or lattice Y is, in a sense, determined up to an orthomorphism from the family of the kernels of the strata πT of T with π ranging over all band projections on Y . Similar reasoning was involved in [4] to characterize order bounded disjointness preserving bilinear operators. Unfortunately, Theorem 3.4 in [4] is erroneous and this note aims to give correct statement and proof of this result. Unexplaine <...> or. We denote the Boolean algebra of band projections in X by P(X). Recall that a linear operator T : X →Y is said to be disjointness preserving if x ⊥ y implies Tx ⊥ Ty for all x, y ∈ X. A bilinear operator B : X Ч Y → Z is called disjointness preserving (a lattice bimorphism) if the linear operators B(x, ·) : y → B(x, y) (y ∈ Y ) and B(·, y) : x → B(x, y) (x ∈ X) are disjointness preserving for all x ∈ X and y ∈ Y (lattice homomorphisms for all x ∈ X+ and y ∈ Y+). Denote Xπ := {ker(πB(·, y)) : y ∈ Y } and Yπ := {ker(πB(x, ·)) : x ∈ X}. Clearly, Xπ and Yπ are vector subspaces of X and Y , respectively. Now we state the main result of the note. Theorem. Assume that X, Y , and Z are vector lattices with Z having the projection property. For an order bounded bilinear operator B : X Ч Y → Z the following assertions are equivale <...> ideals X0 ⊂ X and Y0 ⊂ Y . (iii) There exist lattice homomorphisms g : X → R and h : Y → R such that either Similarly, β(x, ·) is disjointness preserving for all x ∈ X and thus (ii) =⇒(i). The <...> he role of a field of reals within V(B). The descending functor sends every internal from X ЧY to R↓ and from X∧ ЧY ∧,R]] ∈ V(B) stand for the sets respectively of all maps modi <...>
Владикавказский_математический_журнал_№1_2015.pdf
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