\mathfrak R\), isomorphic to \(A\), on the one hand, and the classes of \(\cong\)-equivalent linear functionals from \(L^*(T)\), on the other.

The following theorem gives a simple geometric criterion for strong decomposability of annihilator c.B.a.’s.

Theorem 3. Let \(A\) be a semisimple c.B.a.; let \(T\) and \(S\) be the sets defined above. In order that every algebraically decomposable annihilator c.B.a. \(\mathfrak A\) with \(\mathfrak A/\mathfrak R\), isomorphic to \(A\), be also strongly decomposable, it is necessary and sufficient that, for some constant \(C_5\), the inclusion
\(S \subseteq C_5 T\) hold.

Algebras for which such a constant exists will be called algebras possessing the \(S\)-property.

As is known, for semisimple c.B.a.’s Gelfand’s theorem \((^4)\) holds: algebraically isomorphic semisimple c.B.a.’s are also topologically isomorphic. Theorem 3 makes it possible to describe, among the algebraically decomposable annihilator c.B.a.’s, the class of those possessing an analogous property. Namely, the following holds.

Corollary. Let \(\mathfrak A\) be an algebraically decomposable annihilator c.B.a. In order that every c.B.a. algebraically isomorphic to \(\mathfrak A\) be also topologically isomorphic to \(\mathfrak A\), it is necessary and sufficient that the factor algebra \(A=\mathfrak A/\mathfrak R\) possess the \(S\)-property.

Remark. Trivial examples of c.B.a.’s possessing the \(S\)-property are all c.B.a.’s with identity, as well as all closed ideals of the algebras \(C(\Omega)\) of all continuous functions on a bicompactum. A less trivial example is the algebra \(l_1\) of absolutely convergent series with coordinatewise multiplication. On the other hand, the algebras \(l_p\) for \(1<p<\infty\), as well as the maximal ideals of the algebras \(D_n\) of \(n\)-times differentiable functions of one or several variables, certainly do not possess the \(S\)-property.

In conclusion we formulate a criterion for strong decomposability of a c.b.a. with an arbitrary one-dimensional radical.

Theorem 4. Let \(A\) be a semisimple c.b.a., and let \(A'\) be the c.b.a. obtained from \(A\) by adjoining an identity (if \(A\) itself has an identity, \(A'=A\)). In order that every algebraically decomposable c.b.a. \(\mathfrak A\) with one-dimensional radical \(\mathfrak R\) such that \(\mathfrak A/\mathfrak R\) is isomorphic to \(A\), be also strongly decomposable, it is necessary and sufficient that all maximal ideals of the algebra \(A'\) have the \(S\)-property.

In conclusion, the author expresses his gratitude to M. A. Naimark for his attention to the work.

Moscow State University
named after M. V. Lomonosov

Received
31 I 1964

REFERENCES

1. W. G. Bade, P. C. Curtis, Am. J. Math., 82, 851 (1960).
2. A. Khelemskii, Vestn. Moskovsk. Univ., No. 5 (1964).
3. A. G. Kurosh, Lectures on General Algebra, Moscow, 1962, pp. 17–18.
4. I. M. Gelfand, D. A. Raikov, G. E. Shilov, Commutative Normed Rings, Moscow, 1960, p. 70.