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 real reflexive Banach space with a weakly continuous duality mapping . Let be a nonempty weakly closed star-shaped (with respect to ) subset of . Let = be a uniformly continuous semigroup of asymptotically nonexpansive self-mappings of , which is uniformly continuous at zero. We will show that the implicit iteration scheme: , for all , converges strongly to a common fixed point of the semigroup for some suitably chosen parameters and . Our results extend and improve corresponding ones of Suzuki (2002), Xu (2005), and Zegeye and Shahzad (2009).

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

Let E and E be two Hilbert E -modules over E -algebras E and E , respectively. Let E be a surjective linear isometry from E onto E and E a map from E into E . We will prove in this paper that if the E -algebras E and E are commutative, then E preserves the inner products and E is a module map, ie, there exists a E -isomorphism E between the E -algebras such that <div class="gsh_dspfr"> E

Let 1 p+. We show that the positive part of the closed unit ball of a non-commutative L p-space, as a metric space, is a complete Jordan-invariant for the underlying von Neumann algebra.