Let A 1, A 2 be standard operator algebras on complex Banach spaces X 1, X 2, respectively. For k 2, let (i 1,, i m) be a sequence with terms chosen from {1,, k}, and define the generalized Jordan product T 1 T k= T i 1 T i m+ T i m T i 1 on elements in A i. This includes the usual Jordan product A 1 A 2= A 1 A 2+ A 2 A 1, and the triple {A 1, A 2, A 3}= A 1 A 2 A 3+ A 3 A 2 A 1. Assume that at least one of the terms in (i 1,, i m) appears exactly once. Let a map : A 1 A 2 satisfy that ( (A 1) (A k))= (A 1 A k) whenever any one of A 1,, A k has rank at most one. It is shown in this paper that if the range of contains all operators of rank at most three, then must be a Jordan isomorphism multiplied by an m th root of unity. Similar results for maps between self-adjoint operators acting on Hilbert spaces are also obtained.

We investigate Lamé systems in periodically perforated domains, and establish quantitative homogenization results in the setting where the domain is clamped at the boundary of the holes. Our method is based on layer potentials and it provides a unified proof for various regimes of hole-cell ratios (the ratio between the size of the holes and the size of the periodic cells), and, more importantly, it yields natural correctors that facilitate error estimates. A key ingredient is the asymptotic analysis for the rescaled cell problems, and this is studied by exploring the convergence of the periodic layer potentials for the Lamé system to those in the whole space when the period tends to infinity.

We show in this paper that every bijective linear isometry between the continuous section spaces of two non-square Banach bundles gives rise to a Banach bundle isomorphism. This is to support our expectation that the geometric structure of the continuous section space of a Banach bundle determines completely its bundle structures. We also describe the structure of an into isometry from a continuous section space into an other. However, we demonstrate by an example that a non-surjective linear isometry can be far away from a subbundle embedding.

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.

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