Reflexivity and approximate reflexivity for bounded Boolean algebras of projections

The ideal center of the dual of a Banach lattice

“…This means that Z (E ) leaves invariant each w * -closed C(K ) -invariant subspace of E and commutes with m * (C(K ) ). Then Lemma 5 [7] implies that Z (E ) ⊂ m * (C(K ) ). This completes the proof.…”

“…In particular the proof uses an analogue of the Factorization Theorem of Lozanovsky on Banach function spaces [10], [5] that is proved in [1] for Banach C(K )-modules. For an earlier version of Arenson's Theorem in the resricted setting of Bade's Theorem see ([14], Theorem 2), for a weaker version see ( [7], Lemma 5). This latter result states (in the same circumstances as Arenson's Theorem): If an operator T on E leaves invariant each w * -closed C(K ) -invariant subspace of E and commutes with m * (C(K ) ) then T is in m * (C(K ) ).…”

On Reflexivity of Scalar-Type Spectral Operators

“…Since C(K) is an AM-space with unit, it is isomorphic to C(S) with S a hyperstonian [1,Theorem 15.7], [6], [7]. It is well-known that X' is a locally convex C(5)-module [5], [6].…”

“…It is well-known that X' is a locally convex C(5)-module [5], [6]. From [5], [6], [7], it is known that the linear mapping m* : C(K)" -> L(X') defined by m*{a)x' = a.x', for each a € C(K)", x' 6 X' satisfies the following properties:…”

Characterization of the Dual Center of Barrelled Spaces

“…is known as Arens product [3], [4], and [5]. It is well-known that C(K) is an AM-space with unit and the second dual C(K)" of C(K) is a Dedekind complete AM-space with unit [5].…”