Using elementary arguments, we shall show that every bounded disjointness preserving linear functional of the Wiener ring L1 (Z) assumes the form (f)=

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).

In this paper, we give a complete description of the structure of zero product and orthogonality preserving linear maps between W*-algebras. In particular, two W*-algebras are *-isomorphic if and only if there is a bijective linear map between them preserving their zero product or orthogonality structure in two directions. It is also the case when they have equivalent linear and left (right) ideal structures.

For each real α,0≤α<1, we give examples of endomorphisms in dimension one with infinite topological entropy which are α‑Hölder; and for each real p,1≤p<∞, we also give examples of endomorphisms in dimension one with infinite topological entropy which are (1,p)-Sobolev. These examples are constructed within a family of endomorphisms with infinite topological entropy and which traverse all α-Hölder and (1,p)-Sobolev classes. Finally, we also give examples of endomorphisms, also in dimension one, which lie in the big and little Zygmund classes, answering a question of M. Benedicks.

We state a conjecture for the formulas of the depth 4 low-weight rotational eigenvectors and their corresponding eigenvalues for the 3^G subfactor planar algebras. We prove the conjecture in the case when |G| is odd. To do so, we find an action of G on the reduced subfactor planar algebra at f^{(2)}, which is obtained from shading the planar algebra of the even half. We also show that this reduced subfactor planar algebra is a Yang-Baxter planar algebra.