Jordan Banach Algebras Research Articles

Gleason-Kahane-Zelazko Theorem in Jordan Banach algebras

We review the celebrated Gleason-Kahane-Zelazko and Kowalski-Slodkowski theorems from the setting of associative Banach algebras to the wider class of nonassociative Jordan Banach algebras. We introduce the notion of almost multiplicative linear functionals in Jordan Banach algebras and prove a theorem extending a former result of B.E. Johnson for Banach algebras by employing the more recent concept of condition spectrum. We show how to rediscover the Gleason-Kahane-Zelazko theorem for Jordan Banach algebras from the corresponding version for almost multiplicative linear functionals.

Lie–Trotter Formulae in Jordan–Banach Algebras with Applications to the Study of Spectral-Valued Multiplicative Functionals

Lie–Trotter Formulae in Jordan–Banach Algebras with Applications to the Study of Spectral-Valued Multiplicative Functionals

Suzuki type estimates for exponentiated sums and generalized Lie-Trotter formulas in JB-algebras

Lie-Trotter-Suzuki product formulas are ubiquitous in quantum mechanics, computing, and simulations. Approximating exponentiated sums with such formulas are investigated in the JB-algebraic setting. We show that the Suzuki type approximation for exponentiated sums holds in JB-algebras, we give explicit estimation formulas, and we deduce three generalizations of Lie-Trotter formulas for arbitrary number elements in such algebras. We also extended the Lie-Trotter formulas in a Jordan Banach algebra from three elements to an arbitrary number of elements.

A note on (ψ+,ψ−)-derivations of Banach–Jordan systems with nonzero socle

The main objective of this research paper consists in introducing the concept of [Formula: see text]-derivations acting on Banach–Jordan pairs and Banach–Jordan algebras and giving an automatic continuity result of the operators in question under some algebraic conditions. Concretely, we prove that [Formula: see text] -derivations defined from a Banach–Jordan pair [Formula: see text] into a strongly prime Banach–Jordan pair [Formula: see text] are automatically continuous under the assumptions that the socle [Formula: see text] is nonzero and [Formula: see text] is an isomorphism from [Formula: see text] into [Formula: see text]. Similar results for Banach–Jordan algebras hold to be true.

Automatic continuity of Jordan almost multiplicative maps on special Jordan Banach algebras

Automatic continuity of Jordan almost multiplicative maps on special Jordan Banach algebras

Surjective isometries between sets of invertible elements in unital Jordan-Banach algebras

Let M and N be complex unital Jordan-Banach algebras, and let M−1 and N−1 denote the sets of invertible elements in M and N, respectively. Suppose that M⊆M−1 and N⊆N−1 are clopen subsets of M−1 and N−1, respectively, which are closed for powers, inverses and products of the form Ua(b). In this paper we prove that for each surjective isometry Δ:M→N there exists a surjective real-linear isometry T0:M→N and an element u0 in the McCrimmon radical of N such that Δ(a)=T0(a)+u0 for all a∈M. Assuming that M and N are unital JB⁎-algebras we establish that for each surjective isometry Δ:M→N the element Δ(1)=u is a unitary element in N and there exist a central projection p∈M and a complex-linear Jordan ⁎-isomorphism J from M onto the u⁎-homotope Nu⁎ such thatΔ(a)=J(p∘a)+J((1−p)∘a⁎), for all a∈M. Under the additional hypothesis that there is a unitary element ω0 in N satisfying Uω0(Δ(1))=1, we show the existence of a central projection p∈M and a complex-linear Jordan ⁎-isomorphism Φ from M onto N such thatΔ(a)=Uw0⁎(Φ(p∘a)+Φ((1−p)∘a⁎)), for all a∈M.

Additivity of Quadratic Maps on JB Algebras

The problem of automatic additivity of Jordan type homomorphisms has received a great deal of attention in theory of operator algebras as well in ring theory. We study additivity of quadratic maps, that is the maps between Jordan Banach algebras that preserve the quadratic product (a, b) → aba. The main result shows that any continuous quadratic bijective map between unital Jordan Banach algebras that is linear on associative subalgebras is automatically additive. This contributes to the Borification program in quantum theory and Mackey-Gleason problem on linearity of quasi-linear maps.

Scalar characterization in Banach Jordan algebras

Using a Diagonalization Theorem obtained when the spectrum is Lipschitzian, we extend a result of G. Braatvedt on scalar characterization in Banach algebras to Banach-Jordan algebras. We also establish that any element of a semisimple Banach-Jordan algebra with the property that all elements in some neighbourhood of the identity are spectrally invariant under multiplication by the quadratic U operator, has analogs with the identity.

Spectral characterization of the kh-socle in Banach Jordan algebras

Let J be a unital semisimple Banach Jordan algebra. In this paper, we will give some new spectral characterizations of the kh-socle in J, kh(soc(J)), using the perturbation of the full spectrum theory.

Topological radicals, VI. Scattered elements in Banach Jordan and associative algebras

Topological radicals, VI. Scattered elements in Banach Jordan and associative algebras

Bell Correlated and EPR States in the Framework of Jordan Algebras

We study Bell inequalities and EPR states in the context of Jordan algebras. We show that the set of states violating Bell inequalities across two operator commuting nonmodular Jordan Banach algebras is norm dense in the global state space. It generalizes hitherto known results in quantum field theory in several directions. We propose new Jordan quantity for incommensurable observables in a given state, introduce the concept of EPR state for Jordan structures, and study relationship between EPR states and Bell correlated states. Our analysis shows crucial role of spin factors and Pauli spin matrices for studying noncommutative properties of states and observables.

Retraction Note to: Approximation of Jordan homomorphisms in Jordan-Banach algebras

Retraction Note to: Approximation of Jordan homomorphisms in Jordan-Banach algebras

Martingale ergodic processes in Jordan Banach algebras

Martingale ergodic processes in Jordan Banach algebras

Reduction of Lie–Jordan Banach algebras and quantum states

In this paper, it is shown that the concept of dynamical correspondence for Jordan Banach algebras is equivalent to a Lie structure compatible with the Jordan one. Then a theory of reduction of Lie–Jordan Banach algebras in the presence of quantum constraints is presented and compared to the standard reduction of C*-algebras of observables of a quantum system. The space of states of the reduced Lie–Jordan Banach algebra is characterized in terms of Dirac states on the physical algebra of observables and its GNS representations described in terms of states on the unreduced algebra.

FOLDING AND UNFOLDING QUANTUM STATES

The reduction of a quantum system ("folding" a quantum system) is described as the reduction of its Lie–Jordan Banach algebra of observables with respect to Lie–Jordan Banach subalgebras and Jordan ideals. The space of states of the reduced Lie–Jordan Banach algebra is described in terms of unreduced states ("unfolding" states) as well as the GNS construction. A few examples are discussed.

Approximation of Jordan homomorphisms in Jordan-Banach algebras

Using the direct method based on the Hyers-Ulam-Rassias stability, we investigate and prove the Hyers-Ulam stability of Jordan homomorphisms in Jordan-Banach algebras for the functional equation

On Jordan Banach Algebras with Countably Generated Inner Ideals

In this paper, we show that if J is a complex Jordan Banach algebra in which inner ideals are countably generated, then J is finite dimensional. We also study the real case.

Generalized Hermitian Algebras

We refer to the real Jordan Banach algebra of bounded Hermitian operators on a Hilbert space as a Hermitian algebra. In this paper we define and launch a study of a class of generalized Hermitian (GH) algebras. Among the examples of GH-algebras are ordered special Jordan algebras, JW-algebras, and AJW-algebras, but unlike these more restricted cases, a GH-algebra is not necessarily a Banach space and its lattice of projections is not necessarily complete. In this paper we develop the basic theory of GH-algebras, identify their unit intervals as effect algebras, and observe that their projection lattices are sigma-complete orthomodular lattices. We show that GH-algebras are spectral order-unit spaces and that they admit a substantial spectral theory.

Lie triple ideals and Lie triple epimorphisms on Jordan and Jordan–Banach algebras

A linear subspace $M$ of a Jordan algebra $J$ is said to be a Lie triple ideal of $J$ if $[M,J,J] \subseteq M$, where $[\cdot ,\cdot ,\cdot ]$ denotes the associator. We show that every Lie triple ideal $M$ of a nondegenerate Jordan algebra $J$

Trace and determinant in Jordan-Banach algebras

Using an appropriate definition of the multiplicity of a spectral value, we introduce a new definition of the trace and determinant of elements with finite spectrum in Jordan-Banach algebras. We first extend a result obtained by J. Zemanek in the associative case, on the connectedness of projections which are close to each other spectrally (Theorem 2.3). Secondly we show that the rank of the Riesz projection associated to a finite-rank element a and a finite subset of its spectrum is equal to the sum of the multiplicities of these spectral values (Theorem 2.6). Then we turn to the study of properties such as linearity and continuity of the trace and multiplicativity of the determinant.