The structure of compact disjointness preserving operators on continuous functions

Abstract

The duals of C 0 (a, b) and C[a, b] with respect to disjointness preserving groups are characterized. A. Plessner's result (1929) about the translation group is extended. A Wiener-Young type theorem for disjointness preserving groups is obtained. It is assumed that the reader is familiar with the notions of a Banach lattice, positive operator and disjointness preserving operator. [AB], [LZ] and [MN] are good references for this material. We also assume familiarity with the theory of C 0-semigroups and groups. [Pa] and [Na] should provide all necessary information when needed. Throughout −∞ ≤ a < b ≤ +∞, X is either (a, b) or [a, b], where by [−∞, b] we mean (−∞, b], and by [a, +∞] we mean [a, +∞). Definition. A function f : X → C is called vanishing at infinity if ∀ε > 0 ∃K ⊂ X, K compact such that ∀x ∈ X \ K |f (x)| < ε. We denote the lattice of all continuous functions f : X → C vanishing at infinity by C 0 (X), the lattice of all bounded continuous functions f : X → C by C b (X), the lattice of all continuous functions f : X → C with compact support by C c (X), and the lattice of all regular complex-valued Borel measures on X with finite variation by M (X). Let {T (t)} t∈R be a C 0-group on a Banach space E. Definition. The group dual of E with respect to {T (t)} t∈R , denoted E and pronounced E-sun, is defined in the following way: E = { u * ∈ E * : lim t→0 T * (t)u * − u * = 0 }. [vN] is an excellent source of information about the semigroup and group duals of Banach spaces and related subjects. Definition. A group {T (t)} t∈R on a Banach lattice E is called disjointness preserving if ∀t ∈ R T (t) is a disjointness preserving operator. Let {T (t)} t∈R be a disjointness preserving C 0-group on C 0 (X). It follows from [Na, B-II.3.8] that ∃ϕ : R × X → X a continuous flow ∃q : R → C b (X) a

Weakly compact orthogonality preservers on C*-algebras

“…Whenever A is an abelian C * -algebra, every irreducible finite type I von Neumann factor in A * * is isomorphic to C. Thus, the description of the weakly compact disjointness preserving linear operators between commutative C * -algebras obtained by Lin and Wong [27] follows now as consequence of our Theorem 8 (see also Remark 9).…”

“…Lin and Wong proved in [27,Theorem 2.6] that T being compact is equivalent to T being weakly compact. On the other hand, every compact operator between Banach spaces factorises compactly through some closed subspace of c 0 (compare [13, Exercise 6, (iii), Page 15]).…”

“…On the other hand, every compact operator between Banach spaces factorises compactly through some closed subspace of c 0 (compare [13, Exercise 6, (iii), Page 15]). Corollary 4.1 in [27] shows that a compact orthogonality preserving operator between two abelian C * -algebras actually admits a compact factorisation through the whole c 0 . Our next corollary extends these results to the setting of general C * -algebras.…”

“…Lin and Wong [27] considered weakly compact orthogonality preserving operators between abelian C * -algebras (i.e. C 0 (L)-spaces).…”

Characterizations of Disjointness preserving operators on vector-valued function spaces

“…It was shown that a bounded disjointness preserving operator is a weighted composition operator. In the recent paper [9], a concrete representation is given for compact, weakly compact and completely continuous disjointness preserving operators from C 0 (X) into C 0 (Y ).…”