Real Jordan Algebras Research Articles

Information Geometry, Jordan Algebras, and a Coadjoint Orbit-Like Construction

Jordan algebras arise naturally in (quantum) information geometry, and we want to understand their role and their structure within that framework. Inspired by Kirillov's discussion of the symplectic structure on coadjoint orbits, we provide a similar construction in the case of real Jordan algebras. Given a real, finite-dimensional, formally real Jordan algebra ${\mathcal J}$, we exploit the generalized distribution determined by the Jordan product on the dual ${\mathcal J}^{\star}$ to induce a pseudo-Riemannian metric tensor on the leaves of the distribution. In particular, these leaves are the orbits of a Lie group, which is the structure group of ${\mathcal J}$, in clear analogy with what happens for coadjoint orbits. However, this time in contrast with the Lie-algebraic case, we prove that not all points in ${\mathcal J}^{*}$ lie on a leaf of the canonical Jordan distribution. When the leaves are contained in the cone of positive linear functionals on ${\mathcal J}$, the pseudo-Riemannian structure becomes Riemannian and, for appropriate choices of ${\mathcal J}$, it coincides with the Fisher-Rao metric on non-normalized probability distributions on a finite sample space, or with the Bures-Helstrom metric for non-normalized, faithful quantum states of a finite-level quantum system, thus showing a direct link between the mathematics of Jordan algebras and both classical and quantum information geometry.

Some Inequalities over the Eigenvalues of a Strongly Regular Graph

Let’s consider a primitive strongly regular graph G. In this paper, we establish some inequalities over the spectrum of G in the environment of a real finite dimensional Euclidean Jordan algebra A associated with G recurring to a spectral analysis of some elements of A and recurring to a spectral analysis of the Generalized Krein parameters of G.

On the structure group of an infinite dimensional JB-algebra

We extend several results for the structure group of a real Jordan algebra V, to the setting of infinite dimensional JB-algebras. We prove that the structure group Str(V), the cone preserving group G(Ω) and the automorphism group Aut(V) of the algebra V are embedded Banach-Lie groups of GL(V), and that each of the inclusions Aut(V)⊂G(Ω)⊂Str(V) are of embedded Banach-Lie subgroups. We give a full description of the components of Str(V) via cones, isotopes and central projections. We apply these results to V=B(H)sa the special JB-algebra of self-adjoint operators on an infinite dimensional complex Hilbert space, describing the groups Str(V),G(Ω),Aut(V), their Banach-Lie algebras and their connected components. We show that the action of the unitary group of H on Aut(V) has smooth local cross sections, thus Aut(V) is a smooth principal bundle over the unitary group, with structure group S1.

Random projections for conic programs

We discuss the application of random projections to conic programming: notably linear, second-order and semidefinite programs. We prove general approximation results on feasibility and optimality using the framework of formally real Jordan algebras. We then discuss some computational experiments on randomly generated semidefinite programs in order to illustrate the practical applicability of our ideas.

Maps preserving the $c$-numerical radius of products for operators in $\mathfrak{B}(H)$

Let 𝔅(ℋ) and 𝔅s(ℋ) be the algebra of all bounded linear operators on a complex Hilbert space ℋ and the real Jordan algebra of all self-adjoint operators on ℋ, respectively. Suppose 𝒲 and 𝒱 are subsets of 𝔅(ℋ) or 𝔅s(ℋ) and F(⋅) is the c-numerical radius or the diameter of an operator. Under an assumption on 𝒲 and 𝒱, characterizations are obtained for surjective maps Φ:𝒲→𝒱 satisfying F ( A B ) = F ( Φ ( A ) Φ ( B ) ) ( A , B ∈ 𝒲 ) . When dimℋ≥3, to establish the proofs, some general results are obtained for functions F:𝔅(ℋ)→[0,+∞) satisfying (P1) F(UAU∗)=F(A) for all A∈𝔅(ℋ) and unitary U on ℋ; (P2) For A∈𝔅(ℋ), F(A)=0 if and only if A is a multiple of the identity; (P3) There are nonnegative real numbers α,β with α2+β2≠0 such that F(T)=α∥T∥+β|tr(T)| for each rank-one T∈𝔅(ℋ).

Graph immersions with parallel cubic form

We consider non-degenerate graph immersions into affine space An+1 whose cubic form is parallel with respect to the Levi-Civita connection of the affine metric. There exists a correspondence between such graph immersions and pairs (J,γ), where J is an n-dimensional real Jordan algebra and γ is a non-degenerate trace form on J. Every graph immersion with parallel cubic form can be extended to an affine complete symmetric space covering the maximal connected component of zero in the set of quasi-regular elements in the algebra J. It is an improper affine hypersphere if and only if the corresponding Jordan algebra is nilpotent. In this case it is an affine complete, Euclidean complete graph immersion, with a polynomial as globally defining function. We classify all such hyperspheres up to dimension 5. As a special case we describe a connection between Cayley hypersurfaces and polynomial quotient algebras. Our algebraic approach can be used to study also other classes of hypersurfaces with parallel cubic form.

Commutativity in Jordan operator algebras

While Jordan algebras are commutative, operator commutativity will behave badly for a general non-associative Jordan algebra. When the product operators of two elements commute, the elements are said to operator commute. In some Jordan algebras operator commutation can be badly behaved, for instance having elements a and b operator commute, while a2 and b do not operator commute. In this paper we study JB-algebras, real Jordan algebras which are also Banach spaces in a compatible manner. These include for instance the spaces of self-adjoint elements of unital C⁎-algebras. We show that elements a and b in a JB-algebra operator commute if and only if they generate an associative subalgebra of mutually operator commuting elements, and hence operator commutativity in JB-algebras is as well-behaved as it can be. Letting Ua denote the quadratic operator of a, we also show that positive a and b operator commute if and only if Uab2=Uba2. We use this result to conclude that the unit interval of a JB-algebra is a sequential effect algebra as defined by Gudder and Greechie.

The orthogonality relation classifies finite-dimensional formally real simple Jordan algebras

It is shown that a finite-dimensional formally real simple Jordan algebra is completely determined by the relation of Jordan-orthogonality.Communicated by Prof. Alberto Elduque

A simple and quantum-mechanically motivated characterization of the formally real Jordan algebras

Quantum theory’s Hilbert space apparatus in its finite-dimensional version is nearly reconstructed from four simple and quantum-mechanically motivated postulates for a quantum logic. The recon- struction process is not complete, since it excludes the two-dimensional Hilbert space and still includes the exceptional Jordan algebras, which are not part of the Hilbert space apparatus. Options for physically meaningful potential generalizations of the apparatus are discussed.

Orthomaps on formally real simple Jordan algebras

Orthomaps on formally real simple Jordan algebras

Conformally Covariant Bi-differential Operators on a Simple Real Jordan Algebra

Abstract For a simple real Jordan algebra V, a family of bi-differential operators from $\mathcal{C}^\infty (V\times V)$ to $\mathcal{C}^\infty (V)$ is constructed. These operators are covariant under the rational action of the conformal group of V. They generalize the classical Rankin–Cohen brackets (case $V=\mathbb{R}$).

Binomial Hadamard Series and Inequalities over the Spectra of a Strongly Regular Graph

Let G be a primitive strongly regular graph of order n and A is adjacency matrix. In this paper we first associate to A a real 3-dimensional Euclidean Jordan algebra with rank three spanned by In and the natural powers of A that is a subalgebra of the Euclidean Jordan algebra of symmetric matrix of order n. Next we consider a basis that is a Jordan frame of . Finally, by an algebraic asymptotic analysis of the second spectral decomposition of some Hadamard series associated to A we establish some inequalities over the spectra and over the parameters of a strongly regular graph.

Exotic Matrix Models: the Albert Algebra and the Spin Factor

The matrix models attached to real symmetric matrices and the complex/quaternionic Hermitian matrices have been studied by many authors. These models correspond to three of the simple formally real Jordan algebras over R. Such algebras were classified by Jordan, von Neumann, and Wigner in the 30s, and apart from these three there are two others: (i) the spin factor L_{1,n}, an algebra built on R^{n+1}, and (ii) the Albert algebra A of 3 by 3 Hermitian matrices over the octonions. In this paper we investigate the matrix models attached to these remaining cases.

Infinite dimensional Jordan algebras and symmetric cones

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.

A Royal Road to Quantum Theory (or Thereabouts), Extended Abstract

A representation of finite-dimensional probabilistic models in terms of formally real Jordan algebras is obtained, in a strikingly easy way, from simple assumptions. This provides a framework in which real, complex and quaternionic quantum mechanics can be treated on an equal footing, and allows some (but not too much) room for other alternatives. This is based on earlier work (arXiv:1206:2897), but the development here is further simplified, and also extended in several ways. I also discuss the possibilities for organizing probabilistic models, subject to the assumptions discussed here, into symmetric monoidal categories, showing that such a category will automatically have a dagger-compact structure. (Recent joint work with Howard Barnum and Matthew Graydon (arXiv:1507.06278) exhibits several categories of this kind.)

The variety of three-dimensional real Jordan algebras

In this paper, we study the variety [Formula: see text] of three-dimensional Jordan algebras over the field of real numbers. We establish the list of [Formula: see text] non-isomorphic Jordan algebras and describe the irreducible components of [Formula: see text] proving that it is the union of Zariski closures of the orbits of eight rigid algebras.

Exceptional quantum geometry and particle physics

Based on an interpretation of the quark–lepton symmetry in terms of the unimodularity of the color group SU(3) and on the existence of 3 generations, we develop an argumentation suggesting that the “finite quantum space” corresponding to the exceptional real Jordan algebra of dimension 27 (the Euclidean Albert algebra) is relevant for the description of internal spaces in the theory of particles. In particular, the triality which corresponds to the 3 off-diagonal octonionic elements of the exceptional algebra is associated to the 3 generations of the Standard Model while the representation of the octonions as a complex 4-dimensional space C⊕C3 is associated to the quark–lepton symmetry (one complex for the lepton and 3 for the corresponding quark). More generally it is suggested that the replacement of the algebra of real functions on spacetime by the algebra of functions on spacetime with values in a finite-dimensional Euclidean Jordan algebra which plays the role of “the algebra of real functions” on the corresponding almost classical quantum spacetime is relevant in particle physics. This leads us to study the theory of Jordan modules and to develop the differential calculus over Jordan algebras (i.e. to introduce the appropriate notion of differential forms). We formulate the corresponding definition of connections on Jordan modules.

Generalized Krein Parameters of a Strongly Regular Graph

We consider the real three-dimensional Euclidean Jordan algebra associated to a strongly regular graph. Then, the Krein parameters of a strongly regular graph are generalized and some generalized Krein admissibility conditions are deduced. Furthermore, we establish some relations between the classical Krein parameters and the generalized Krein parameters.

Centro-affine hypersurface immersions with parallel cubic form

We consider non-degenerate centro-affine hypersurface immersions in $$\mathbb R^n$$ whose cubic form is parallel with respect to the Levi-Civita connection of the affine metric. There exists a bijective correspondence between homothetic families of proper affine hyperspheres with center in the origin and with parallel cubic form, and Köchers conic $$\omega $$ -domains, which are the maximal connected sets consisting of invertible elements in a real semi-simple Jordan algebra. Every level surface of the $$\omega $$ function in an $$\omega $$ -domain is an affine complete, Euclidean complete proper affine hypersphere with parallel cubic form and with center in the origin. On the other hand, every proper affine hypersphere with parallel cubic form and with center in the origin can be represented as such a level surface. We provide a complete classification of proper affine hyperspheres with parallel cubic form based on the classification of semi-simple real Jordan algebras. Centro-affine hypersurface immersions with parallel cubic form are related to the wider class of real unital Jordan algebras. Every such immersion can be extended to an affine complete one, whose conic hull is the connected component of the unit element in the set of invertible elements in a real unital Jordan algebra. Our approach can be used to study also other classes of hypersurfaces with parallel cubic form.

Maps Preserving Peripheral Spectrum of Generalized Jordan Products of Self-Adjoint Operators

LetA1andA2 be standard real Jordan algebras of self-adjoint operators on complex Hilbert spacesH1 and H2, respectively. For k≥2, let (i1,…,im)be a fixed sequence withi1,…,im∈{1,…,k}and assume that at least one of the terms in(i1,…,im)appears exactly once. Define the generalized Jordan product T1∘T2∘⋯∘Tk=Ti1Ti2⋯Tim+Tim⋯Ti2Ti1on elements in Ai. LetΦ:A1→A2be a map with the range containing all rank-one projections and trace zero-rank two self-adjoint operators. We show thatΦsatisfies thatσπ(Φ(A1)∘⋯∘Φ(Ak))=σπ(A1∘⋯∘Ak)for allA1,…,Ak, whereσπ(A)stands for the peripheral spectrum ofA, if and only if there exist a scalarc∈{-1,1}and a unitary operatorU:H1→H2such thatΦ(A)=cUAU*for allA∈A1, orΦ(A)=cUAtU*for allA∈A1, whereAtis the transpose ofAfor an arbitrarily fixed orthonormal basis ofH1. Moreover,c=1whenevermis odd.