arXiv subjects

Narayan Rakshit

Publications and source records attributed to Narayan Rakshit.

8 recordsLinked to original sources

arXiv2024

Contractive representations of odometer semigroup

Given a natural number $n \geq 1$, the odometer semigroup $O_n$, also known as the adding machine or the Baumslag-Solitar monoid with two generators, is a well-known object in group theory. This paper examines the odometer semigroup in relation to representations of bounded linear operators. We focus on noncommutative operators and prove that contractive representations of $O_n$ always admit to nicer representations of $O_n$. We give a complete description of representations of $O_n$ on the Fock space and relate it to the odometer lifting and subrepresentations of $O_n$. Along the way, we also classify Nica covariant representations of $O_n$.

math.FA↗

arXiv2022

Orthogonal decompositions and twisted isometries II

We classify tuples of (not necessarily commuting) isometries that admit von Neumann-Wold decomposition. We introduce the notion of twisted isometries for tuples of isometries and prove the existence of orthogonal decomposition for such tuples. The former classification is partially inspired by a result that was observed more than three decades ago by Gaspar and Suciu. And the latter result generalizes Popovici's orthogonal decompositions for pairs of commuting isometries to general tuples of twisted isometries which also includes the case of tuples of commuting isometries. Our results unify all the known orthogonal decomposition related results in the literature.

math.FA↗

arXiv2022

Orthogonal decompositions and twisted isometries

Let $n > 1$. Let $\{U_{ij}\}_{1 \leq i < j \leq n}$ be $\binom{n}{2}$ commuting unitaries on some Hilbert space $\mathcal{H}$, and suppose $U_{ji} := U_{ij}^*$, $1 \leq i < j \leq n$. An $n$-tuple of isometries $V = (V_1, \ldots ,V_n)$ on $\mathcal{H}$ is called $\mathcal{U}_n$-twisted isometry with respect to $\{U_{ij}\}_{i<j}$ (or simply $\mathcal{U}_n$-twisted isometry if $\{U_{ij}\}_{i<j}$ is clear from the context) if $V_i$'s are in the commutator $\{U_{st}: s \neq t\}'$, and $V_i^*V_j=U_{ij}^*V_jV_i^*$, $i \neq j$ We prove that each $\mathcal{U}_n$-twisted isometry admits a von Neumann-Wold type orthogonal decomposition, and prove that the universal $C^*$-algebra generated by $\mathcal{U}_n$-twisted isometry is nuclear. We exhibit concrete analytic models of $\mathcal{U}_n$-twisted isometries, and establish connections between unitary equivalence classes of the irreducible representations of the $C^*$-algebras generated by $\mathcal{U}_n$-twisted isometries and the unitary equivalence classes of the non-zero irreducible representations of twisted noncommutative tori. Our motivation of $\mathcal{U}_n$-twisted isometries stems from the classical rotation $C^*$-algebras, Heisenberg group $C^*$-algebras, and a recent work of de Jeu and Pinto.

math.OA↗

arXiv2022

Poisson boundary on full Fock space

This article is devoted to studying the non-commutative Poisson boundary associated with $\Big(B\big(\mathcal{F}(\mathcal{H})\big), P_ω\Big)$ where $\mathcal{H}$ is a separable Hilbert space (finite or infinite-dimensional), $\dim \mathcal{H} > 1$, with an orthonormal basis $\mathcal{E}$, $B\big(\mathcal{F}(\mathcal{H})\big)$ is the algebra of bounded linear operators on the full Fock space $\mathcal{F}(\mathcal{H})$ defined over $\mathcal{H}$, $ω= \{ω_e : e \in \mathcal{E} \}$ is a sequence of positive real numbers such that $\sum_e ω_e = 1$ and $P_ω$ is the Markov operator on $B\big(\mathcal{F}(\mathcal{H})\big)$ defined by \begin{align*} P_ω(x) = \sum_{e \in \mathcal{E}} ω_e l_e^* x l_e, \ x \in B\big(\mathcal{F}(\mathcal{H})\big), \end{align*} where, for $e \in \mathcal{E}$, $l_e$ denotes the left creation operator associated with $e$. The non-commutative Poisson boundary associated with $\Big(B\big(\mathcal{F}(\mathcal{H})\big), P_ω\Big)$ turns out to be an injective factor of type $III$ for any choice of $ω$. Moreover, if $\mathcal{H}$ is finite-dimensional, we completely classify the Poisson boundary in terms of its Connes $S$-invarinat and curiously they are type $III _{λ}$ factors with $λ$ belonging to a certain small class of algebraic numbers.

math.OA↗

arXiv2020

Orthogonality in Banach Spaces via projective tensor product

Let $X$ be a complex Banach space and $x,y\in X$. By definition, we say that $x$ is Birkhoff-James orthogonal to $y$ if $ \|x+λy\|_{X} \geq \|x\|_{X}$ for all $λ\in \mathbb{C}$. We prove that $x$ is Birkhoff-James orthogonal to $y$ if and only if there exists a semi-inner product $φ$ on $X$ such that $\|φ\| = 1$, $φ(x,x)=\|x\|^2$ and $φ(x,y)=0$. A similar result holds for $C^*$-algebras. A key point in our approach to orthogonality is the representations of bounded bilinear maps via projective tensor product spaces.

math.FA↗

arXiv2020

A caricature of dilation theory

We present a set-theoretic version of some basic dilation results of operator theory. The results we have considered are Wold decomposition, Halmos dilation, Sz. Nagy dilation, inter-twining lifting, commuting and non-commuting dilations, BCL theorem etc. We point out some natural generalizations and variations.

math.OA↗

arXiv2018

Tube representations and twisting of graded categories

We study deformation of tube algebra under twisting of graded monoidal categories. When a tensor category $\mathcal{C}$ is graded over a group $Γ$, a torus-valued 3-cocycle on $Γ$ can be used to deform the associator of $\mathcal{C}$. Based on a natural Fell bundle structure of the tube algebra over the action groupoid of the adjoint action of $Γ$, we show that the tube algebra of the twisted category is a 2-cocycle twisting of the original one.

math.QA↗

arXiv2016

Tube algebra of group-type subfactors

We describe the tube algebra and its representations in the cases of diagonal and Bisch-Haagerup subfactors possibly with a scalar 3-cocycle obstruction. We show that these categories are additively equivalent to the direct product over conjugacy classes of representation category of a centralizer subgroup (corresponding to the conjugacy class) twisted by a scalar 2-cocycle obtained from the 3-cocycle obstruction.

math.OA↗