For Banach lattices X and Y, let X | | Y and X | | Y denote the positive projective and injective tensor products of X and Y, respectively. In this paper, we characterize the inheritance of reflexivity and containment of copies of c 0, 1, by X | | Y and X | | Y from X and Y, when one of them is an atomic Banach lattice. In this case, we also give an affirmative answer to an open question of Jeurnink.

A unital ring is called clean (resp. strongly clean) if every element can be written as the sum of an invertible element and an idempotent (resp. an invertible element and an idempotent that commutes). T.Y. Lam proposed a question: which von Neumann algebras are clean as rings? In this paper, we characterize strongly clean von Neumann algebras and prove that all finite von Neumann algebras and all separable infinite factors are clean.

In this paper, we first prove a general fixed point theorem for nonlinear mappings in a Banach space. Then we prove a nonlinear mean convergence theorem of Baillons type and a weak convergence theorem of Manns type for 2-generalized nonspreading mappings in a Banach space.

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.

Abstract Let T: V W be a surjective real linear isometry between full real Hilbert C-modules over real C-algebras A and B, respectively. We show that the following conditions are equivalent:(a) T is a 2-isometry;(b) T is a complete isometry;(c) T preserves ternary products;(d) T preserves inner products;(e) T is a module map. When A and B are commutative, we give a full description of the structure of T.