CC BY
9
Владикавказский математический журнал 2008, Том 10, Выпуск 2, С. 30-31
УДК 512.555+517.982
AN ALMOST F-ALGEBRA MULTIPLICATION EXTENDS FROM A MAJORIZING SUBLATTICE
A. G. Kusraev
It is proved that an almost /-algebra multiplication defined on a majorizing sublattice of a Dedekind
complete vector lattice can be extended to the whole vector lattice.
Mathematics Subject Classification (2000): 06F25, 46A40.
Key words: almost /-algebra, /-algebra, vector lattice, majorizing sublattice, lattice homomorphism,
positive operator, square of a vector lattice.
A lattice ordered algebra (E, •) is called an almost f -algebra if x Л y = 0 implies x • y = 0 for all x, y G E or equivalently |x| • |x| = x • x for every x G E (cp. [2]). C. B. Huijsmans in [6] posed the question of whether the multiplication of an almost f-algebra can be extended to its Dedekind completion. G. Buskes and A. van Rooij in [4, Theorem 10] answered in the affirmative to the question and this result raises naturally another question: Can an almost f-algebra multiplication given on a majorizing vector sublattice of a Dedekind complete vector lattice be extended to an almost f-algebra multiplication on the ambient vector lattice? A positive answer was announced in [8, Corollary 7] by the author:
Theorem. Let E be a majorizing sublattice of a Dedekind complete vector lattice E and simultaneously an almost f-algebra. Then E can be endowed with an almost f algebra multiplication that extends the multiplication on E.
The aim of this note is to present the proof. Our reasoning is along the same lines as in [4] and rely upon a general structure theorem for almost f-algebras saying that they are actually distorted f-algebras as was shown in [4, Theorem 2]. All vector lattices and lattice ordered algebras are assumed to be Archimedean.
Recall that an f-algebra is a lattice-ordered algebra whenever xЛy = 0 implies (a • x^y = 0 and (x • a) Л y = 0 (or equivalently (x • y) Л a = 0 provided that x Л a = 0 or y Л a = 0) for all x, y G A and a G A+. It is well known that an f-algebra multiplication is commutative [2] and order continuous [9].
For an arbitrary vector lattice E there exists a (essentially unique) pair (E0,0) such that E0 is a vector lattice, 0 is a symmetric lattice bimorphism from E x E to E0 and the following universal property holds: for every symmetric lattice bimorphism b from E x E to some vector lattice F there exists a unique lattice homomorphism Ф5 : E0 ^ F with b = Ф&0. This notion was introduced by G. Buskes and A. van Rooij, see [5] and [3]. The said universal property remains valid if we replace b and Ф5 by a positive orthosymmetric ( = xЛy = 0 ^ b(x, y) = 0) bilinear operator and a positive linear operator provided that F is uniformly complete, see [5, Theorem 9] and [3, Theorem 3.1].
© 2008 Kusraev A. G.
An almost f -algebra multiplication extends from a majorizing sublattice
31
We now present the needed structure result from [4, Theorem 2]. Let E be an arbitrary vector lattice and h a lattice homomorphism from E into a Dedekind complete semiprime f-algebra G with a multiplication o. Then F := h(E) is a sublattice of G and in view of [3, Proposition 2.5] F0 can be considered also as a sublattice of G with (x, y) ^ x o y instead of 0. Denote by F(2) the linear hull of {x o y : u, v G F}. Then F0 is the sublattice of G generated by F(2) and F(2) is uniformly closed in F0.
Let a positive linear operator $ from F(2) to E and an element w G G are such that h$(u) = w o u for all u G F(2). Of course, one can consider $ as a positive operator from F0 to Eru, the uniform completion of E. Put x • y := $(hx o hy) (x, y G E). Then (E, •) is an almost f-algebra. Indeed, (x, y) ^ x • y can be taken as an almost f-algebra multiplication, since evidently x A y = 0 implies hx o hy = 0, whence x • y = 0 and its associativity is also easily seen:
(x • y) • z = $(w o hx o hy o hz) = x • (y • z).
It is proved in [4, Theorem 2] that every Archimedean almost f-algebra arises in this way.
< Proof of the Theorem. Let E be a majorizing sublattice of a Dedekind complete vector lattice E and also an almost f-algebra under a multiplication •. According to [4, Theorem 2] one can choose F, G, o, h, and $ as above. By [1, Theorem 7.17] or [7, Theorem 3.3.11 (2)] there exists a lattice homomorphism h from E onto a sublattice F C G extending h. Moreover, F is a majorizing and order dense sublattice of F. By [3, Proposition 2.7] F0 is also a majorizing and order dense sublattice of (i^)0. According to [1, Theorem 2.8] or [7, Theorem 3.1.7] the positive operator $ from F0 to Eru C E has a positive extension $ from(F^)0 to E. Now, h$ is obviously an extension of h$ and it remains to ensure that h$ (u) = w o u (u G (F^)0), since in this event an almost f-algebra multiplication on E can be defined by x • y := $(h(x) oh(y)) (x, y G E) as was observed above. For a fixed u G (F^)0 take arbitrarily u;, u" G F0 such that u' ^ u ^ u//. Then w o u/ = h$ (u;) ^ h$ (u) ^ h$ (u") = w o u'/ and thus, by order continuity of f-algebra multiplication, sup{w o u/} = h$(u) = inf{w o u//} = w o u. >
References
1. Aliprantis C. D., Burkinshaw O. Positive Operators.—New York: Acad. press, 1985.—xvi+367 p.
2. Birkhoff G. Lattice Theory.—Providence (RI): Amer. Math. Soc., 1967.—(Colloq. Publ., № 25.)
3. Buskes G., Kusraev A. G. Representation and extension of orthoregular bilinear operators // Vladikavkaz Math. J.—2007.—V. 9, № 1.—P. 16-29.
4. Buskes G., van Rooij A. Almost /-algebras: structure and Dedekind completion // Positivity.—2000.— V. 4, № 3.—P. 233-243.
5. Buskes G., van Rooij A. Squares of Riesz spaces // Rocky Mountain J. Math.—2001.—V. 31, № 1.— P. 45-56.
6. Huijsmans C. B. Lattice-Ordered-Algebras and and /-algebras: A Survey // In: Stidies in Economic Theory.—Berlin etc.: Springer, 1990.—V. 2.—P. 151-169.
7. Kusraev A. G. Dominated Operators.—Dordrecht: Kluwer, 2000.—446 p.
8. Kusraev A. G. On the structure of orthosymmetric bilinear operators in vector lattices // Dokl. RAS.— 2006.—V. 408, № 1.—P. 25-27.—In Russian.
9. de Pagter B. /-algebras and orthomorphisms.—Leiden, 1981.—(Thesis).
Received Mart 24, 2008.
Anatoly G. Kusraev
Institute of Applied Mathematics and Informatics Vladikavkaz Science Center of the RAS Vladikavkaz, 362040, RUSSIA E-mail: [email protected]
