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 be a locally compact Hausdorff space. We show that any local -linear map (where" local" is a weaker notion than -linearity) between Banach -modules are" nearly -linear" and" nearly bounded". As an application, a local -linear map between Hilbert -modules is automatically -linear. If, in addition, contains no isolated point, then any -linear map between Hilbert -modules is automatically bounded. Another application is that if a sequence of maps between two Banach spaces" preserve -sequences"(or" preserve ultra- -sequences"), then is bounded for large enough and they have a common bound. Moreover, we will show that if is a bijective" biseparating" linear map from a" full" essential Banach -module into a" full" Hilbert -module (where is another locally compact Hausdorff space), then is" nearly bounded"(in fact, it is automatically bounded if or contains no isolated point) and there exists a homeomorphism such that ( ).

Let H: Mm Mm be a holomorphic function of the algebra Mm of complex m m matrices. Suppose that H is orthogonally additive and orthogonally multiplicative on self-adjoint elements. We show that either the range of H consists of zero trace elements, or there is a scalar sequence {n} and an invertible S in Mm such that

We describe the structure of all bijective maps on the cone of positive definite operators acting on a finite and at least two-dimensional complex Hilbert space which preserve the quantum 2 -divergence for some 2 . We prove that any such transformation is necessarily implemented by either a unitary or an antiunitary operator. Similar results concerning maps on the cone of positive semidefinite operators as well as on the set of all density operators are also derived.

In this paper, we introduce and study a triple hierarchical variational inequality (THVI) with constraints of minimization and equilibrium problems. More precisely, let Fix ( T ) be the fixed point set of a nonexpansive mapping, let Fix ( T ) be the solution set of a mixed equilibrium problem (MEP), and let <i></i> be the solution set of a minimization problem (MP) for a convex and continuously Frechet differential functional in Hilbert spaces. We want to find a solution Fix ( T ) of a variational inequality with a variational inequality constraint over the intersection of Fix ( T ) , Fix ( T ) , and <i></i>. We propose a hybrid iterative algorithm with regularization to compute approximate solutions of the THVI, and we present the convergence analysis of the proposed iterative algorithm.