MATHEMATICS

A. I. VEKSLER

ON LINEAR STRUCTURES WITH A SUFFICIENT SET OF MAXIMAL \(l\)-IDEALS

(Presented by Academician A. I. Mal'tsev, 5 I 1963)

Some properties are studied of Archimedean* \(K\)-lineals (linear structures) in which every \(l\)-ideal is embedded in a maximal one.

In the theory of \(l\)-groups and, in particular, \(K\)-lineals, the concept of an \(l\)-ideal plays a significant role. We shall everywhere have in mind only proper \(l\)-ideals. There is a certain analogy between ideals in rings (and algebras) and \(l\)-ideals in \(l\)-groups and \(K\)-lineals. It is well known that the presence of a multiplicative identity in a ring (or algebra) ensures the possibility of embedding every ideal in a maximal one. In the case of a \(K\)-lineal (or \(l\)-group), the analogous property for \(l\)-ideals is ensured by the presence of a strong unit**. In the general case this is, generally speaking, not so. For example, in the Archimedean \(K\)-lineal of all functions continuous on \((0,1]\), the subset of bounded functions is an \(l\)-ideal that is not embedded in any maximal one.

Let us introduce the basic definition. We first recall that an \(l\)-ideal, or normal sublineal, of a \(K\)-lineal \(X\) is any linear subset \(I \subset X\) such that, if \(x \in I\), \(y \in X\), and \(|y| \leq |x|\), then \(y \in I\) (\(I\) is automatically a substructure of \(X\)). We shall use only the first of these terms.

Definition. An Archimedean \(K\)-lineal will be called a \(K\)-lineal with a sufficient set of maximal \(l\)-ideals, or an \(MK\)-lineal, if every one of its \(l\)-ideals can be embedded in a maximal one. It is clear that a similar definition can also be given for an arbitrary \(l\)-group.

It is \(MK\)-lineals that are mainly studied in this work. All \(K\)-lineals occurring below will, without any reservations, be considered Archimedean. In particular, by a \(K\)-lineal of bounded elements we shall mean an Archimedean \(K\)-lineal with a strong unit.

As was noted above, a \(K\)-lineal of bounded elements is an \(MK\)-lineal. Thus the concept of an \(MK\)-lineal is a generalization of the concept of a \(K\)-lineal of bounded elements. In particular, an \(MK\)-lineal may even have no unit.

Let us note that in the case when a \(K\)-lineal is simultaneously an algebra, the theory of ordinary \(l\)-ideals differs essentially from the theory of \(l\)-ideals that are at the same time ideals of the algebra (or, more simply in the present case, ring ideals). Such ideals were studied, for example, in \((^5)\), and there they were also called \(l\)-ideals. In particular, it turns out

* A \(K\)-lineal \(X\) is called Archimedean if from \(x, y \in X\) and \(0 \leq nx \leq y\) for every natural \(n\) it follows that \(x = 0\).

** An element \(1\) of an \(l\)-group \(X\) is called a unit if \(1 \wedge |x| > 0\) for every \(x \ne 0\), \(x \in X\); a unit is called strong if \(|x| \leq n1\) for some natural \(n = n(x)\). A strong unit belongs to no proper \(l\)-ideal, and this makes it possible to apply Zorn’s theorem to the set of all proper \(l\)-ideals ordered by inclusion; from the latter it follows that every \(l\)-ideal is embeddable in a maximal one.

different classes of \(K\)-lineals—algebras having a sufficient set of maximal \(l\)-ideals in the usual sense and in the sense of \({}^{(5)}\).

We shall use mainly the terminology of \({}^{(4)}\). Some concepts introduced in \({}^{(2)}\) will also be used.

1. Let \(X\) be a \(K\)-lineal, \(\hat X\) its \(K\)-completion, i.e., a \(K\)-space which is the completion of \(X\) by the method of sections (\({}^{(4)}\), p. 126). It is well known that \(\hat X\) is realized as a certain normal\(^*\) set of continuous functions on a uniquely determined, up to homeomorphism, extremally disconnected bicompact space \(Q\), which may take the values \(\pm\infty\) on nowhere dense subsets of \(Q\) (\({}^{(4)}\), p. 155). Therefore \(X\) too is realized as a certain set \(\mathfrak C_\infty(Q)\) of such functions (not normal, if \(X\) is not a \(K\)-space). This bicompact space \(Q=Q(X)\) in \({}^{(2)}\) was called the canonical bicompact space for \(X\). The main result of this part is a characterization of an \(M K\)-lineal in terms of its realization in the form \(\mathfrak C_\infty(Q)\) on \(Q(X)\).

Preliminarily, using the terminology and results of \({}^{(2)}\), we introduce some concepts.

On \(Q(X)\) there may be points \(t'\) such that for every function from \(\mathfrak C_\infty(Q)\)

\[ t' \in \overline{\{t \in Q : x(t) \ne 0\}} . \]

For any such point, under every realization of \(X\) on \(Q(X)\),\(^ {**}\) one always has \(x(t')=0\) for each of the functions considered in this realization—images of elements of \(X\). It is clear that such points are, in a certain sense, superfluous on \(Q(X)\).

Definition. The set \(S=S(X)\) of all points of the canonical bicompact space which do not possess the property just considered, with the topology induced from \(Q(X)\), will be called the proper space for \(X\). In the case where \(X\) is a \(K\)-space, this concept coincides with the concept of proper space from \({}^{(8)}\).

The proper space \(S(X)\), generally speaking, is smaller than the canonical bicompact space \(Q(X)\), coinciding with it, as is not difficult to verify, if and only if \(X\) has a unit.

Let \(t \in S(X)\). Then there exists \(x_0>0\) such that, for some realization, \(x_0(t)>0\). The ratio \(\dfrac{x(t)}{x_0(t)}\) has the same value for all realizations for which \(x_0(t)\ne0,\infty\); in \({}^{(2)}\) the numerical value of this ratio was denoted by \(\dfrac{x}{x_0}[t]\).

Definition. A point \(t_0 \in S(X)\) will be called strong if, for some \(x_0>0\) and every \(x \in X\), one always has

\[ \left|\frac{x}{x_0}\right|[t_0] < +\infty . \]

The element \(x_0\) itself is called a strong local unit (s.l.u.) at \(t_0\). If there is no such \(x_0\), the point will be called weak.

We shall call a \(K\)-lineal \(X\) a \(K\)-lineal with a sufficient set of strong local units (s.s.s.l.u. \(K\)-lineal) if \(S(X)\) consists only of strong points (and, consequently, for every \(t \in S(X)\) there is an s.l.u.).

Theorem 1. For every strong point \(t_0\), the set

\[ M_0=\left\{x\in X:\frac{x}{x_0}[t_0]=0\right\}, \]

where \(x_0\) is an s.l.u. at \(t_0\), is a maximal \(l\)-ideal of the \(K\)-lineal \(X\). Conversely, every maximal \(l\)-ideal in \(X\) is representable in this form.

\(^*\) A set of continuous functions is called normal if, together with every continuous function, it contains also every other function not larger in modulus.

\(^ {**}\) Different realizations of \(X\) on \(Q(X)\) are possible. Each realization is determined by the choice of an element from a maximal extension (universal completion according to \({}^{(8)}\)) \(\hat X\), which is realized as the function identically equal to one.

Theorem 2. In order that a \(K\)-lineal be an \(MK\)-lineal, it is necessary and sufficient that it be a d.m.s.l.e. \(K\)-lineal.

1. Here we formulate two theorems concerning the relation between maximal \(l\)-ideals in a \(K\)-lineal \(X\) and in its \(K\)-completion \(\hat X\).

Theorem 3. In order that a \(K\)-lineal \(X\) be an \(MK\)-lineal, it is necessary and sufficient that its \(K\)-completion \(\hat X\) be an \(MK\)-lineal.

It is not hard to see that always \(S(\hat X)=S(X)\). Let now \(M_0\) be a maximal \(l\)-ideal of \(X\), considered in Theorem 1, and let \(\hat M_0\) be the corresponding maximal \(l\)-ideal in \(\hat X\). The natural correspondence \(\hat M_0 \to M_0\) associates with each maximal \(l\)-ideal of \(\hat X\) a maximal \(l\)-ideal in \(X\). In this case \(M_0=\hat M_0\cap X\). However, this correspondence need not be one-to-one (for \(t_1\ne t_0\) always \(\hat M_1\ne \hat M_0\), while at the same time it may happen that \(M_1=M_0\), i.e. the correspondence established in Theorem 1 between maximal \(l\)-ideals and strong points is, generally speaking, not single-valued if \(X\) is not a \(K\)-space). If the correspondence \(\hat M_0\to M_0\) is nevertheless one-to-one, we shall say that the sets of maximal \(l\)-ideals in \(X\) and in \(\hat X\) coincide.

Let us recall a few more definitions. A component of a \(K\)-lineal \(X\) is any set \(X'\subset X\) which is the disjoint complement of some \(E\subset X\). It is easily proved \(\left({}^{3},\text{ p. }13;\ \text{see also }{}^{6},\text{ p. }352\right)\) that this definition is equivalent to the usual definition of a component in a \(K\)-space \(\left({}^{4},\text{ p. }97\right)\). If for every \(x\in X\) and for every component \(X'\) there exists a projection of \(x\) onto \(X'\) (for \(x>0\) this means the existence of \(\sup\{x'\in X':0\le x'\le x\}\)), then \(X\) is called a \(K\)-lineal with projections.

Theorem 4. Let \(X\) be an \(MK\)-lineal. In order that the sets of maximal \(l\)-ideals in \(X\) and in \(\hat X\) coincide, it is necessary and sufficient that \(X\) be a \(K\)-lineal with projections.*

1. It is known (see \(\left({}^{1}\right)\), Theorems 2 and 3; \(\left({}^{3}\right)\), Theorems 3.5 and 3.6) that for a \(K\)-lineal of bounded elements \(X\) the following assertions are equivalent.

A. For some \(K\)-space \(Y\), the spaces \((X\to Y)_r\) and \((\hat X\to Y)_r\) of regular operators from \(X\) to \(Y\) and from \(\hat X\) to \(Y\) coincide (i.e. every regular operator from \(X\) to \(Y\) admits a unique extension to all of \(\hat X\)).

B. For every \(Y\), \((X\to Y)_r\) and \((\hat X\to Y)_r\) coincide.

C. For some \(Y\), the following criterion of disjointness of regular operators from \((X\to Y)_r\) is satisfied.

If \(U_1,U_2\in (X\to Y)_r\), then, for their disjointness, it is sufficient that there exist two mutually complementary components \(X_1\) and \(X_2\) such that
\[ U_1(X_1)=U_2(X_2)=\{0\}. \]

D. For every \(Y\) in \((X\to Y)_r\), the criterion of disjointness is satisfied.

E. \(X\) is a \(K\)-lineal with projections.

In the case of an arbitrary \(K\)-lineal, assertion E implies all the others, but follows from none of them.

However, for an \(MK\)-lineal \(X\) the corresponding conclusion is valid; that is, we have

Theorem 5. For an \(MK\)-lineal \(X\), assertions A, B, C, D, E are equivalent.

Along the way the following lemma is established, which is also of independent interest.**

* Some characteristics of arbitrary \(K\)-lineals with projections are given in \(\left({}^{2,6}\right)\).

** For a \(K\)-lineal of bounded elements, the assertion of the lemma was essentially established in \(\left({}^{7}\right)\).

Lemma. Let \(X\) be a \(K\)-lineal, let \(x_0\) be a weak order unit, and let \(t_0 \in S(X)\). Then the functional

\[ f_{t_0,x_0}(x)=\frac{x}{x_0}[t_0] \]

is discrete*. Conversely, every positive discrete functional has the indicated form.

1. As the simplest example of an \(MK\)-lineal \(Z\) that is not a \(K\)-lineal of bounded elements, let us take the so-called Kaplansky \(K\)-lineal—the set of all functions on \([0,1]\), each of which is continuous except possibly for a finite number of jumps of the second kind. It can be shown that every maximal \(l\)-ideal in \(Z\) is the set of all functions from \(Z\) that are finite at a certain point. It can also be shown that all functions from an arbitrary \(l\)-ideal are finite (or even equal to 0) at at least one point. Hence it is easily seen that \(Z\) is an \(MK\)-lineal.

\(Z\) does not possess some of the properties that hold in the \(K\)-lineal of bounded elements. In \(Z\), as is not difficult to verify, the theorem on an annihilating sequence does not hold; in \(Z\), evidently, there does not exist a sufficient set of discrete functionals.

One can verify that if \(X\) is some \(MK\)-lineal, then, for example, so is each of its components onto which a projection can be carried out. If, moreover, \(X\) is a continuous \(K\)-lineal with projections, then every minimal simple** \(l\)-ideal of it is an \(MK\)-lineal in which there is no unit.

The author expresses gratitude to Prof. B. Z. Vulikh for discussion of the manuscript of the article and to D. A. Vladimirov for checking the proof.

Leningrad Textile Institute
named after S. M. Kirov

Received
23 XI 1962

CITED LITERATURE

1. A. I. Veksler, Izv. vyssh. uchebn. zaved., Mathematics, No. 1, 48 (1960).
2. A. I. Veksler, Sibirsk. matem. zhurn., 3, No. 1, 7 (1962).
3. A. I. Veksler, Dissertation, Leningrad State Pedagogical Institute named after A. I. Herzen, 1959.
4. B. Z. Vulikh, Introduction to the Theory of Partially Ordered Spaces, 1961.
5. M. Henriksen, D. G. Johnson, Fund. Math., 50, No. 1, 73 (1961).
6. D. G. Johnson, J. E. Kist, Arch. d. Math., 12, No. 5, 349 (1961).
7. M. G. Krein, S. G. Krein, Matem. sborn., 13, No. 1, 1 (1943).
8. H. Nakano, Modern Spectral Theory, Tokyo, 1950.

* An element \(x\) of a \(K\)-lineal \(X\) is called discrete if from \(|y|\le |x|\), \(y\in X\), it follows that \(y=\alpha x\) for some real \(\alpha\). A regular functional is discrete if it is discrete as an element of the \(K\)-space of regular functionals.

** An \(l\)-ideal \(I\) is called simple if from \(x\wedge y\in I,\ x\notin I\) it follows that \(y\in I\). A simple \(l\)-ideal is called minimal if it contains no other simple \(l\)-ideal.