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Sruthymurali

Publications and source records attributed to Sruthymurali.

8 recordsLinked to original sources
arXiv2025

Noncommutative BKW-Operators

Inspired by the classical Bohman-Korovkin-Wulbert (BKW) operators, we initiate a study of noncommutative BKW-operators. Let $A$ be a unital $C^*$-algebra, and $S$ be a set of generators of $A$. A unital completely positive (UCP)-map $\phi: A\rightarrow B(H)$ is said to be a \textit{noncommutative BKW-operator} for $S$ with respect to norm or weak operator topology (WOT) or strong operator topology (SOT) if for any sequence of UCP-maps $\phi_n:A\rightarrow B(H)$, $n=1,2,...,$ $\lim_{n\rightarrow \infty}\phi_n(s)=\phi(s),\forall ~s\in S$ in norm (or WOT or SOT) $\Rightarrow \lim_{n\rightarrow \infty}\phi_n(a)=\phi(a), \forall ~a\in A$ in norm (or WOT or SOT, respectively). We identify a connection between noncommutative BKW-operators and the unique CP-extension of UCP-maps. We have discussed several examples and explored different notions of noncommutative BKW-operators and their interconnections. Additionally, we introduce the concept of hyperrigidity with respect to a UCP-map and characterize it along the lines of Arveson. Although independent yet related to noncommutative BKW-operators, we provide a noncommutative version of operator version of the Korovkin theorem recently proposed by D. Popa.

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arXiv2025

Understanding Quantum Instruments Through the Analysis of $C^*$-Convexity and Their Marginals

Quantum instruments are mathematical devices introduced to describe the conditional state change during a quantum process. They are completely positive map valued measures on measurable spaces. We may also view them as non-commutative analogues of joint probability measures. We analyze the $C^*$-convexity structure of spaces of quantum instruments. A complete description of the $C^*$-extreme instruments in finite dimensions has been established. Further, the implications of $C^*$-extremity between quantum instruments and their marginals has been explored.

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arXiv2025

Higher reflections and entropy of canonical shifts for inclusions of $C^*$-algebras with finite Watatani index

Given a unital inclusion of simple $C^*$-algebras equipped with a conditional expectation of index-finite type, we study Fourier transforms and rotation operators and introduce the reflection operators on the relative commutants. We prove that the reflections are unital, involutive, $*$-preserving anti-homomorphisms that preserve certain Markov-type traces. As an application, we prove Fourier theoretic inequalities on the higher relative commutants and refine the existing constant in Young's inequality as presented in the current literature. By employing the reflection operators, we define a canonical shift on the von Neumann algebra generated by the relative commutants. We establish a connection between the Connes-St{\o}rmer entropy of the canonical shift and the minimal Watatani index.

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arXiv2022

Fourier theoretic inequalities for inclusion of simple C*-algebras

This paper originates from a naive attempt to establish various non-commutative Fourier theoretic inequalities for an inclusion of simple C*-algebras equipped with a conditional expectation of index-finite type. In this setting, we discuss the Hausdorff-Young inequality and Young's inequality. As a consequence, we prove the Hirschman-Beckner uncertainty principle and Donoho-Stark uncertainty principle. Our results generalize some of the results of Jiang, Liu and Wu [Noncommutative uncertainty principle, J. Funct. Anal., 270(1): 264--311, 2016].

math.OA↗
arXiv2022

KMS states on $C_c^{*}(\mathbb{N}^2)$

Let $C_c^{*}(\mathbb{N}^{2})$ be the universal $C^{*}$-algebra generated by a semigroup of isometries $\{v_{(m,n)}: m,n \in \mathbb{N}\}$ whose range projections commute. We analyse the structure of KMS states on $C_{c}^{*}(\mathbb{N}^2)$ for the time evolution determined by a homomorphism $c:\mathbb{Z}^{2} \to \mathbb{R}$. In contrast to the reduced version $C_{red}^{*}(\mathbb{N}^{2})$, we show that the set of KMS states on $C_{c}^{*}(\mathbb{N}^{2})$ has a rich structure. In particular, we exhibit uncountably many extremal KMS states of type I, II and III.

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arXiv2021

Intermediate planar algebra revisited, II

We describe the subfactor planar algebra of an intermediate subfactor $N\subset Q \subset M$ of an extremal subfactor $N\subset M$ of finite Jones index which is not necessarily irreducible.

math.OA↗
arXiv2019

Planar algebras, quantum information theory and subfactors

We define generalised notions of biunitary elements in planar algebras and show that objects arising in quantum information theory such as Hadamard matrices, quantum latin squares and unitary error bases are all given by biunitary elements in the spin planar algebra. We show that there are natural subfactor planar algebras associated with biunitary elements.

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