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Multiplication in Vector Lattices MULTIPLICATION IN VECTOR LATTICES NORMAN M. RICE 1. Introduction. B. Z. Vulih has shown (13) how an essentially unique intrinsic multiplication can be defined in a Dedekind complete vector lattice L having a weak order unit. Since this work is available only in Russian, a brief outline is given in § 2 (cf. also the review by E. Hewitt (4), and for details, consult (13) or (11)). In general, not every pair of elements in L will have a product in L. In § 3 we discuss certain properties which ensure that, in fact, the multiplication will be universally defined, and it turns out that L can always be embedded as an order-dense order ideal in a larger space L# which has these properties. It is then possible to define multiplication in spaces without a unit. In § 4 we show that if L has a normal integral 0, then 4> can be extended to a normal integral on a larger space Li(<t>) in L#, and £i(0) may be regarded as an abstract integral space. In § 5 a very general form of the Radon-Niko- dym theorem is proved, and in § 6 this is used to give a relatively simple proof of a theorem of Segal giving a necessary and sufficient condition for the Radon-Nikodym theorem to hold in a measure space. 2. Multiplication in spaces with a unit. Let L be a vector lattice which is Dedekind complete (i.e., every set which is bounded above has a least upper bound) and has a weak order unit 1 (i.e., inf (1, x) > 0, whenever x > 0). An element e G L is called unitary if inf(e, 1 — e) = 0. (These correspond, roughly, to characteristic functions.) e will always denote a unitary element, and U(L, 1) = U(L) will denote the set of unitary elements. It is easy to see that any set E of unitary elements is bounded below by 0 and above by 1 so that sup(E) and inf(E) exist, and it is not hard to show that sup(E) and inf(E) are also unitary, so we can conclude (and this will be useful in § 6) that U(L) is a complete Boolean algebra. For any xÇl the characteristic element of x is defined to be s(x) = supn'mî(n \x\, 1). s(x) is always a unitary element, s(x) = 0 if and only if x = 0, s (ax) = s(x) for any real number a ^ 0, andx_L y (i.e., inf (\x\, \y\) = 0) if and only if s(x) _L s(y). Freudenthal has shown (3) that for every 0 g x £ L there exists a largest unitary element e such that e ^ x, and that e = 1 — s[(l — x)+\. It follows Received January 4, 1967. The contents of this paper are largely derived from the author's Ph.D. thesis, written under the supervision of Professor W. A. J. Luxemburg and presented to the California Institute of Technology in 1966. 1136 Downloaded from https://www.cambridge.org/core. 29 Sep 2021 at 00:13:52, subject to the Cambridge Core terms of use. VECTOR LATTICES 1137 that if we define ka{x) = 1 — s[{al — x)+], then a-ka{x) ^ x for any 0 ^ x G L, and &a(x) is the largest unitary element with this property. Freudenthal showed that it also follows that if 0 < x G L, then there exists 0 < e G U{L) and a > 0 such that ae ^ x. Vulih used these results to show that any 0 ^ x G L can be achieved as the supremum of all the linear combinations of unitary elements that lie below it. For applications in § 6 we shall need a somewhat stronger result. THEOREM 2.1. If 0 rg x G L, /Aew x = sup{r£r(x): rational r ^ 0}. Proof. Since rkr{x) S x for every r, there exists y = sup{rkr{x)} ^ x. Suppose x — y > 0; then there exists e > 0 and a > 0 such that x—3/ > 2ae. Let 5 = supjfr': b'e S x} ^ 2a, and let r be a rational number such that b — a ^ r ^ b. Then 0 < re ^ x, therefore e ^ &r(x), and hence rg S rkr{x) ^ 3;. But then (b + a)g = (b — a)e + 2ae ^ re + 2ae ^ y + 2ae ^ x, contra­ dicting the maximality of b. We now define multiplication: (i) If e, e' G U{L), the product ee' is defined by ee' = inf(e, e'). (ii) If x ^ 0 and y ^ 0, the product X3/ is defined by xy = sup {abee'i 0 ^ ae S x, 0 t^ be' S y} if this supremum exists. xy is not defined if the supremum does not exist. (iii) In general, the product xy is denned by xy = x+y+ — x+3>_ — X-y+ + x_j_ if all the products on the right exist. Note. Vulih's definition of multiplication in (13) is formally somewhat a different. For x, y ^ 0, if 0 ^ x' — £ x<?x ^ x and 0^3/ = X^Me'M g y are two finite sums, he defines x'y' to be Ex.^W/»» and then defines X3> to be sup {x'y : 0 g x' ^ x, 0 ^ 3/ S y} if this supremum exists. He shows, how­ ever, that the particular representation of x' as a finite sum does not affect the product x'y', and with this observation it is easy to see that his definition of xy coincides with the one given above; for we may write x' and y' in such a way that they have disjoint summands, so that ^a,\b ^e'^ has disjoint summands and hence equals supx,M{ax&Me\e'M}- We list below some of the properties of the multiplication. M(i) xl always exists and equals x. M (ii) If xy exists, then yx exists and equals xy. M (iii) If xy, {xy)z, and yz all exist, then x(yz) exists and equals {xy)z. M(iv) If xy and xz exist, then x(y + z) exists and equals xy + xz. M(v) If xy exists and a is real, then {ax)y exists and equals a{xy). Downloaded from https://www.cambridge.org/core. 29 Sep 2021 at 00:13:52, subject to the Cambridge Core terms of use. 1138 NORMAN M. RICE M(vi) If x, y ^ 0 and xy exists, then xy ^ 0. M(vii) If xy exists, and |x'| :§ \x\ and \y'\ S \y\, then x'y' exists. M (viii) x _L y if and only if xy exists and equals 0. It can be shown that any (partially defined) multiplication in L which satisfies the above eight properties must in fact be identical to the Vulih multiplication. Remark. The uniqueness referred to above depends, of course, on the unit 1 (cf. property M(i)). In general, two elements which have a certain prod­ uct with respect to one unit will have a different product (or none at all) with respect to another unit. However, there is a connecting formula (cf. 11, Theorem 5.3): let 1 and V be two units of L\ denote the product of x and y with respect to 1 by xy, and the product with respect to V by x * y; if xy and x * y both exist, then V {x * y) = xy. Some further properties of the multiplication are the following. M(ix) If xy exists, then s(xy) = inf(s(x), s(y)). M (x) If x rjs y, then there exists e > 0 and a > 0 such that xe ^ ye + ae. M(xi) For any element x ^ 0 and any integer n > 0 there is a unique positive nth root of x, i.e., a unique y ^ 0 such that yn exists and equals x. M(xii) Let {xa} and {ya\ be two nets in L indexed by the same directed set. Suppose (0)-lim(xa) = x, (0)-lim(;>/«) = y, xaya exists for each a, and there exists z Ç L such that \xaya\ ^ z for all a. Then the product xy exists in L, and (0)-lim(xaya) = xy. M(xiii) Since xe + x(l — *) = x with x(l — e) Ç {e}x and xe 6 {^}J--L, we see that xe is the component of x in [e], the normal subspace of L generated by e (cf. 2, Chapter II, § 1.5). Vulih defines the inverse of an element x to be an element y (if such exists) such that s(y) = s(x) and x^ = s(x). He denotes the inverse of x by x-1, and proves, for instance: I(i) If x ^ 0 and x~l exists, then x_1 ^ 0, I(ii) If xy = s(x), then x_1 exists and x~l = y-s(x), I(iii) Let x = y + z, where y 1. z. If x-1 exists, then y~l and z-1 exist, and x_1 = 3>_1 + z~1. Conversely, if y~l and z~x exist, then x_1 exists, _1 l I(iv) If x exists, and \y\ ^ |x| and s(y) = s(x)y then y~ exists and ly-1] ^ Ix-1!. Remark. Vulih's proof of I (iv) can be considerably simplified by noting the following criterion (cf. 11, Theorem 4.2): for x ^ 0, let S = {y ^ 0: s (y) ^ s(x), and xy S s(x)}; then x_1 exists if and only if sup (S) exists, and in this case x_1 = sup (S). 3. Rings, and extensions to rings.
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