Asian-European Journal of Mathematics

We examine Ptak's inequality in a real Banach *-algebra which is hermitian and skew hermitian, and give conditions for a real Banach *-algebra to be isomorphic to a C* algebra.

We show that every simple abelian real Hilbert ternary algebra is isomorphic to the algebra of Hilbert-Schmidt operators between two real, complex or quaternionic Hilbert spaces, up to a positive multiple of the inner product.

We survey some recent results on the isometries of weighted spaces of holomorphic functions defined on an open subset of ℂⁿ. We will see that these isometries are determined by a subgroup of the automorphisms on a distinguished subset of the domain. We will look for weights with 'large' groups of isometries and observe that in certain circumstances the group of isometries determines the weight.

We obtain a complete characterization of all orthogonality preserving operators from a JB*-algebra to a JB*-triple. If T : J → E is a bounded linear operator from a JB*-algebra (respectively, a C*-algebra) to a JB*-triple and h denotes the element T**(1), then T is orthogonality preserving, if and only if, T preserves zero-triple-products, if and only if, there exists a Jordan *-homomorphism such that S(x) and h operator commute and T(x) = h•_r(h) S(x), for every x ∈ J, where r(h) is the range tripotent of h, is the Peirce-2 subspace associated to r(h) and •_r(h) is the natural product making a JB*-algebra.

This characterization culminates the description of all orthogonality preserving operators between C*-algebras and JB*-algebras and generalizes all the previously known results in this line of study.

We introduce a class of Riemannian symmetric spaces, called Jordan symmetric spaces, which correspond to real Jordan triple systems and may be infinite dimensional. This class includes the symmetric R-spaces as well as the Hermitian symmetric spaces.

In this article we show that holomorphic Fredholm-valued mappings defined on connected pseudo-convex domains in Banach spaces with unconditional basis always have meromorphic generalised inverses. We show they have holomorphic generalised inverses if and only if the kernels have the same dimension at all points in Ω.

We give an elementary alternative proof of the Bernstein inequalities and the Szegö inequalities for trigonometric polynomials or polynomials.

We continue the study of the relations between the set Tri(A) of all tripotents in an arbitrary JBW*-triple A and the set of weak*-closed inner ideals in A. Particular attention is paid to the subset of those weak*-closed inner ideals that are principal.

In this paper, we give a complete description of the structure of separating linear maps between continuous fields of Banach spaces. Some automatic continuity results are obtained.

Involutive automorphism, or bijective triple homorphisms of order two, on a JBW*-triple are in a one-to-one correspondence with involutive gradings and bicontractive projections. Such mappings are always isometric and weak*-continuous. This paper analyses the algebraic kernels of the 1-eigenspace B₊ and -1-eigenspace B_- of involutive automorphisms. It is shown that B₊ and B_- give rise to several decompositions of A and, remarkably, Ker(B₊) and Ker(B_-) are Peirce inner ideals.

In this note, we look at homotopes of Jordan triple structures and show that, following a renorming, an isotope of a JB*-triple is also a JB*-triple. We also provide a proof of the Russo—Dye theorem for JBW*-triples.

Much of classical vector measure theory can be interpreted as the study of weakly compact operators on commutative function algebras. Non-commutative measure theory can be thought of as a similar study, where the domain algebras are replaced by non-commutative operator algebras. Hypermeasure theory is something beyond this, where we replace operator algebras by more general classes of Banach space.

We discuss several open problems on spectrally bounded operators, some new, some old, adding in a few new insights.

Let E be a complex Banach space with a Schauder basis and let G(E; r) be the Grassmann manifold of all r-dimensional complex linear subspaces in E. Let (ω, φ) be a Riemann domain over G(E; r) with ω ≠ G(E; r). Then we show that ω is a domain of existence if and only if ω is pseudoconvex.

Peirce projections P₁, P_1/2 and P₀ are one of the fundamental technical tools in the theory of JB*-triples. It is well known that all three of them are contractive. We show that the sum of two Peirce projections need not be contractive. We also give the upper estimate for the norm of such a sum, valid in all JC*-triples, and present a conjecture of the exact value of this norm, based on some numerical experiments.