It is shown that the operator space generated by peripheral eigenvectors of a unital completely positive map on a von Neumann algebra has a C^*-algebra structure. This extends the notion of non-commutative Poisson boundary by including the point spectrum of the map contained in the unit circle. The main ingredient is dilation theory. This theory provides a simple formula for the new product. The notion has implications to our understanding of quantum dynamics. For instance, it is shown that the peripheral Poisson boundary remains invariant in discrete quantum dynamics.

We identify and characterize unital completely positive (UCP) maps on finite dimensional C⁎-algebras for which the Choi-Effros product extended to the space generated by peripheral eigenvectors matches with the original product. We analyze a decomposition of general UCP maps in finite dimensions into persistent and transient parts. It is shown that UCP maps on finite dimensional C⁎-algebras with spectrum contained in the unit circle are ⁎-automorphisms.

We introduce and study the notion of k-strong convexity in Banach spaces. It is a generalization of the notion of strong convexity first studied by Fan and Glicksberg. A Banach space is said to be k-strongly convex if it is reflexive, k-strictly convex and has the Kadec-Klee property. We use the idea of k-dimensional diameter to give several characterizations of k-strong convexity. Further, we study k-strict convexity and k-strong convexity in some products of Banach spaces. Finally, we give characterizations of k-uniform convexity that distinguish it from k-strong convexity.

We prove a best proximity point version of Krasnoselskii’s fixed point theorem. As a consequence we obtain the existence of best proximity points of relatively u-continuous maps for a pair of compact convex sets.

Using the concept of asymptotic center we obtain the existence of fixed points having preassigned location for a wider class of asymptotic nonexpansive mappings in a uniformly convex Banach space. This generalization leads us to get a recent result of Alfuraidan and Khamsi for continuous monotone asymptotic nonexpansive mappings as well as the classical fixed-point result of Geobel and Kirk for asymptotic nonexpansive mappings in a uniformly convex Banach space. Also we prove a fixed-point theorem for order preserving continuous maps on a quasiordered closed convex subset of a uniformly convex Banach sapce having monotone norm.

In a recent paper Shunmugaraj and Thota characterized uniform convexity in terms of property UC. In this paper we introduce a more general notion called property k-UC and characterize k-uniform rotundity in terms of property k-UC. We also obtain some results related to k-uniform rotundity and property k-UC. Some results obtained in this paper generalize some results recently obtained by Shunmugaraj and Thota as well as a result of Suzuki et al.