Ternary Rings Of Operators Research Articles

Anti-C*-algebras

We introduce a class of Banach algebras that we call anti-C*-algebras. We show that the normed standard embedding of a C*-ternary ring is the direct sum of a C*-algebra and an anti-C*-algebra. We prove that C*-ternary rings and anti-C*-algebras are semisimple. We give two new characterizations of C*-ternary rings which are isomorphic to a TRO (ternary ring of operators), providing answers to a query raised by Zettl (Adv Math 48(2): 117–143, 1983), and we propose some problems for further study.

Contractively decomposable projections on noncommutative Lp-spaces

We describe and characterize the contractively decomposable projections on noncommutative Lp-spaces. Our result relies on a new lifting result for decomposable maps of independent interest and on some tools from ergodic theory. Our theorem is new even for finite-dimensional Schatten spaces. Our description allows us to connect this topic with W⁎-ternary rings of operators and a slight generalization of our result for more general projections makes JBW⁎-triples appear in this context. We also prove that all rectangular Lp-spaces associated with W⁎-ternary rings of operators arise as contractively decomposable complemented subspaces of noncommutative Lp-spaces. Finally, we introduce a notion of Lp-space associated to each σ-finite JBW⁎-triple and we explain the link with the context of this paper.

Extreme states and boundary representations of operator spaces in ternary rings of operators

This paper explores different notions of extreme states on operator systems and on operator spaces. We prove that boundary points of the generalized state space of an operator system in finite dimensions are precisely the restrictions of boundary representations of the generated [Formula: see text]-algebra. We study weak TRO-extreme states on operator spaces in ternary rings of operators as a non self-adjoint version of [Formula: see text]-extreme states on operator systems in [Formula: see text]-algebras. Rectangular boundary point for a rectangular matrix convex set is introduced and its connection with boundary representations of the associated operator space is established in finite dimensional setting. Also, we show that the set of all rectangular boundary points is the minimal spanning set for a compact rectangular matrix convex set in finite dimensions.

Extrem states, operator spaces and ternary rings of operators

In this survey article on extreme states of operator spaces in C¡-algebras and related ternary ring of operators an extension result for rectangular operator extreme states on operator spaces in ternary rings of operators is discussed. We also observe that in the spacial case of operator spaces in rectangular matrix spaces, rectangular extreme states are conjugates of inclusion or identity maps implemented by isometries or unitaries. A characterization result for operator spaces of matrices for which the inclusion map is an extreme state is deduced using the above mentioned results.

Extreme states on operator spaces in ternary rings of operators

Extreme states on operator spaces in ternary rings of operators

Editorial

Richard Martin Timoney (17 July 1953 – 1 January 2019) EDITORIAL Martin Mathieu Richard M. Timoney was a professor at Trinity College, The University of Dublin. He passed away, far too soon, at the start of this year leaving an immense mathematical legacy behind. Seán Dineen’s article in this volume pays tribute to Richard’s wide-ranging interests, which he pursued with numerous co-authors from around the world. At a conference entitled “Analysis, Geometry, and Algebra” held in Trinity College in May 2019, many of those mathematicians gathered together with his colleagues, some of his seven PhD students, friends and Richard’s family to celebrate his life and work. A comprehensive obituary appeared in Irish Math. Soc. Bulletin 83, Summer 2019, 29–36, also written by Seán. The articles in the present volume, by some of the speakers at the conference in Richard’s honour, reflect his journey through mathematics, from the topics in his PhD thesis to the most recent developments in operator spaces and ternary rings of operators, which now have to be published posthumously. We are grateful to Richard for dedicating his professional life to Mathematics, not only to research and teaching but also his many other efforts to allow Mathematics to flourish in Ireland and beyond. We all miss him a lot. Belfast, November 2019 ∗Corresponding author, e-mail: m.m@qub.ac.uk doi: https://dx.doi.org/10.3318/PRIA.2019.119.13 Cite as follows: Martin Mathieu, Editorial, Mathematical Proceedings of the Royal Irish Academy 119A (2019), i; https://dx.doi.org/10.3318/PRIA.2019.119.13 Mathematical Proceedings of the Royal Irish Academy, 119A (2), i (2019) © Royal Irish Academy ...

Corrigendum: Derivations on ternary rings of operators

Corrigendum: Derivations on ternary rings of operators

On the structure of weakly closed ternary rings of operators

On the structure of weakly closed ternary rings of operators

On decompositions of ternary rings of operators and their morphisms

On decompositions of ternary rings of operators and their morphisms

Capacity Estimates via Comparison with TRO Channels

A ternary ring of operators (TRO) in finite dimensions is a diagonal sum of spaces of rectangular matrices. TRO as operator space corresponds to quantum channels that are diagonal sums of partial traces, which we call TRO channels. TRO channels admits simple, single-letter capacity formula. Using operator space and complex interpolation techniques, we give perturbative capacities estimates for a wider class of quantum channels by comparison to TRO channels. Our estimates applies mainly for quantum and private capacity and also strong converse rates. The examples includes random unitary from group representations which in general are non-degradable channels.

Derivations on ternary rings of operators

To each projection $p$ in a $C^*$-algebra $A$ we associate a family of derivations on $A$, called $p$-derivations, and relate them to the space of triple derivations on $p A (1-p)$. We then show that every derivation on a ternary ring of operators is spatial and we investigate whether every such derivation on a weakly closed ternary ring of operators is inner.

Actions of locally compact (quantum) groups on ternary rings of operators, their crossed products, and generalized Poisson boundaries

Actions of locally compact groups and quantum groups on W∗-ternary rings of operators are discussed, and related crossed products are introduced. The results generalize those for von Neumann algebraic actions with proofs based mostly on passing to the linking von Neumann algebra. They are motivated by the study of fixed-point spaces for convolution operators generated by contractive, not necessarily positive measures, both in the classical and in the quantum context.

Čebyšëv subspaces of JBW∗-triples

We describe the one-dimensional Cebysev subspaces of a JBW∗-triple M by showing that for a non-zero element x in M, $\mathbb{C}x$ is a Cebysev subspace of M if and only if x is a Brown-Pedersen quasi-invertible element in M. We study the Cebysev JBW∗-subtriples of a JBW∗-triple M. We prove that for each non-zero Cebysev JBW∗-subtriple N of M, exactly one of the following statements holds: We also provide new examples of Cebysev subspaces of classic Banach spaces in connection with ternary rings of operators.

The Universal Enveloping Ternary Ring of Operators of a JB*-Triple System

AbstractWe associate to every JB*-triple system a so-called universal enveloping ternary ring of operators (TRO). We compute the universal enveloping TROs of the finite dimensional Cartan factors.

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.

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.

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

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.

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.