Let A , A be two regular commutative unital Banach algebras such that A is integral over A . In 2003, Dawson and Feinstein showed that the topological stable rank A whenever A . In this note, we investigate whether we will have A in general. For instance, when A is a commutative unital A -algebra, we show that A , and the equality holds at least when the integral extension is separable. In general, A and A have the same Bass stable ranks A .

A convex subset B of a real locally convex space B is said to have the separation property if it can be separated from every closed convex subset B of B , which is disjoint from B , by a closed hyperplane. The strong separation theorem says that if B is weakly compact, then it has the separation property. In this paper, we present two versions of the converse and discuss an application of them. For example, we prove that a normed space is reflexive if and only if its closed unit ball has the separation property. Results in this paper can be considered as supplements of the famous theorem of James.

Using elementary arguments, we shall show that every bounded disjointness preserving linear functional of the Wiener ring L1 (Z) assumes the form (f)=

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.

In this paper, we introduce a broad class of nonlinear mappings in a Hilbert space which covers nonexpansive mappings, nonspreading mappings, hybrid mappings and contractive mappings. Then we prove fixed point theorems for the class of such mappings. Using these results, we prove well-known and new fixed point theorems in a Hilbert space.