Let X be a compact Hausdorff space and C(X) the space of continuous functions defined on X. There are three versions of the Banach-Stone theorem. They assert that the Banach space geometry, the ring structure, and the lattice structure of C(X) determine the topological structure of X, respectively. In particular, the lattice version states that every disjointness preserving linear bijection T from C(X) onto C(Y) is a weighted composition operator Tf = h . f which provides a homeomorphism from Y onto X. In this note, we manage to use basically algebraic arguments to give this lattice version a short new proof. In this way, all three versions of the Banach-Stone theorem are unified in an algebraic framework such that different isomorphisms preserve different ideal structures of C(X).

Let X and X be compact Hausdorff spaces, and X , X be Banach lattices. Let X denote the Banach lattice of all continuous X -valued functions on X equipped with the pointwise ordering and the sup norm. We prove that if there exists a Riesz isomorphism X such that X is non-vanishing on X if and only if X is non-vanishing on X , then X is homeomorphic to X , and X is Riesz isomorphic to X . In this case, X can be written as a weighted composition operator: X , where X is a homeomorphism from X onto X , and X is a Riesz isomorphism from X onto X for every X in X . This generalizes some known results obtained recently.

Let A be a A -algebra with real rank zero. Let A and A be Hilbert A -modules with A being full. Suppose that A is a linear map preserving orthogonality, ie,

The notion of generating topologies, introduced by Stephani El0], is useful for studying the injective hull of an operator ideal. Using Randtke's idea (see [8, p. 90] or [12,(3.2. 1) and (3.2. 7)]), we can characterize generating topologies in terms of seminorms which satisfy some expected properties (see Lemma 3.3 and Theorems 3.4 and 3.9). By a well-known and useful idea of Grothendieck, the dual notions of generating topologies and ideal-topologies, the so-called generating bornologies, are given and studied in Sect. 4. In term of ideal-bornologies, the surjective hull of an operator ideal on Banach spaces is given (see Lemma 4.8 and Theorem 4.10). In terms of these two dual concepts, we are able to classify locally convex spaces, and to study their dual results. For instance, we show that if 9.1 is a symmetric (resp. completely symmetric) operator ideal on Banach spaces then a Banach space E is an 9~-topological

Suppose<i>X</i>and<i>Y</i>are locally compact Hausdorff spaces,<i>E</i>and<i>F</i>are Banach spaces, and<i>F</i>is strictly convex. We show that every linear isometry<i>T</i>from<i>C</i>₀(<i>X</i>,<i>E</i>)<i>into</i><i>C</i>₀(<i>Y</i>,<i>F</i>) is essentially a weighted composition operator<i>Tf</i>(<i>y</i>)=<i>h</i>(<i>y</i>)(<i>f</i>((<i>y</i>))).