Let T be a locally compact Hausdorff space, called base space. Suppose for each t in T there is a (real or complex) Banach space Et. A vector field x is an element in the product space t T Et, that is, x (t) Et, for all t T.

In this paper, we try to attack a conjecture of Araujo and Jarosz that every bijective linear map between C-algebras, with both and its inverse 1 preserving zero products, arises from an algebra isomorphism followed by a central multiplier. We show it is true for CCR C-algebras with Hausdorff spectrum, and in general, some special C-algebras associated to continuous fields of C-algebras.

New spaces of Lipschitz type on metric-measure spaces are introduced and they are shown to be just the well-known Besov spaces or Triebel–Lizorkin spaces when the smooth index is less than 1. These theorems also hold in the setting of spaces of homogeneous type, which include Euclidean spaces, Riemannian manifolds and some self-similar fractals. Moreover, the relationships amongst these Lipschitz-type spaces, Hajłasz–Sobolev spaces, Korevaar–Schoen–Sobolev spaces, Newtonian Sobolev space and Cheeger–Sobolev spaces on metric-measure spaces are clarified, showing that they are the same space with equivalence of norms. Furthermore, a Sobolev embedding theorem, namely that the Lipschitz-type spaces with large orders of smoothness can be embedded in Lipschitz spaces, is proved. For metric-measure spaces with heat kernels, a Hardy–Littlewood–Sobolev theorem is establish, and hence it is proved that Lipschitz-type spaces with small orders of smoothness can be embedded in Lebesgue spaces.

It has been a successful practice to define a canonical pre-ordering on a normed space using the inclusion of faces of its closed dual unit ball. This pre-ordering reflects some geometric property in a natural way. In this article, we will give an algebraic description of this pre-ordering in the case of complex C*-algebras as well as that of their self-adjoint parts. In developing our theory we introduce the essential support of an element, which is closely related to the notion of peak projections studied recently by Blecher and Hay. As applications, we give some interesting facts about weak*-closed faces, and will identify the quasi-maximal elements and the quasi-minimal elements with respects to this pre-ordering. They are closely related to the extreme points and the smooth points of the unit sphere of the C*-algebra.

Let and be a real -uniformly smooth Banach space, be a nonempty closed convex subset of and be a Lipschitz continuous mapping. Let and be bounded sequences in and and be real sequences in satisfying some restrictions. Let be the sequence generated from an arbitrary by the Ishikawa iteration process with errors: , , . Sufficient and necessary conditions for the strong convergence to a fixed point of is established.