Characterizations of Morita equivalent inverse semigroups

We prove that four different notions of Morita equivalence for inverse semigroups motivated by C^∗-algebra theory, topos theory, semigroup theory and the theory of ordered groupoids are equivalent. We also show that the category of unitary actions of an inverse semigroup is monadic over the category of étale actions. Consequently, the category of unitary actions of an inverse semigroup is equivalent to the category of presheaves on its Cauchy completion. More generally, we prove that the same is true for the category of closed actions, which is used to define the Morita theory in semigroup theory, of any semigroup with right local units.     

Amenability and uniqueness

The main result of this paper is a characterization of properly infinite injective von Neumann algebras and of nuclear C^∗$C^{\ast }$-algebras by using a uniqueness theorem, based on generalizations of Voiculescu’s famous Weyl–von Neumann theorem.     

AF inverse monoids and the structure of countable MV-algebras

This paper is a further contribution to the developing theory of Boolean inverse monoids. These monoids should be regarded as non-commutative generalizations of Boolean algebras; indeed, classical Stone duality can be generalized to this non-commutative setting to yield a duality between Boolean inverse monoids and a class of étale topological groupoids. MV-algebras are also generalizations of Boolean algebras which arise from many-valued logics. It is the goal of this paper to show how these two generalizations are connected. To do this, we define a special class of Boolean inverse monoids having the property that their lattices of principal ideals naturally form an MV-algebra. We say that an arbitrary MV-algebra can be co-ordinatized if it is isomorphic to an MV-algebra arising in this way. Our main theorem is that every countable MV-algebra can be so co-ordinatized. The particular Boolean inverse monoids needed to establish this result are examples of what we term AF inverse monoids and are the inverse monoid analogues of AF $C^{?}$-algebras. In particular, they are constructed from Bratteli diagrams as direct limits of finite direct products of finite symmetric inverse monoids.     

K-theory and exact sequences of partial translation algebras

In an earlier paper, the authors introduced partial translation algebras as a generalisation of group C^?${\mathrm{C}}^{?}$-algebras. Here we establish an extension of partial translation algebras, which may be viewed as an excision theorem in this context. We apply this general framework to compute the K-theory of partial translation algebras and group C^?${\mathrm{C}}^{?}$-algebras in the context of almost invariant subspaces of discrete groups. This generalises the work of Cuntz, Lance, Pimsner and Voiculescu. In particular we provide a new perspective on Pimsner's calculation of the K-theory for a graph product of groups.     

Asymptotic patterns of a structured population diffusing in a two-dimensional strip

In this paper, we derive a population model for the growth of a single species on a two-dimensional strip with Neumann and Robin boundary conditions. We show that the dynamics of the mature population is governed by a reaction–diffusion equation with delayed global interaction. Using the theory of asymptotic speed of spread and monotone traveling waves for monotone semiflows, we obtain the spreading speed c^∗$c^{\ast }$, the non-existence of traveling waves with wave speed 0<c<c^∗$0<c<c^{\ast }$, and the existence of monotone traveling waves connecting the two equilibria for c≥c^∗$c\ge c^{\ast }$.     

A generalized Cuntz–Krieger uniqueness theorem for higher-rank graphs

We present a uniqueness theorem for k  -graph C^?$C^{?}$-algebras that requires neither an aperiodicity nor a gauge invariance assumption. Specifically, we prove that for the injectivity of a representation of a k  -graph C^?$C^{?}$-algebra, it is sufficient that the representation be injective on a distinguished abelian C^?$C^{?}$-subalgebra. A crucial part of the proof is the application of an abstract uniqueness theorem, which says that such a uniqueness property follows from the existence of a jointly faithful collection of states on the ambient C^?$C^{?}$-algebra, each of which is the unique extension of a state on the distinguished abelian C^?$C^{?}$-subalgebra.     

The K-theoretical range of Cuntz–Krieger algebras

We augment Restorff's classification of purely infinite Cuntz–Krieger algebras by describing the range of his invariant on purely infinite Cuntz–Krieger algebras. We also describe its range on purely infinite graph C^?$C^{?}$-algebras with finitely many ideals, and provide ‘unital’ range results for purely infinite Cuntz–Krieger algebras and unital purely infinite graph C^?$C^{?}$-algebras.     

A note on spectral triples and quasidiagonality

Spectral triples (of compact type) are constructed on arbitrary separable quasidiagonal C^*$C^{*}$-algebras. On the other hand an example of a spectral triple on a non-quasidiagonal algebra is presented.     

A groupoid approach to discrete inverse semigroup algebras

Let K be a commutative ring with unit and S an inverse semigroup. We show that the semigroup algebra KS can be described as a convolution algebra of functions on the universal étale groupoid associated to S by Paterson. This result is a simultaneous generalization of the author's earlier work on finite inverse semigroups and Paterson's theorem for the universal C^∗-algebra. It provides a convenient topological framework for understanding the structure of KS, including the center and when it has a unit. In this theory, the role of Gelfand duality is replaced by Stone duality.Using this approach we construct the finite dimensional irreducible representations of an inverse semigroup over an arbitrary field as induced representations from associated groups, generalizing the case of an inverse semigroup with finitely many idempotents. More generally, we describe the irreducible representations of an inverse semigroup S that can be induced from associated groups as precisely those satisfying a certain “finiteness condition.” This “finiteness condition” is satisfied, for instance, by all representations of an inverse semigroup whose image contains a primitive idempotent.     

The maximum of a Lévy process reflected at a general barrier

We investigate the reflection of a Lévy process at a deterministic, time-dependent barrier and in particular properties of the global maximum of the reflected Lévy process. Under the assumption of a finite Laplace exponent, ψ(θ)$\psi \left(\theta \right)$, and the existence of a solution θ^∗>0${\theta }^{\ast }>0$ to ψ(θ)=0$\psi \left(\theta \right)=0$ we derive conditions in terms of the barrier for almost sure finiteness of the maximum. If the maximum is finite almost surely, we show that the tail of its distribution decays like Kexp(−θ^∗x)$Kexp(-{\theta }^{\ast }x)$. The constant K$K$ can be completely characterized, and we present several possible representations. Some special cases where the constant can be computed explicitly are treated in greater detail, for instance Brownian motion with a linear or a piecewise linear barrier. In the context of queuing and storage models the barrier has an interpretation as a time-dependent maximal capacity. In risk theory the barrier can be interpreted as a time-dependent strategy for (continuous) dividend pay out.     

New fundamental relation of hyperrings

We prove that θ^∗${\theta }^{\ast }$ as previously defined [Int. J. Contemp. Math. Sci. 5 (2010) 721] is the smallest equivalence relation such that the quotient structure R/θ^∗$R/{\theta }^{\ast }$ is a commutative fundamental ring. We also investigate some properties with respect to the commutative fundamental relation θ^∗${\theta }^{\ast }$ on a hyperring R$R$.     

Regularity of semigroups generated by Lévy type operators via coupling

By adopting the coupling method, we obtain new verifiable sufficient conditions about the _Cb(R^d)$C_b\left({\mathbb{R}}^d\right)$-Feller continuity, the Lipschitz continuity and the strong Feller continuity of the semigroups associated with Lévy type operators. These results easily apply to jump–diffusion processes and stochastic differential equations driven by Lévy processes. Our results also yield the criterion for the e$e$-property (namely the characterization about the equi-continuity of semigroups acting on bounded Lipschitz functions) of Lévy type operators, and show that both genuine Lévy processes and the Ornstein–Uhlenbeck type processes are e$e$-processes.     

Étale groupoids and their quantales

We establish close and previously unknown relations between quantales and groupoids. In particular, to each étale groupoid, either localic or topological, there is associated a unital involutive quantale. We obtain a bijective correspondence between localic étale groupoids and their quantales, which are given a rather simple characterization and here are called inverse quantal frames. We show that the category of inverse quantal frames is equivalent to the category of complete and infinitely distributive inverse monoids, and as a consequence we obtain a (non-functorial) correspondence between these and localic étale groupoids that generalizes more classical results concerning inverse semigroups and topological étale groupoids. This generalization is entirely algebraic and it is valid in an arbitrary topos. As a consequence of these results we see that a localic groupoid is étale if and only if its sublocale of units is open and its multiplication map is semiopen, and an analogue of this holds for topological groupoids. In practice we are provided with new tools for constructing localic and topological étale groupoids, as well as inverse semigroups, for instance via presentations of quantales by generators and relations. The characterization of inverse quantal frames is to a large extent based on a new quantale operation, here called a support, whose properties are thoroughly investigated, and which may be of independent interest.     

Cone isomorphisms and almost surjective operators

Let E$E$ be a Banach lattice and F$F$ a Banach space. A bounded linear operator T:E→F$T:E\to F$ is an isomorphism on the positive cone of E$E$ if and only if T^∗$T^{\ast }$ is almost surjective. A dual version of this theorem holds also. A bounded linear operator T:F→E$T:F\to E$ is almost surjective if and only if T^∗$T^{\ast }$ is an isomorphism on the positive cone of F^∗$F^{\ast }$.