Positivity (2005) 9:397–399 © Springer 2005
A Dedekind σ -Complete Banach Lattice
into which all Bounded Operators are Regular
Z. ERCAN and S. ONAL
Department of Mathematics, Middle East Technical University, 06531 Ankara, Turkey.
Received: 24 June 2004; accepted 25 January 2005
Abstract. We give an answer to a question of Abramovich–Wickstead asked in [Quart. J.
Math. Oxford (2), 44 (1993), 257–270] under the axiom ‘measurable cardinal exists’.
Mathematics Subject Classiﬁcation (2000): Primary. 47B60
Key words: BIR-space, Banach lattices, measurable cardinal
We refer to standard texts  and  for Banach lattice theory. A compact
Hausdorff space K is called Stonean (quasi-Stonean) if the closure of any
open set (countable union of open sets) is also open, respectively.
Let E and F be Banach lattices. L(E, F ) denotes the space of bounded
operators from E into F and L
(E, F ) stands for the regular operators
from E into F . A Banach lattice F is called a BIR-space if
L(E, F ) = L
(E, F ),
for each Banach lattice E. The notion of a BIR-space was introduced in
 and has been studied in ,  and . In particular, in , it is proved
that a Dedekind complete Banach lattice E is a BIR-space if and only if
it is a unital AM-space.
For convience, we call a compact Hausdorff space K a BIR-space if the
Banach lattice of real valued continuous functions on K, C(K),isaBIR-
space. In  the relationship between Stonean spaces, quasi-Stonean spaces
and BIR-spaces was studied and the following theorem is ‘proved’
THEOREM 1. There exists a quasi-Stonean BIR-space which is not Stonean.
In  (p. 269), it is claimed that the proof of the above theorem is
incorrect and it remained as an open question since then. In this paper we
show that the proof is correct under the axiom ‘measurable cardinal exists’.
Recall that an uncountable cardinal κ is said to be measurable provided