Pure Maps between Euclidean Jordan Algebras

Positive and doubly stochastic maps, and majorization in Euclidean Jordan algebras

In this paper we generalize the major results of Andersson and Perlman on LCI models to the setting of symmetric cones and give an explicit closed form formula for the estimate of the covariance matrix in the generalized LCI models that we define. To this end, we replace the cone $H_I^+(\mathbb{R})$ sitting inside the Jordan algebra of symmetric real $I \times I$-matrices by the symmetric cone $\Omega$ of an Euclidean Jordan algebra $V$. We introduce the Andersson-Perlman cone $\Omega(\mathscr{K}\subseteq\Omega$ which generalizes $\mathscr{P}(\mathscr{K})\subseteq H_I^+(\mathscr{R})$. We prove several characterizations and properties of $\Omega(\mathscr{K})$ which allows us to recover, though with different proofs, the main results of Andersson and Perlman regarding $\mathscr{P}(\mathscr{K})$. The new lattice conditional independence models are defined, assuming that the Euclidean Jordan algebra $V$ has a symmetric representation. Using standard results from the theory of Jordan algebras, we can reduce the general model to the case where $V$ is the Jordan algebra of Hermitian matrices over the real, complex or quaternionic numbers, and $\Omega$ is the corresponding cone of positive-definite matrices. Our main statistical result is a closed-form formula for the estimate of the covariance matrix in the generalized LCI model. We also give the likelihood ratio test for testing a given model versus another one, nested within the first.

Every symmetric cone $K$ arises as the cone of squares in aEuclidean Jordan algebra $V$. As $V$ is a real inner-productspace, we may denote by $\mathrm{Isom}(V)$ its group of isometries. Thegroups $\mathrm{JAut}(V)$ of its Jordan-algebra automorphisms and$\mathrm{Aut}(K)$ of the linear cone automorphisms are then related. Forcertain inner products,\begin{equation*}\mathrm{JAut}(V)=\mathrm{Aut}(K) \cap \mathrm{Isom}(V).\end{equation*}We characterize the inner products for which this holds.

In This Chapter We Introduce Euclidean Jordan Algebras. As Will Be Seen In Chapter Viii, These are the same as the formally real Jordan algebras more usually considered in the literature. For such algebras we prove a ‘spectral theorem’, which, in the case of the algebra of real symmetric matrices, specializes to the usual spectral theorem. Then we prove a fundamental result which establishes a one-to-one correspondence between symmetric cones and Euclidean Jordan algebras. A consequence of this is the unique decomposition of every symmetric cone into a direct product of irreducible ones. Finally, for symmetric cones, we give an independent description in Jordan algebra terms of all the geometric and analytic objects introduced in Chapter I.

Furber and Jacobs have shown in their study of quantum computation that the category of commutative C*-algebras and PU-maps (positive linear maps which preserve the unit) is isomorphic to the Kleisli category of a comonad on the category of commutative C*-algebras with MIU-maps (linear maps which preserve multiplication, involution and unit). [Furber and Jacobs, 2013] In this paper, we prove a non-commutative variant of this result: the category of C*-algebras and PU-maps is isomorphic to the Kleisli category of a comonad on the subcategory of MIU-maps. A variation on this result has been used to construct a model of Selinger and Valiron's quantum lambda calculus using von Neumann algebras. [Cho and Westerbaan, 2016]

The exceptional euclidean Jordan algebra $J_3^8$, consisting of $3\times 3$ hermitian octonionic matrices, appears to be tailor made for the internal space of the three generations of quarks and leptons. The maximal rank subgroup of the authomorphism group $F_4$ of $J_3^8$ that respects the lepton-quark splitting is $(SU(3)_c\times SU(3)_{ew})/\mathbb{Z}_3$. Its restriction to the special Jordan subalgebra $J_2^8\subset J_3^8$, associated with a single generation of fundamental fermions, is precisely the symmetry group $S(U(3)\times U(2))$ of the Standard Model. The Euclidean extension $\mathcal{H}_{16}(\mathbb{C})\otimes \mathcal{H}_{16}(\mathbb{C})$ of $J_2^8$, the subalgebra of hermitian matrices of the complexification of the associative envelope of $J_2^8$, involves 32 primitive idempotents giving the states of the first generation fermions. The triality relating left and right $Spin(8)$ spinors to 8-vectors corresponds to the Yukawa coupling of the Higgs boson to quarks and leptons.

Let F be a closed sublattice of a Banach lattice G. We show that any positive linear mapping from F into L 1 ( μ ) {L^1}(\mu ) or C ( X ) C(X) for a Stonian space X has a positive norm preserving extension to G. A dual result for positive norm preserving liftings is also established. These results are applied to obtain extension and lifting theorems for order summable and majorizing linear mappings. We also obtain some partial results concerning positive extensions and liftings of compact linear mappings.

A note on inequalities for positive linear maps

Given a family of completely positive maps, indexed by a group, from aC*-algebra into itself, we are concerned with its dilation to a group of *-automorphisms on a larger algebra. A Schwarz-type inequality forn-positive *-linear mappings from an involutive algebra into the bounded linear operators on a hilbert space is obtained. Strongly continuous one-parameter semigroups and groups onC*-algebras, which have certain positivity properties, are studied.

A weighted complementarity problem is to find a pair of vectors belonging to the intersection of a manifold and a cone such that the product of the vectors in a certain algebra equals a given weight vector. If the weight vector is zero, we get a complementarity problem. Examples of such problems include the Fisher market equilibrium problem and the linear programming and weighted centering problem. In this paper we consider the weighted horizontal linear complementarity problem in the setting of Euclidean Jordan algebras and establish some existence and uniqueness results. For a pair of linear transformations on a Euclidean Jordan algebra, we introduce the concepts of $$\mathbf{R}_0$$ , $$\mathbf{R}$$ , and $$\mathbf{P}$$ properties and discuss the solvability of wHLCPs under nonzero (topological) degree conditions. A uniqueness result is stated in the setting of $${\mathbb {R}}^{n}$$ . We show how our results naturally lead to interior point systems.

Let A and B be unital JB-algebras. We study the extremal structure of the convex set S (A,B) of all identity preserving positive linear maps from A to B. We show that every unital Jordan homomorphism from A to B is an extreme point of S (A,B). An extreme point of S (A,B) need not be a homomorphism and we show that, given A, every extreme point of S (A,B) is a homomorphism for any B if, and only if, dim A ≤ 2. We also determine when S (A,B) is a simplex.

Shannon like inequalities for f-connections of positive linear maps and positive operators

We generalize two classes of D-type positive linear maps by permutations. For the first class, we discuss their indecomposability and give conditions when they are indecomposable and decomposable. For the other class, we show that they are atomic.

Global and local continuity conditions for the output von Neumann entropy for positive maps between Banach spaces of trace-class operators in separable Hilbert spaces are obtained. Special attention is paid to completely positive maps: infinite dimensional quantum channels and operations.It is shown that as a result of some specific properties of the von Neumann entropy (as a function on the set of density operators) several results on the output entropy of positive maps can be obtained, which cannot be derived from the general properties of entropy type functions. In particular, it is proved that global continuity of the output entropy of a positive map follows from its finiteness. A characterization of positive linear maps preserving continuity of the entropy (in the following sense: continuity of the entropy on an arbitrary subset of input operators implies continuity of the output entropy on this subset) is obtained. A connection between the local continuity properties of two completely positive complementary maps is considered.Bibliography: 21 titles.

For a Euclidean Jordan algebra V with the corresponding symmetric cone $\Omega$, we consider the semigroup $\Gamma\sb{\Omega}$ of elements in the automorphism group $G(T\sb{\Omega})$ of the tube domain $V$ + $i\Omega$ which can be extended to $\Omega$ and maps $\Omega$ into itself. A study of this semigroup was first worked out by Koufany in connection to Jordan algebra theory and Lie theory of semigroups. In this work we give a new proof of Koufany's results and generalize up to infinite dimensional Jordan algebras, so called $JB$-algebras. One of the nice examples of the semigroup $\Gamma\sb{\Omega}$ is from the Jordan algebra $Sym(n,\IR$) of symmetric matrices. However, $V\sb{\sigma}$ the set of all self-adjoint operators on $\IR\sp{n}$ with respect to a non-degenerated symmetric bilinear form $\sigma$, is a non-Euclidean Jordan algebra with a cone $\Omega\sb{\sigma}$ which is isomorphic to $\Omega$ of the symmetric cone of $Sym(n,\IR)$. We get an isomorphism of the automorphism groups between two tube domains which also induces an isomorphism between two Lie semigroups. The Lorentzian cone, which is one of the irreducible symmetric cones, is an essential tool in the study of semigroups in Mobius and Lorentzian geometry. J. D. Lawson studied the Mobius and Lorentzian semigroups with an Ol'shanskii decomposition even in the infinite dimensional cases. We study these semigroups via a Jordan algebra theory.