We recall the definition and properties of an algebra cone C of a complex unital Banach algebra A. It can be shown that C induces on A an ordering which is compatible with the algebraic structure of A, and A is then called an ordered Banach algebra. The Banach algebra ℒ(E) of all bounded linear operators on a complex Banach lattice E is an example of an ordered Banach algebra, and an interesting aspect of research in ordered Banach algebras is that of investigating in an ordered Banach algebra-context certain problems that originated in ℒ(E). In this paper we investigate the problems of providing conditions under which (1) a positive element a with spectrum consisting of 1 only will necessarily be greater than or equal to 1, and (2) f (a) will be positive if a is positive, where f (a) is the element defined by the holomorphic functional calculus.

We prove metric regularity results for both single-valued maps and set-valued maps defined between metric spaces.

In this paper, we discover a sufficient and necessary condition under which two geodesic segments joining the base point and another point in an infinite-dimensional Teichmüller space are the same.

An Abelian torsion group is determined by its automorphism group if and only if its locally cyclic component is determined by its automorphism group. We describe the locally cyclic groups that are determined by their automorphism groups.

For a pythagorean field F with semiordering Q and associated preordering T, it is shown that the Witt ring WT (F) is isomorphic to the Witt ring W (K) whre K is a closure of F with respect to Q. For an arbitrary preordering T, it is shown how the covering number of T relates to the construction of WT (F).

Let G be a finite p-solvable group for a fixed prime p. We determine the structure of G when the set of p-regular conjugacy class sizes of G is {1, m} for an arbitrary integer m > 1.

The sequence space m (ø), introduced and studied by W.L.C. Sargent in 1960, is closely related to the space ℓp. In this paper we obtain an explicit formula for the Hausdorff measure of noncompactness of any bounded subset in m (ø). We also show that m (ø)enjoys the weak Banach-Saks property, while C (m (ø)) = 2. This shows that the condition C (X) < 2, known to be sufficient for the space X to have the weak Banach-Saks property, is not a necessary one.

In [1] there is an error, as pointed out to us by J.M. Isidro. In that paper we consider the set Rm of all m-rotations, that is, Rm = {czm: |c| = 1}; the set Aut() of the automorphisms of the unit disc  and the set of the m-automorphisms of , Autm () = {ψ ∘ R ∘ ϕ : ψ, ϕ ∈ Aut(), R ∈ Rm}. We asserted that

where ϕa(Z) = (a − z)/(1 − āz). Equality (1) is not true. For instance, it suffices to consider f (z) = ϕ¼ ∘ ∘ z2 ∘ ϕ½. We have that f ∈ Aut2 (), f (0) = (0) and, after calculations, f (z) = (7z2 − 6z)/(6z − 7). Then it is evident that will never hold f (z) = ϕ0 ∘ R ∘ ϕ0 with R ∈ R2, because ϕ0 is equal to identity.

In this note, we prove that if a torsion group G has an Abelian subgroup B such that G/B is Abelian and R is a G-adapted ring with the property that R (G/B) has only trivial units then G has the normaliser property in RG.