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Sandipan De

Publications and source records attributed to Sandipan De.

8 recordsLinked to original sources
arXiv2024

Isometric pairs with compact and normal cross-commutator

We represent and classify pairs of commuting isometries $(V_1, V_2)$ acting on Hilbert spaces that satisfy the condition \[ [V_1^*, V_2] = \text{compact and normal}, \] where $[V_1^*, V_2] := V_1^* V_2 - V_2 V_1^*$ is the cross-commutator of $(V_1, V_2)$. The precise description of such pairs also gives a complete and concrete set of unitary invariants. The basic building blocks of representations of such pairs consist of four distinguished pairs of commuting isometries. One of them relies on some peculiar examples of invariant subspaces tracing back to Rudin's intricate constructions of analytic functions on the bidisc. Along the way, we present a rank formula for a general pair of commuting isometries that looks to be the first of its kind.

math.FA↗
arXiv2022

Pairs of projections and commuting isometries

It is known that the non-zero part of compact defect operators of Berger-Coburn-Lebow pairs (BCL pairs in short) of isometries are diagonal operators of the form \[ \begin{bmatrix} I_1 & & & \\ & D & & \\ & & - I_2 & \\ & & & - D \\ \end{bmatrix}, \] where $I_1$ and $I_2$ are the identity operators and $D$ is a positive contractive diagonal operator. We discuss the question of constructing an irreducible BCL pair from a diagonal operator of the above type. The answer to this question is sometimes in the affirmative and sometimes in the negative. This also answers a part of the question raised by He, Qin, and Yang. Our explicit constructions of BCL pairs yield concrete examples of pairs of commuting isometries.

math.FA↗
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

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↗
arXiv2019

From a Kac algebra subfactor to Drinfeld double

Given a finite-index and finite-depth subfactor, we define the notion of \textit{quantum double inclusion} - a certain unital inclusion of von Neumann algebras constructed from the given subfactor - which is closely related to that of Ocneanu's asymptotic inclusion. We show that the quantum double inclusion when applied to the Kac algebra subfactor $R^H \subset R$ produces Drinfeld double of $H$ where $H$ is a finite-dimensional Kac algebra acting outerly on the hyperfinite $II_1$ factor $R$ and $R^H$ denotes the fixed-point subalgebra. More precisely, quantum double inclusion of $R^H \subset R$ is isomorphic to $R \subset R \rtimes D(H)^{cop}$ for some outer action of $D(H)^{cop}$ on $R$ where $D(H)$ denotes the Drinfeld double of $H$.

math.OA↗
arXiv2019

Quantum double inclusions associated to a family of Kac algebra subfactors

In \cite{Sde2018} we defined the notion of \textit{quantum double inclusion} associated to a finite-index and finite-depth subfactor and studied the quantum double inclusion associated to the Kac algebra subfactor $R^H \subset R$ where $H$ is a finite-dimensional Kac algebra acting outerly on the hyperfinite $II_1$ factor $R$ and $R^H$ denotes the fixed-point subalgebra. In this article we analyse quantum double inclusions associated to the family of Kac algebra subfactors given by $\{ R^H \subset R \rtimes \underbrace{H \rtimes H^* \rtimes \cdots}_{\text{$m$ times}} : m \geq 1 \}$. For each $m > 2$, we construct a model $\mathcal{N}^m \subset \mathcal{M}$ for the quantum double inclusion of $\{ R^H \subset R \rtimes \underbrace{H \rtimes H^* \rtimes \cdots}_{\text{$m-2$ times}} \}$ with $\mathcal{N}^m = ((\cdots \rtimes H^{-2} \rtimes H^{-1}) \otimes (H^m \rtimes H^{m+1} \cdots))^{\prime \prime}, \mathcal{M} = (\cdots \rtimes H^{-1} \rtimes H^0 \rtimes H^1 \rtimes \cdots)^{\prime \prime}$ and where for any integer $i$, $H^i$ denotes $H$ or $H^*$ according as $i$ is odd or even. In this article, we give an explicit description of $P^{\mathcal{N}^m \subset \mathcal{M}}$ ($m > 2$), the subfactor planar algebra associated to $\mathcal{N}^m \subset \mathcal{M}$, which turns out to be a planar subalgebra of $^{*(m)}\!P(H^m)$ (the adjoint of the $m$-cabling of the planar algebra of $H^m$). We then show that for $m > 2$, depth of $\mathcal{N}^m \subset \mathcal{M}$ is always two. Observing that $\mathcal{N}^m \subset \mathcal{M}$ is reducible for all $m > 2$, we explicitly describe the weak Hopf $C^*$-algebra structure on $(\mathcal{N}^m)^{\prime} \cap \mathcal{M}_2$, thus obtaining a family of weak Hopf $C^*$-algebras starting with a single Kac algebra $H$.

math.OA↗
arXiv2016

Planar algebras, cabling and the Drinfeld double

We produce an explicit embedding of the planar algebra of the Drinfeld double of a finite-dimensional, semisimple and cosemisimple Hopf algebra $H$ into the two-cabling of the planar algebra of the dual Hopf algebra $H^*$ and characterise the image.

math.QA↗