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S. Sundar

Publications and source records attributed to S. Sundar.

At least 37 records · Page 2Linked to original sources
arXiv2020

On the existence of $E_{0}$-semigroups -- the multiparameter case

Let $P \subset \mathbb{R}^{d}$ be a closed convex cone. Assume that $P$ is pointed, i.e. the intersection $P \cap -P=\{0\}$ and $P$ is spanning, i.e. $P-P=\mathbb{R}^{d}$. Denote the interior of $P$ by $Ω$. Let $E$ be a product system over $Ω$. We show that there exists an infinite dimensional separable Hilbert space $\mathcal{H}$ and a semigroup $α:=\{α_x\}_{x \in P}$ of unital normal $*$-endomorphisms of $B(\mathcal{H})$ such that $E$ is isomorphic to the product system associated to $α$.

math.OA↗
arXiv2020

Product systems associated to compound Poisson Processes

In this paper, we consider a simple test case of multiparameter product systems that arise out of random measures. We associate a product system to a stationary Poisson process and a stationary compound Poisson process. We show that the resulting $E_0$-semigroups are CCR flows.

math.OA↗
arXiv2020

An improved mathematical model for the sedimentation of microplastic particles in a lid-driven cavity with obstacle

In this paper, we developed a mathematical model for the sedimentation process of small particles in a lid-driven cavity flow with an obstacle to model the transport of microplastic particles in rivers. A stationary incompressible Navier-Stokes simulation at moderate Reynolds-numbers provides the background flow field. Spherical particles are injected into this flow field and their equation of motion is solved to determine the number of particles that sediment on the surface of the obstacle. To capture the effect of typical biological organisms and biofilms on the bottom surface of river beds, the obstacle can exert an attraction force onto the particles. Various simulations and parameter studies are carried out to determine the influence of the obstacle geometry, particle densities, and the attraction force on the sedimentation process.

physics.flu-dyn↗
arXiv2020

An asymmetric multiparameter CCR flow

In this note, we exhibit an example of a multiparameter CCR flow which is not cocycle conjugate to its opposite. This is in sharp contrast to the one parameter situation

math.OA↗
arXiv2019

Arveson's characterisation of CCR flows: the multiparameter case

In this paper, we revisit Arveson's characterisation of CCR flows in terms of decomposibility of the product system in the multiparameter context. We show that a multiparameter $E_0$-semigroup is a CCR flow if and only if it is decomposable and admits a unit. In contrast to the one parameter situtation, we exhibit uncountably many examples of decomposable $E_0$-semigroups which do not admit any unit. As applications, we show that for a pure isometric representation $V$, the associated CCR flow $α^{V}$ remembers the unitary equivalence class of $V$. We also compute the positive contractive local cocycles and projective local cocycles of a CCR flow. A necessary and a sufficient condition for a CCR flow to be prime is obtained.

math.OA↗
arXiv2019

CCR flows associated to closed convex cones

Let $P$ be a closed convex cone in $\mathbb{R}^{d}$ which we assume to be spanning and pointed i.e. $P-P=\mathbb{R}^{d}$ and $P \cap -P=\{0\}$. In this article, we consider CCR flows over $P$ associated to isometric representations that arises out of $P$-invariant closed subsets, also called as $P$-modules, of $\mathbb{R}^{d}$. We show that for two $P$-modules the associated CCR flows are cocycle conjugate if and only if the modules are translates of each other.

math.OA↗
arXiv2018

E-semigroups over closed convex cones

We initiate a study of E-semigroups over convex cones. We prove a structure theorem for E-semigroups which leave the algebra of compact operators invariant. Then we study in detail the CCR flows, E$_0$semigroups constructed from isometric representations, by describing their units and gauge groups. We exhibit an uncountable family of $2-$parameter CCR flows, containing mutually non-cocycle-conjugate E$_0$-$semigroups.

math.OA↗
arXiv2017

$E_{0}^{P}$-semigroups and product systems

Let $P$ be a closed convex cone in $\mathbb{R}^{n}$. Assume that $P$ is spanning i.e. $P-P=\mathbb{R}^{n}$ and pointed i.e. $P \cap -P=\{0\}$. Let $α:=\{α_{x}:x \in P\}$ be a $σ$-weakly continuous family of unital normal endomorphisms on $B(H)$. Denote the "product system" associated to $α$ by $\mathcal{E}_α$. We show that $\mathcal{E}_α$ is a concrete product system and $α$, up to cocycle conjugacy, can be recovered completely from $\mathcal{E}_α$

math.OA↗
arXiv2016

On the Wiener-Hopf compactification of a Symmetric Cone

Let V be a finite dimensional real Euclidean Jordan algebra with the identity element 1. Let Q be the closed convex cone of squares. We show that the Wiener- Hopf compactification of Q is the interval (1-Q) \cap (-1+Q). As a consequence, we deduce that the K-groups of the Wiener-Hopf C^{*}-algebra associated to Q are trivial.

math.OA↗
arXiv2016

Fractional Brownian motion time-changed by gamma and inverse gamma process

Many real time-series exhibit behavior adequate to long range dependent data. Additionally very often these time-series have constant time periods and also have characteristics similar to Gaussian processes although they are not Gaussian. Therefore there is need to consider new classes of systems to model these kind of empirical behavior. Motivated by this fact in this paper we analyze two processes which exhibit long range dependence property and have additional interesting characteristics which may be observed in real phenomena. Both of them are constructed as the superposition of fractional Brownian motion (FBM) and other process. In the first case the internal process, which plays role of the time, is the gamma process while in the second case the internal process is its inverse. We present in detail their main properties paying main attention to the long range dependence property. Moreover, we show how to simulate these processes and estimate their parameters. We propose to use a novel method based on rescaled modified cumulative distribution function for estimation of parameters of the second considered process. This method is very useful in description of rounded data, like waiting times of subordinated processes delayed by inverse subordinators. By using the Monte Carlo method we show the effectiveness of proposed estimation procedures.

physics.data-an↗
arXiv2016

Toeplitz algebras associated to Endomorphisms of Ore semigroups

In this paper, we consider the Toeplitz algebra associated to actions of Ore semigroups on $C^{*}$-algebras. In particular, we consider injective and surjective actions of such semigroups. We use the theory of groupoid dynamical systems to represent the Toeplitz algebra as a groupoid crossed product. We also discuss the K-theory of the Toeplitz algebra in some examples. For instance, we show that for the semigroup of positive matrices, the K-theory of the associated Toeplitz algebra vanishes.

math.OA↗
arXiv2015

On a construction due to Khoshkam and Skandalis

In this paper, we consider the Wiener Hopf algebra, denoted $\mathcal{W}(A,P,G,α)$, associated to an action of a discrete subsemigroup $P$ of a group $G$ on a $C^{*}$-algebra $A$. We show that $\mathcal{W}(A,P,G,α)$ can be represented as a groupoid crossed product. As an application, we show that when $P=\mathbb{F}_{n}^{+}$, the free semigroup on $n$ generators, the $K$-theory of $\mathcal{W}(A,P,G,α)$ and the $K$-theory of $A$ coincides.

math.OA↗
arXiv2015

C*-algebras associated to topological Ore semigroups

Let $G$ be a locally compact group and $P \subset G$ be a closed Ore semigroup containing the identity element. Let $V: P \to B(\clh)$ be a representation such that for every $a \in P$, $V_{a}$ is an isometry and the final projections of $\{V_{a}: a \in P\}$ commute. In this article, we study the $C^{*}$-algebra $\mathcal{W}_{V}(P,G)$, generated by $\{\int f(a)V_{a} da: f \in L^{1}(P)\}$. We show that there exists a universal $C^{*}$-algebra, which admits a groupoid description, of which $\mathcal{W}_{V}(P,G)$ is a quotient. If $P=G$, then this universal algebra is just $C^{*}(G)$.

math.OA↗
arXiv2014

Groupoids associated to semigroup actions

In this paper, we consider topological semigroup actions on compact topological spaces. Under mild assumptions on the semigroup and the action, we construct a semi-direct product groupoid with a Haar system. We also show that it is equivalent to a transformation groupoid. We apply this construction to the Wiener-Hopf $C^{*}$-algebras.

math.OA↗
arXiv2012

Cuntz-Li relations, Inverse semigroups and Groupoids

In this paper we show that the universal C*-algebra satisfying the Cuntz-Li relations is generated by an inverse semigroup of partial isometries. We apply Exel's theory of tight representations to this inverse semigroup. We identify the universal C*-algebra as the C*-algebra of the tight groupoid associated to the inverse semigroup.

math.OA↗
arXiv2012

A computation with the Connes-Thom isomorphism

Let $A \in M_{n}(\mathbb{R})$ be an invertible matrix. Consider the semi-direct product $\mathbb{R}^{n} \rtimes \mathbb{Z}$ where $\mathbb{Z}$ acts on $\mathbb{R}^{n}$ by matrix multiplication. Consider a strongly continuous action $(α,τ)$ of $\mathbb{R}^{n} \rtimes \mathbb{Z}$ on a $C^{*}$-algebra $B$ where $α$ is a strongly continuous action of $\mathbb{R}^{n}$ and $τ$ is an automorphism. The map $τ$ induces a map $\widetildeτ$ on $B \rtimes_α \mathbb{R}^{n}$. We show that, at the $K$-theory level, $τ$ commutes with the Connes-Thom map if $\det(A)>0$ and anticommutes if $\det(A)<0$. As an application, we recompute the $K$-groups of the Cuntz-Li algebra associated to an integer dilation matrix.

math.OA↗