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 $phi$ and $psi$ of $X$ and $Y$, respectively, and ternary rings of operators $M_1, M_2$ such that $phi (X)= [M_2^*psi (Y)M_1]^{-w^*}$ and $psi (Y)=[M_2phi (X)M_1^*].$ 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. We provide examples motivated by CSL algebra theory.

Jordan isomorphism of purely infinite C*-algebras

We prove that every unital bounded linear mapping from a unital purely infinite C*-algebra of real rank zero into a unital Banach algebra which preserves elements of square zero is a Jordan homomorphism. This entails a description of unital surjective spectral isometries as the Jordan isomorphisms in this setting.

Isomorphism between ice and silica

Both ice and silica crystallize into solid-state structures composed of tetrahedral building units that are joined together to form an infinite four-connected net. Mathematical considerations suggest that there is a vast number of such nets and thus potential crystal structures. It is therefore perhaps surprising to discover that, despite the differences in the nature of interatomic interactions in these materials, a fair number of commonly observed ice and silica phases are based on common nets. Here we use computer simulation to investigate the origin of this symmetry between the structures formed for ice and silica and to attempt to understand why it is not complete. We start from a comparison of the dense phases and then move to the relationship between the different open (zeolitic and clathratic) structures formed for both materials. We show that there is a remarkably strong correlation between the energetics of isomorphic silica and water ice structures and that this correlation arises because of the strong link between the total energy of a material and its local geometric features. Finally, we discuss a number of as yet unsynthesized low-energy structures which include a phase of ice based on quartz, a silica based on the structure of ice VI, and an ice clathrate that is isomorphic to the silicate structure nonasil.

An isomorphism between group C*-algebras of ax + b-like groups

We give a necessary and sufficient condition for two ax+b-like groups to have isomorphic C*-algebras. In particular, we show that there are many non-isomorphic ax+b -like Lie groups having isomorphic group C*-algebras.

On the derived category of a regular toric scheme

Let X be a quasi-compact scheme, equipped with an open covering by affine schemes U s = Spec A s . A quasi-coherent sheaf on X gives rise, by taking sections over the U s , to a diagram of modules over the coordinate rings A s , indexed by the intersection poset S of the covering. If X is a regular toric scheme over an arbitrary commutative ring, we prove that the unbounded derived category of quasi-coherent sheaves on X can be obtained from a category of Sop-diagrams of chain complexes of modules by inverting maps which induce homology isomorphisms on hyper-derived inverse limits. Moreover, we show that there is a finite set of weak generators, one for each cone in the fan S. The approach taken uses the machinery of Bousfield–Hirschhorn colocalisation of model categories. The first step is to characterise colocal objects; these turn out to be homotopy sheaves in the sense that chain complexes over different open sets U s agree on intersections up to quasi-isomorphism. In a second step it is shown that the homotopy category of homotopy sheaves is equivalent to the derived category of X.

Double complexes and vanishing of Novikov cohomology

We consider non-standard totalisation functors for double complexes, involving left or right truncated products. We show how properties of these imply that the algebraic mapping torus of a self map h of a cochain complex of finitely presented modules has trivial negative Novikov cohomology, and has trivial positive Novikov cohomology provided h is a quasi-isomorphism. As an application we obtain a new and transparent proof that a finitely dominated cochain complex over a Laurent polynomial ring has trivial (positive and negative) Novikov cohomology.

Policy Transfer and the Sustainability of Public Private Partnership in Ireland::A conceptual analysis

There has been private sector involvement in the delivery of public services in the Irish State since its foundation. This involvement was formalised in 1998 when Public Private Partnership (PPP) was officially introduced. Ireland is a latecomer to PPP and, prior to the credit crisis, was seen as a ‘rapid follower’ relying primarily on the UK PPP model in the procurement of infrastructure in transport, education, housing/urban regeneration and water/wastewater.  PPP activity in Ireland stalled during the credit crisis, and some projects were cancelled, but it has taken off again recently with part of the Infrastructure and Capital Investment Plan 2016 – 2021 to be delivered through PPP showing continuing political commitment to PPP.  Ireland’s interest in PPP cannot be explained by economic rationale alone, as PPP was initiated during a period of prosperity. We consider three alternative explanations: voluntary adoption – where the UK model was closely followed; coercive adoption – where PPP policy was forced upon Ireland; and institutional isomorphism – where institutional creation and change was promoted to aid public sector organisations in gaining institutional legitimacy. We find evidence of all three patterns, with coercive adoption becoming more relevant in recent years. Ireland’s rapid uptake of PPP differs from other European countries, mostly because when PPP was introduced in 1998, the Irish State was in an economic position where it could have directly procured necessary infrastructure. This paper therefore asks why PPP was adopted and how this adoption pattern has affected the sustainability of PPP in Ireland.  This paper defines PPP; examines the background to the PPP approach adopted in Ireland; outlines the theoretical framework of the paper: transfer theory and institutional theory; discusses the methodology; reports on findings and gives conclusions.   

The evolution of public–private partnership in Ireland: a sustainable pathway?

Ireland is a latecomer to Public Private Partnership (PPP) having only adopted it in 1998. Prior to the credit crisis, Ireland followed the UK model with PPPs being implemented in transport, education, housing/urban regeneration and water/wastewater. Having stalled during the credit crisis, PPP has been reactivated recently with the domestic infrastructure stimulus programme . The focus of this paper is on Ireland as a younger participant in PPP and the nexus between adoption patterns and sustainability characteristics of Irish PPP. Using document analysis and exploratory interviews, the paper examines the reasons for Ireland’s interest in PPP which cannot be attributed to economic rationales alone. We consider three explanations: voluntary adoption – where the UK model was closely followed as part of a domestic modernisation agenda; coercive adoption – where PPP policy was forced upon public sector organisations; and institutional isomorphism – where institutional creation and change around PPP was promoted to help public sector organisations gain institutional legitimacy. We find evidence of all three patterns with coercive adoption becoming more relevant in recent years, which is likely to affect sustainability adversely unless incentives for voluntary adoption are strengthened and institutional capacity building is boosted.

Orbital and strongly orbital spaces

We say that a (countably dimensional) topological vector space X is orbital if there is T∈L(X) and a vector x∈X such that X is the linear span of the orbit {Tnx:n=0,1,…}. We say that X is strongly orbital if, additionally, x can be chosen to be a hypercyclic vector for T. Of course, X can be orbital only if the algebraic dimension of X is finite or infinite countable. We characterize orbital and strongly orbital metrizable locally convex spaces. We also show that every countably dimensional metrizable locally convex space X does not have the invariant subset property. That is, there is T∈L(X) such that every non-zero x∈X is a hypercyclic vector for T. Finally, assuming the Continuum Hypothesis, we construct a complete strongly orbital locally convex space.

As a byproduct of our constructions, we determine the number of isomorphism classes in the set of dense countably dimensional subspaces of any given separable infinite dimensional Fréchet space X. For instance, in X=ℓ2×ω, there are exactly 3 pairwise non-isomorphic (as topological vector spaces) dense countably dimensional subspaces.

Three dimensional Sklyanin algebras and Gröbner bases

We consider Sklyanin algebras $S$ with 3 generators, which are quadratic algebras over a field $\K$ with $3$ generators $x,y,z$ given by $3$ relations $pxy+qyx+rzz=0$, $pyz+qzy+rxx=0$ and $pzx+qxz+ryy=0$, where $p,q,r\in\K$. this class of algebras has enjoyed much attention. In particular, using tools from algebraic geometry, Feigin, Odesskii \cite{odf}, and Artin, Tate and Van Den Bergh, showed that if at least two of the parameters $p$, $q$ and $r$ are non-zero and at least two of three numbers $p^3$, $q^3$ and $r^3$ are distinct, then $S$ is Artin--Schelter regular. More specifically, $S$ is Koszul and has the same Hilbert series as the algebra of commutative polynomials in 3 indeterminates (PHS). It has became commonly accepted that it is impossible to achieve the same objective by purely algebraic and combinatorial means like the Groebner basis technique. The main purpose of this paper is to trace the combinatorial meaning of the properties of Sklyanin algebras, such as Koszulity, PBW, PHS, Calabi-Yau, and to give a new constructive proof of the above facts due to Artin, Tate and Van Den Bergh. Further, we study a wider class of Sklyanin algebras, namely
the situation when all parameters of relations could be different. We call them generalized Sklyanin algebras. We classify up to isomorphism all generalized Sklyanin algebras with the same Hilbert series as commutative polynomials on
3 variables. We show that generalized Sklyanin algebras in general position have a Golod–Shafarevich Hilbert series (with exception of the case of field with two elements).