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.

Let <i>M</i> be a compact connected (topological) manifold of finite- or infinite-dimension <i>n</i>. Let 0 <i>r</i> 1 be arbitrary but fixed. We construct in this paper a space-filling curve <i>f</i> from [0,1] onto <i>M</i>, under which <i>M</i> is the image of a compact set <i>A</i> of Hausdorff dimension <i>r</i>. Moreover, the restriction of f to <i>A</i> is one-to-one over the image of a dense subset provided that 0 <i>r</i> log|2^<i>n</i>/log(2^<i>n</i> + 2). The proof is based on the special case where <i>M</i> is the Hilbert cube [0,1].

We define type A, type B, type C as well as C*-semi-finite C*-algebras.

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.

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.