Unique geodesics for Thompson's metric

In this paper a geometric characterization of the unique geodesics in Thompson's metric spaces is presented. This characterization is used to prove a variety of other geometric results. Firstly, it will be shown that there exists a unique Thompson's metric geodesic connecting $x$ and $y$ in the cone of positive self-adjoint elements in a unital $C^*$-algebra if, and only if, the spectrum of $x^{-1/2}yx^{-1/2}$ is contained in $\{1/\beta,\beta\}$ for some $\beta\geq 1$. A similar result will be established for symmetric cones. Secondly, it will be shown that if $C^\circ$ is the interior of a finite-dimensional closed cone $C$, then the Thompson's metric space $(C^\circ,d_C)$ can be quasi-isometrically embedded into a finite-dimensional normed space if, and only if, $C$ is a polyhedral cone. Moreover, $(C^\circ,d_C)$ is isometric to a finite-dimensional normed space if, and only if, $C$ is a simplicial cone. It will also be shown that if $C^\circ$ is the interior of a strictly convex cone $C$ with $3\leq \dim C<\infty$, then every Thompson's metric isometry is projectively linear.Comment: 30 page

Order isomorphisms between cones of JB-algebras

In this paper we completely describe the order isomorphisms between cones of atomic JBW-algebras. Moreover, we can write an atomic JBW-algebra as an algebraic direct summand of the so-called engaged and disengaged part. On the cone of the engaged part every order isomorphism is linear and the disengaged part consists only of copies of $\mathbb{R}$. Furthermore, in the setting of general JB-algebras we prove the following. If either algebra does not contain an ideal of codimension one, then every order isomorphism between their cones is linear if and only if it extends to a homeomorphism, between the cones of the atomic part of their biduals, for a suitable weak topology

Midpoints for Thompson's metric on symmetric cones

We characterise the affine span of the midpoints sets, $M(x,y)$, for Thompson's metric on symmetric cones in terms of a translation of the zero-component of the Peirce decomposition of an idempotent. As a consequence we derive an explicit formula for the dimension of the affine span of $M(x,y)$ in case the associated Euclidean Jordan algebra is simple. In particular, we find for $A$ and $B$ in the cone positive definite Hermitian matrices that $dim(aff M(A,B)) = q^2$, where $q$ is the number of eigenvalues $\mu$ of $A^{-1}B$, counting multiplicities, such that $\mu ≠ max\{\lambda_+(A^{-1}B),\lambda_-(A^{-1}B)^{-1}\},$ where $\lambda_+(A^{-1}B) := max\{\lambda:\lambda \in \sigma(A^{-1}B)\}$ and $\lambda_-(A^{-1}B) := min\{\lambda:\lambda \in \sigma(A^{-1}B)\}$. These results extend work by Y. Lim [18]

Order isomorphisms on order intervals of atomic JBW-algebras

In this paper a full description of order isomorphisms between effect algebras of atomic JBW-algebras is given. We will derive a closed formula for the order isomorphisms on the effect algebra of type I factors by proving that the invertible part of the effect algebra of a type I factor is left invariant. This yields an order isomorphism on the whole cone, for which a characterisation exists. Furthermore, we will show that the obtained formula for the order isomorphism on the invertible part can be extended to the whole effect algebra again. As atomic JBW-algebras are direct sums of type I factors and order isomorphisms factor through the direct sum decomposition, this yields the desired description.Comment: 17 page

Series and power series on universally complete complex vector lattices

In this paper we prove an nth root test for series as well as a Cauchy–Hadamard type formula and Abel's' theorem for power series on universally complete Archimedean complex vector lattices. These results are aimed at developing an alternative approach to the classical theory of complex series and power series using the notion of order convergence

Differentiable, Holomorphic, and Analytic Functions on Complex Φ\Phi-Algebras

Using the notion of order convergent nets, we develop an order-theoretic approach to differentiable functions on Archimedean complex $\Phi$-algebras. Most notably, we improve the Cauchy-Hadamard formulas for universally complete complex vector lattices given by both authors in a previous paper in order to prove that analytic functions are holomorphic in this abstract setting

Order theoretical structures in atomic JBW-algebras: disjointness, bands, and centres

Every atomic JBW-algebra is known to be a direct sum of JBW-algebra factors of type I. Extending Kadison's anti-lattice theorem, we show that each of these factors is a disjointness free anti-lattice. We characterise disjointness, bands, and disjointness preserving bijections with disjointness preserving inverses in direct sums of disjointness free anti-lattices and, therefore, in atomic JBW-algebras. We show that in unital JB-algebras the algebraic centre and the order theoretical centre are isomorphic. Moreover, the order theoretical centre is a Riesz space of multiplication operators. A survey of JBW-algebra factors of type I is included

Order isomorphisms on order intervals of atomic JBW-algebras

In this paper a full description of order isomorphisms between effect algebras of atomic JBW-algebras is given. We will derive a closed formula for the order isomorphisms on the effect algebra of type I factors by proving that the invertible part of the effect algebra of a type I factor is left invariant. This yields an order isomorphism on the whole cone, for which a characterisation exists. Furthermore, we will show that the obtained formula for the order isomorphism on the invertible part can be extended to the whole effect algebra again. As atomic JBW-algebras are direct sums of type I factors and order isomorphisms factor through the direct sum decomposition, this yields the desired description.http://link.springer.com/journal/202021-07-13hj2020Mathematics and Applied Mathematic

Hilbert and Thompson isometries on cones in JB-algebras

Hilbert's and Thompson's metric spaces on the interior of cones in JB-algebras are important examples of symmetric Finsler spaces. In this paper we characterize the Hilbert's metric isometries on the interiors of cones in JBW-algebras, and the Thompson's metric isometries on the interiors of cones in JB-algebras. These characterizations generalize work by Bosche on the Hilbert and Thompson isometries on symmetric cones, and work by Hatori and Molnar on the Thompson isometries on the cone of positive self-adjoint elements in a unital C* -algebra. To obtain the results we develop a variety of new geometric and Jordan algebraic techniques