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.

Let A be a C*-algebra, L a closed left ideal of A and p the closed projection related to L. We show that for an xp in A**p ( A**/L**) if pAxp pAp and px*xp pAp then xp Ap ( A/L). The proof goes by interpreting elements of A**p (resp. Ap) as admissible (resp. continuous admissible) vector sections over the base space F(p) = { A* : 0, (p) = 1} in the notions developed by Diximier and Douady, Fell, and Tomita. We consider that our results complement both Kadison function representation and Takesaki duality theorem.

Inspired by the recent work of Bekka, we study two reasonable analogues of property (T) for not necessarily unital C-algebras. The stronger one of the two is called property (T) and the weaker one is called property (T e). It is shown that all non-unital C-algebras do not have property (T)(neither do their unitalizations). Moreover, all non-unital -unital C-algebras do not have property (T e).

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.

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.