A version of Kalton’s theorem for the space of regular operators

Abstract

In this note we extend some recent results in the space of regular operators [appeared in Bu and Wong (Indag Math 23:199–213, 2012), Bu et al. (Collect Math 62:131–137, 2011), and Li et al. (Taiwan J Math 16:207–215, 2012)]. Our main result is the following Banach lattice version of a classical result of Kalton: Let \(E\) be an atomic Banach lattice with an order continuous norm and \(F\) a Banach lattice. Then the following are equivalent: (i) \(L^r(E,F)\) contains no copy of \(\ell _\infty \), (ii) \(L^r(E,F)\) contains no copy of \(c_0\), (iii) \(K^r(E,F)\) contains no copy of \(c_0\), (iv) \(K^r(E,F)\) is a (projection) band in \(L^r(E,F)\), (v) \(K^r(E,F)=L^r(E,F)\).