Ternary Rings Research Articles

Inclusions of ternary rings of operators and conditional expectations

AbstractIt is shown that if T is a ternary ring of operators (TRO), X is a nondegenerate sub-TRO of T and there exists a contractive idempotent surjective map P: T → X then P has a unique, explicitly described extension to a conditional expectation between the associated linking algebras. A version of the result for W*-TROs is also presented and some applications mentioned.

The effect of supercritical water on coal pyrolysis and hydrogen production: A combined ReaxFF and DFT study

The reaction mechanism of coal pyrolysis and hydrogen production in supercritical water (SCW) was investigated using the molecular dynamic simulations via the reactive force field (ReaxFF) method combined with the density functional theory (DFT) method. Our calculations present that the water clusters in SCW weaken the C–C bonds in aromatic rings, thus the C(ring)–C(ring) bond cracking energy decreases as much as 287.3kJ/mol and 94.6kJ/mol compared with that in pure coal pyrolysis and in coal pyrolysis in vapor state, respectively. After the aromatic rings break into small cyclic structures, such as quaternary rings and ternary rings, the water clusters in SCW further weaken their C–C ring bonds to induce the small cyclic rings to open. During this process, the water clusters (without any radicals) in SCW turn into H radical-rich water clusters after providing OH radicals to the cyclic rings. This is the main source for the production of hydrogen molecules in SCW–coal system. The combination of H radicals produced by coal with water clusters in SCW is another pathway which forms H radical-rich water clusters. Under the catalysis of water molecules or clusters, H radical-rich water clusters decompose into H2 and OH radicals. These OH radicals further bind with coal intermediates and result in the breaking of coal intermediates into smaller products. Therefore, the cooperative effects between SCW and coal form a virtuous circle, which greatly enhances the reaction rate of coal gasification, promotes the production of small molecules, and increases the yield of hydrogen.

Contractive idempotents on locally compact quantum groups

A general form of contractive idempotent functionals on coamenable locally compact quantum groups is obtained, generalising the result of Greenleaf on contractive measures on locally compact groups. The image of a convolution operator associated to a contractive idempotent is shown to be a ternary ring of operators. As a consequence a one-to-one correspondence between contractive idempotents and a certain class of ternary rings of operators is established.

On non-commutative Minkowski spheres

Abstract The purpose of the following is to try to make sense of the stereo- graphic projection in a non-commutative setup. To this end, we consider the open unit ball of a ternary ring of operators, which naturally comes equipped with a non-commutative version of a hyperbolic metric and ask for a manifold onto which the open unit ball can be mapped so that one might think of this situation as providing a noncommutative analog to mapping the open disk of complex numbers onto the hyperboloid in three space, equipped with the restriction of the Minkowskian metric. We also obtain a related result on the Jordan algebra of self-adjoint operators

ON THE UNIVERSAL TRO OF A JC*-TRIPLE, IDEALS AND TENSOR PRODUCTS

In a development from material introduced in recent work, we discuss the interconnections between ternary rings of operators (TROs) and right C*-algebras generated by JC*-triples, deducing that every JC*-triple possesses a largest universally reversible ideal, that the universal TRO commutes with appropriate tensor products and establishing a reversibility criterion for type I JW*-triples.

Approximate ternary quadratic derivations on ternary Banach algebras and C*-ternary rings

In the current article, we use a fixed point alternative theorem to establish the Hyers-Ulam stability and also the superstability of a ternary quadratic derivation on ternary Banach algebras and C*-ternary rings which is introduced in Shagholi et al.

Controllable assembly of silver(I) complexes by variations of the carboxyl configuration: From a 2-D (45·6)2(418·610) layer to an unusual 3-D 5-connected self-penetrating (44·66)2 network

Abstract The reaction of AgClO4·6H2O with (+/−)-trans-epoxysuccinic acid (H2tes) in the presence of 2,6-dimethylpyridine afforded a three-dimensional (3-D) AgI coordination polymer [Ag2(tes)]∞ (1), which exhibits an unusual 5-connected self-penetrating (44·66)2 topological net (tes = (+/−)-trans-epoxysuccinate). Comparison of the structural differences with our relevant finding, a two-dimensional (2-D) (4,8)-connected (45·6)2(418·610) coordination polymer [Ag4(ces)2]∞ (S1) (ces = cis-epoxysuccinate), suggests that the carboxyl configuration on the ternary ring backbone of H2tes or H2ces ligand plays an important role in the construction of coordination networks.

On the operator space structure of Hilbert spaces

Operator spaces of Hilbertian JC*-triples E are considered in the light of the universal ternary ring of operators (TRO) introduced in recent work. For these operator spaces, it is shown that their triple envelope (in the sense of Hamana) is the TRO they generate, that a complete isometry between any two of them is always the restriction of a TRO isomorphism and that distinct operator space structures on a fixed E are never completely isometric. In the infinite-dimensional cases, operator space structure is shown to be characterized by severe and definite restrictions upon finite-dimensional subspaces. Injective envelopes are explicitly computed.

Operator space structure of JC*-triples and TROs, I

We embark upon a systematic investigation of operator space structure of JC*-triples via a study of the TROs (ternary rings of operators) they generate. Our approach is to introduce and develop a variety of universal objects, including universal TROs, by which means we are able to describe all possible operator space structures of a JC*-triple. Via the concept of reversibility we obtain characterisations of universal TROs over a wide range of examples. We apply our results to obtain explicit descriptions of operator space structures of Cartan factors regardless of dimension

On Approximate "Equation missing" -Ternary "Equation missing" -Homomorphisms: A Fixed Point Approach

AbstractUsing fixed point methods, we prove the stability and superstability of "Equation missing"-ternary additive, quadratic, cubic, and quartic homomorphisms in "Equation missing"-ternary rings for the functional equation "Equation missing", for each "Equation missing".

On planar fuzzy ternary rings

In this paper, we extend the process of coordinatization of projective planes to the fuzzy pro jective planes and introduce notion of fuzzy ternary ring which determines its associated fuzzy pro jective plane. Later, we give some propositions concerned with linearity of the defined fuzzy ternary operation.

Non-commutative generalisations of Urysohn's lemma and hereditary inner ideals

We establish several generalisations of Urysohn's lemma in the setting of JB ∗ -triples which provide full answers to Problems 1.12 and 1.13 in Fernández-Polo and Peralta (2007) [22]. These results extend the previous generalisations obtained by C.A. Akemann, G.K. Pedersen and L.G. Brown in the setting of C ∗ -algebras. A generalised Kadison's transitivity theorem is established for finite sums of pairwise orthogonal compact tripotents in JBW ∗ -triples. We introduce the notion of positively open tripotent in the bidual of a JB ∗ -triple as an extension of a concept which was already considered in the setting of ternary rings of operators. We investigate the connections appearing between positively open tripotents and hereditary inner ideals.

On the Local Lifting Properties of Operator Spaces

Abstract In this paper, we mainly study operator spaces which have the locally lifting property (LLP). The dual of any ternary ring of operators is shown to satisfy the strongly local reflexivity, and this is used to prove that strongly local reflexivity holds also for operator spaces which have the LLP. Several homological characterizations of the LLP and weak expectation property are given. We also prove that for any operator space V, V** has the LLP if and only if V has the LLP and V* is exact.

Stability of Homomorphisms and Derivations on C*-Ternary Rings Associated to an Euler–Lagrange Type Additive Mapping

Let X, Y be Banach spaces and let \(r_1, \ldots, r_n\, \in {\mathbb{R}}\) be given. We prove the Hyers–Ulam–Rassias stability of the following functional equation in Banach spaces: $$ \sum\limits_{{j = 1}}^{n} {f\left( { - r_{j} x_{{j}} + \,\sum\limits_{\begin{subarray}{l} 1 \le i \le n \\ \quad i \ne j \end{subarray} } {r_{i} x_{i} } } \right)\, + \,\sum\limits_{{i = 1}}^{n} {r_{i} f\left( {x_{i} } \right)} \, = \,\sum\limits_{{1 \le i<j \le n}} {f\left( {r_{i} x_{{i\,}} + \,r_{j} x_{j} } \right)} }\quad (0.1)$$ We show that if \(\sum^{n}_{i=1} r_i \neq \frac{n^{2}-3n}{2}\) and ri, rj ≠ 0 for some 1 ≤ i < j ≤ n and a mapping f : X → Y satisfies the functional equation (0.1), then the mapping f : X → Y is Cauchy additive. As an application, we investigate homomorphisms and derivations between C*-ternary rings.

Stable isomorphism of dual operator spaces

We prove that two dual operator spaces X and Y are stably isomorphic if and only if there exist completely isometric normal representations ϕ and ψ of X and Y, respectively, and ternary rings of operators M 1 , M 2 such that ϕ ( X ) = [ M 2 ∗ ψ ( Y ) M 1 ] − w ∗ and ψ ( Y ) = [ M 2 ϕ ( X ) M 1 ∗ ] − w ∗ . We prove that this is equivalent to certain canonical dual operator algebras associated with the operator spaces being stably isomorphic. We apply these operator space results to prove that certain dual operator algebras are stably isomorphic if and only if they are isomorphic. Consequently, we obtain that certain complex domains are biholomorphically equivalent if and only if their algebras of bounded analytic functions are Morita equivalent in our sense. Finally, we provide examples motivated by the theory of CSL algebras.

Projective planes I

Semiadditive rings are defined and their relationship with the projective planes is studied. Free semiadditive rings provide an analogue of the ring of integers and polynomials for the ternary rings. A structure theory for free semiadditive rings is developed. It is shown that each element of a large class of semiadditive rings is obtained from a quotient of a polynomial ring over integers by an additive subgroup, by twisting addition and multiplication. This class includes all planar ternary rings. These methods are being developed to study the well known conjectures that every finite projective plane with no proper subplane is isomorphic to a prime field plane and that the order of a finite projective plane is a power of a prime number. Applications to these problems will be discussed in part II.

On the isotopism classes of finite semifields

A projective plane is called a translation plane if there exists a line L such that the group of elations with axis L acts transitively on the points not on L. A translation plane whose dual plane is also a translation plane is called a semifield plane. The ternary ring corresponding to a semifield plane can be made into a non-associative algebra called a semifield, and two semifield planes are isomorphic if and only if the corresponding semifields are isotopic. In [S. Ball, G. Ebert, M. Lavrauw, A geometric construction of finite semifields, J. Algebra 311 (1) (2007) 117–129] it was shown that each finite semifield gives rise to a particular configuration of two subspaces with respect to a Desarguesian spread, called a BEL-configuration, and vice versa that each BEL-configuration gives rise to a semifield. In this manuscript we investigate the question when two BEL-configurations determine isotopic semifields. We show that there is a one-to-one correspondence between the isotopism classes of finite semifields and the orbits of the action a subgroup of index two of the automorphism group of a Segre variety on subspaces of maximum dimension skew to a determinantal hypersurface.

Stably isomorphic dual operator algebras

We prove that two unital dual operator algebras A, B are stably isomorphic if and only if they are Δ-equivalent (Eleftherakis in J Pure Appl Algebra, ArXiv:math. OA/0607489v4, 2007), if and only if they have completely isometric normal representations α,β on Hilbert spaces H, K respectively and there exists a ternary ring of operators \({\mathcal{M} \subset B(H, K)}\) such that \({\alpha (A) = [\mathcal{M}^* \beta(B)\mathcal{M}]^{-w^*}}\) and \({\beta(B) = [\mathcal{M}\alpha(A)\mathcal{M}^*]^{-w^*}.}\)

On Axiomatizability of Non-Commutative Lp-Spaces

AbstractIt is shown that Schatten p-classes of operators between Hilbert spaces of different (infinite) dimensions have ultrapowers which are (completely) isometric to non-commutative Lp-spaces. On the other hand, these Schatten classes are not themselves isomorphic to non-commutative Lp spaces. As a consequence, the class of non-commutative Lp-spaces is not axiomatizable in the first-order language developed by Henson and Iovino for normed space structures, neither in the signature of Banach spaces, nor in that of operator spaces. Other examples of the same phenomenon are presented that belong to the class of corners of non-commutative Lp-spaces. For p = 1 this last class, which is the same as the class of preduals of ternary rings of operators, is itself axiomatizable in the signature of operator spaces.

A nearfield-free definition of regular Hughes planes and some embeddings of PG(3,q) in Hughes planes of order q 4

We give a nearfield-free definition of some finite and infinite incidence systems by means of half-points and half-lines and show that they are projective planes. We determine a planar ternary ring for these planes and use it to determine the full collineation group and to demonstrate some embeddings of these planes among themselves. We show that these planes include all finite regular Hughes planes and many infinite ones. We also show that PG(3, q) embeds in Hu(q 4) (and show infinite versions of this embedding).