Inverses and regularity of band preserving operators

The following four main results are proved here. Theorem 3.3.For each one-to-one band preserving operatorT:X → Xon a vector lattice its inverseT⁻¹:T(X) → Xis also band preserving. This answers a long standing open question. The situation is quite different if we move from endomorphisms to more general operators. Theorem 4.2.For a vector lattice X the following two conditions are equivalent:

1.

i)|For each one-to-one band preserving operator T:X → X^u from X to its universal completion X^u the inverse T⁻¹ is also band preserving.

2.

ii)|For each non-zero x ? X and each non-zero band U ⊂ {x}^dd there exists a non-zero semi-component of x in U.

Theorem 5.1.For a vector lattice X the following two conditions are equivalent.

1.

i)|Each band preserving operator T:X → X^u is regular.

2.

ii)|The d-dimension of X equals 1.

Corollary 5.9.Let X be a vector sublattice of C(K) separating points and containing the constants, where K is a compact Hausdorff space satisfying any one of the following three conditions: K is metrizable, or connected, or locally connected. Then each band preserving operatorT: X → Xis regular.