Related papers: Sequential Product Spaces are Jordan Algebras
This note adds to the recent spate of derivations of the probabilistic apparatus of finite-dimensional quantum theory from various axiomatic packages. We offer two different axiomatic packages that lead easily to the Jordan algebraic…
It has already been established that the properties required of an abstract sequential product as introduced by Gudder and Greechie are not enough to characterise the standard sequential product $a\circ b = \sqrt{a}b\sqrt{a}$ on an operator…
I explore several related routes to deriving the Jordan-algebraic structure of finite-dimensional quantum theory from more transparent operational or physical principles, mainly involving ideas about the symmetries of, and the correlations…
The famous Koecher-Vinberg theorem characterizes the Euclidean Jordan algebras among the finite dimensional order unit spaces as the ones that have a symmetric cone. Recently Walsh gave an alternative characterization of the Euclidean…
A celebrated result of Koecher and Vinberg asserts the one-one correspondence between the finite dimensional formally real Jordan algebras and Euclidean symmetric cones. We extend this result to the infinite dimensional setting.
Among the many important geometric properties of quantum state space are: transitivity of the group of symmetries of the cone of unnormalized states on its interior (homogeneity), identification of this cone with its dual cone of effects…
The famous Koecher-Vinberg theorem characterises the finite dimensional formally real Jordan algebras among the finite dimensional order unit spaces as the ones that have a symmetric cone. An alternative characterisation of symmetric cones…
Physical theories can be characterized in terms of their state spaces and their evolutive equations. The kinematical structure and the dynamical structure of finite dimensional quantum theory are, in light of the Choi-Jamio{\l}kowski…
Our basic structure is a finite-dimensional complex Hilbert space $H$. We point out that the set of effects on $H$ form a convex effect algebra. Although the set of operators on $H$ also form a convex effect algebra, they have a more…
Let T be be a zero-product preserving bounded linear map between C*-algebras A and B. Here neither A nor B is necessarily unital. In this note, we investigate when T gives rise to a Jordan homomorphism. In particular, we show that A and B…
Given a Jordan algebra $A$ and a vector space $V$, we describe and classify all Jordan algebras containing $A$ as a subalgebra of codimension ${\rm dim}_k (V)$ in terms of a non-abelian cohomological type object ${\mathcal J}_{A} \, (V, \,…
We prove that C*-algebras which, as Banach spaces, are Grothendieck cannot be decomposed into a tensor product of two infinite-dimensional C*-algebras. By a result of Pfitzner, this class contains all von Neumann algebras and their…
We study symmetric continuous bilinear maps $V$ on a C$^*$-algebra $A$ that have the Jordan product property at a fixed element $z\in A$. We show that, whenever $A$ is a finite direct sum or a $c_0$-sum of infinite simple von Neumann…
To each discrete product system E of finite-dimensional Hilbert spaces we associate a C*-algebra O_E. When E is the n-dimensional product system over N, O_E is the Cuntz algebra O_n, and the irrational rotation algebras appear as O_E for…
We prove that the module categories of Noether algebras (i.e., algebras module finite over a noetherian center) and affine noetherian PI algebras over a field enjoy the following product property: Whenever a direct product $\prod_{n \in…
By analogy with the well-established notions of just-infinite groups and just-infinite algebras, in particular $C^*$-algebras, we initiate a study of just-infinite $JB$-algebras, i.e. infinite dimensional $JB$-algebras for which all proper…
We show that the *-algebra of the product of two synchronous games is the tensor product of the corresponding *-algebras. We prove that the product game has a perfect C*-strategy if and only if each of the individual games does, and that in…
We show that the product of any number of sequentially pseudocompact topological spaces is still sequentially pseudocompact. The definition of sequential pseudocompactness can be given in (at least) two ways: we show their equivalence. Some…
We introduce a class of subproduct systems of finite dimensional Hilbert spaces whose fibers are defined by the Jones-Wenzl projections in Temperley-Lieb algebras. The quantum symmetries of a subclass of these systems are the free…
We show that, under suitably general formulations, covering properties, accumulation properties and filter convergence are all equivalent notions. This general correspondence is exemplified in the study of products. Let $X$ be a product of…