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Ved Prakash Gupta

Publications and source records attributed to Ved Prakash Gupta.

At least 19 recordsLinked to original sources
arXiv2026

Regular diagonal subfactors

We show that a diagonal subfactor arising from a finite family of automorphisms of a $II_1$-factor $Q$ is regular precisely when the classes of the defining automorphisms occur with same cardinality and form a subgroup of $\mathrm{Out}(Q)$, a subgroup that happens to be isomorphic to the generalized Weyl group of the subfactor. Moreover, it turns out that the cleanest picture of regularity in diagonal subfactors is graph-theoretic, namely, a diagonal subfactor is regular precisely when its principal graph is a complete, regular, balanced bipartite multigraph, with the generalized Weyl group fixing its size and the common multiplicity of the defining automorphisms determining its regular edge multiplicity. Prior to the characterization of regularity, revisiting Bisch and Popa's observations on the standard invariant and depth of diagonal subfactors, we give an exact criterion for a diagonal subfactor to have any prescribed depth, in terms of a stabilizing sequence of subsets of $\mathrm{Out}(Q)$ consisting of non-reduced alternating words in the classes of the defining automorphisms, which proves useful in the characterization of regularity.

math.OA↗
arXiv2026

Angles Between Intermediate Operator Subalgebras

Motivated by [2] and [5], the notions of interior and exterior angles between a pair of compatible intermediate W*-subalgebras of an inclusion of W*-algebras with a normal conditional expectation with finite probabilistic index are introduced. This is then employed effectively to define the interior angle between a pair of compatible intermediate C*-subalgebras of an inclusion of non-unital C*-algebras with a conditional expectation with finite Watatani index. It is also shown that the interior angle is stable under the minimal tensor product of unital C*-algebras.

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arXiv2025

Relations amongst the distances between $C^{*}$-subalgebras and some canonically associated operator algebras

We prove that the Christensen distance (resp., the Kadison-Kastler distance) between two $C^*$-subalgebras $\mathcal{A}$ and $\mathcal{B}$ of a $C^*$-algebra $\mathcal{C}$ is equal to that between their enveloping von Neumann algebras $\mathcal{A}^{**}$ and $\mathcal{B}^{**}$ (resp., the tensor product algebras $\mathcal{A} \otimes^{\min} \mathcal{D}$ and $\mathcal{B} \otimes^{\min} \mathcal{D}$, for any unital commutative $C^*$-algebra $\mathcal{D}$).

math.OA↗
arXiv2024

Regular inclusions of simple unital $C^*$-algebras

We prove that an inclusion $\mathcal{B} \subset \mathcal{A}$ of simple unital $C^*$-algebras with a finite-index conditional expectation is regular if and only if there exists a finite group $G$ that admits a cocycle action $(α,σ)$ on the intermediate $C^*$-subalgebra $\mathcal{C}$ generated by $\mathcal{B}$ and its centralizer $\mathcal{C}_\mathcal{A}(\mathcal{B})$ such that $\mathcal{B}$ is outerly $α$-invariant and $(\mathcal{B} \subset \mathcal{A}) \cong ( \mathcal{B} \subset \mathcal{C}\rtimes^r_{α, σ} G)$. Prior to this characterization, we prove the existence of two-sided and unitary quasi-bases for the minimal conditional expectation of any such inclusion, and also show that such an inclusion has integer Watatani index and depth at most $2$.

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arXiv2024

On various notions of distance between subalgebras of operator algebras

Given any irreducible inclusion $\mB \subset \mA$ of unital $C^*$-algebras with a finite-index conditional expectation $E: \mA \to \mB$, we show that the set of $E$-compatible intermediate $C^*$-subalgebras is finite, thereby generalizing a finiteness result of Ino and Watatani (from \cite{IW}). A finiteness result for a certain collection of intermediate $C^*$-subalgebras of a non-irreducible inclusion of simple unital $C^*$-algebras is also obtained, which provides a $C^*$-version of a finiteness result of Khoshkam and Mashood (from \cite{KM}). Apart from these finiteness results, comparisons between various notions of distance between subalgebras of operator algebras by Kadison-Kastler, Christensen and Mashood-Taylor are made. Further, these comparisons are used satisfactorily to provide some concrete calculations of distance between operator algebras associated to two distinct subgroups of a given discrete group.

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arXiv2022

On possible values of the interior angle between intermediate subalgebras

We show that all values in the interval $[0,\fracπ{2}]$ can be attained as the interior angle between intermediate subalgebras (as introduced in [3]) of a certain inclusion of simple unital C*-algebras. We also calculate the interior angle between intermediate crossed product subalgebras of any inclusion of crossed product algebras corresponding to any action of a countable discrete group and its subgroups on a unital C*-algebra.

math.OA↗
arXiv2022

On strong Arens irregularity of projective tensor product of Hilbert-Schmidt space

It was shown in [16] that the Banach algebra $A:=S_2(\ell^2)\otimes^γ S_2(\ell^2)$ is not Arens regular, where $S_2(\ell^2)$ denotes the Banach algebra of the Hilbert-Schmidt operators on $\ell^2$. In this article, employing the notion of limits along ultrafilters, we prove that the irregularity of $S_2(\ell^2)\otimes^γ S_2(\ell^2)$ is not strong. Along the way, we provide a class of functionals in $A^{**}$ which lie in the topological center but are not in $A$; and, as a consequence, we deduce that $A^{**}$ is not an annihilator Banach algebra with respect to any of the two Arens products.

math.FA↗
arXiv2022

A note on irreducible quadrilaterals of $II_1$ factors

Given any finite index quadrilateral $(N, P, Q, M)$ of $II_1$-factors, the notions of interior and exterior angles between $P$ and $Q$ were introduced in \cite{BDLR2017}. We determine the possible values of these angles when the quadrilateral is irreducible and the subfactors $N \subset P$ and $N \subset Q$ are both regular in terms of the cardinalities of the Weyl groups of the intermediate subfactors. For a more general quadruple, an attempt is made to determine the values of angles by deriving expressions for the angles in terms of the common norm of two naturally arising auxiliary operators and the indices of the intermediate subfactors of the quadruple. Finally, certain bounds on angles between $P$ and $Q$ are obtained, which enforce some restrictions on the index of $N \subset Q$ in terms of that of $N \subset P$.

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arXiv2021

A few remarks on Pimsner-Popa bases and regular subfactors of depth 2

We prove that a finite index regular inclusion of $II_1$-factors with commutative first relative commutant is always a crossed product subfactor with respect to a minimal action of a biconnected weak Kac algebra. Prior to this, we prove that every finite index inclusion of $II_1$-factors which is of depth $2$ and has simple first relative commutant (respectively, is regular and has commutative or simple first relative commutant) admits a two-sided Pimsner-Popa basis (respectively, a unitary orthonormal basis)

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arXiv2020

Lattice of intermediate subalgebras

Analogous to subfactor theory, employing Watatani's notions of index and $C^*$-basic construction of certain inclusions of $C^*$-algebras, (a) we develop a Fourier theory (consisting of Fourier transforms, rotation maps and shift operators) on the relative commutants of any inclusion of simple unital $C^*$-algebras with finite Watatani index, and (b) we introduce the notions of interior and exterior angles between intermediate $C^*$-subalgebras of any inclusion of unital $C^*$-algebras admitting a finite index conditional expectation. Then, on the lines of [2], we apply these concepts to obtain a bound for the cardinality of the lattice of intermediate $C^*$-subalgebras of any irreducible inclusion as in (a), and improve Longo's bound for the cardinality of intermediate subfactors of an inclusion of type $III$ factors with finite index. Moreover, we also show that for a fairly large class of inclusions of finite von Neumann algebras, the lattice of intermediate von Neumann subalgebras is always finite.

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arXiv2018

On closed Lie ideals of certain tensor products of C*-algebras II

We identify all closed Lie ideals of $A \otimes^α B$ and $B(H) \otimes^α B(H)$, where $\otimes^α$ is either the Haagerup tensor product, the Banach space projective tensor product or the operator space projective tensor product, $A$ is any simple C*-algebra, $B$ is any C*-algebra with one of them admitting no tracial states, and $H$ is an infinite dimensional separable Hilbert space. Further, generalizing a result of Marcoux, we also identify all closed Lie ideals of $A\otimes^{\min} B$, where $A$ is a simple C*-algebra with at most one tracial state and $B$ is any commutative C*-algebra.

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arXiv2018

On Banach space projective tensor product of $C^*$-algebras

We analyze certain algebraic structures of the Banach space projective tensor product of $C^*$-algebras which are comparable with their known counterparts or the Haagerup tensor product and the operator space projective tensor product of $C^*$-algebras. Highlights of this analysis include (a) injectivity of the Banach space projective tensor product when restricted to the tensor products of $C^*$-algebras, (b) detailed structure of closed ideals of $A \otimes_γ B$ in terms of those of $A$ and $B$, (c) identification of certain spaces of ideals of $A \otimes_γ B$ in terms of those of $A$ and $B$ from the perspective of hull-kernel topology, and (d) identification of the center of $A \otimes_γ B$ with $Z(A) \otimes_γ Z(B)$, where $A$ and $B$ are $C^*$-algebras.

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arXiv2017

On closed Lie ideals of certain tensor products of $C^*$-algebras

For a simple $C^*$-algebra $A$ and any other $C^*$-algebra $B$, it is proved that every closed ideal of $A \otimes^{\min} B$ is a product ideal if either $A$ is exact or $B$ is nuclear. Closed commutator of a closed ideal in a Banach algebra whose every closed ideal possesses a quasi-central approximate identity is described in terms of the commutator of the Banach algebra. If $α$ is either the Haagerup norm, the operator space projective norm or the $C^*$-minimal norm, then this allows us to identify all closed Lie ideals of $A \otimes^α B$, where $A$ and $B$ are simple, unital $C^*$-algebras with one of them admitting no tracial functionals, and to deduce that every non-central closed Lie ideal of $B(H) \otimes^α B(H)$ contains the product ideal $K(H) \otimes^α K(H)$. Closed Lie ideals of $A \otimes^{\min} C(X)$ are also determined, $A$ being any simple unital $C^*$-algebra with at most one tracial state and $X$ any compact Hausdorff space. And, it is shown that closed Lie ideals of $A \otimes^α K(H)$ are precisely the product ideals, where $A$ is any unital $C^*$-algebra and $α$ any completely positive uniform tensor norm.

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arXiv2016

Operator System Nuclearity via $C^*$-envelopes

We prove that an operator system is (min, ess)-nuclear if its C*-envelope is nuclear. This allows us to deduce that an operator system associated to a generating set of countable discrete group by Farenick et al. is (min, ess)-nuclear if and only if the group is amenable. We also make a detailed comparison between ess and other operator system tensor products and show that an operator system associated to a minimal generating set of a finitely generated discrete group (resp., a finite graph) is (min, max)-nuclear if and only if the group is of order less than or equal to 3 (resp., every component of the graph is complete).

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arXiv2015

The Functional Analysis of Quantum Information Theory

This book is a compilation of notes from a two-week international workshop on the "The Functional Analysis of Quantum Information Theory" that was held at the Institute of Mathematical Sciences during 26/12/2011-06/01/2012. The workshop was devoted to the mathematical framework of quantized functional analysis (QFA), and aimed at illustrating its applications to problems in quantum communication. The lectures were given by Gilles Pisier (Pierre and Marie Curie University and Texas A&M), K.R. Parthasarathy (ISI Delhi), Vern Paulsen (University of Houston), and Andreas Winter (Universitat Autonoma de Barcelona). Topics discussed include Operator Spaces and Completely bounded maps, Schmidt number and Schmidt rank of bipartite entangled states, Operator Systems and Completely Positive Maps, and, Operator Methods in Quantum Information.

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arXiv2014

Drinfeld center of planar algebra

We introduce fusion, contragradient and braiding of Hilbert affine representations of a subfactor planar algebra $P$ (not necessarily having finite depth). We prove that if $N \subset M$ is a subfactor realization of $P$, then the Drinfeld center of the $N$-$N$-bimodule category generated by $_N L^2 (M)_M$, is equivalent to the category of Hilbert affine representations of $P$ satisfying certain finiteness criterion. As a consequence, we prove Kevin Walker's conjecture for planar algebras.

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arXiv2012

Affine modules and the Drinfeld Center

Given a finite index subfactor, we show that the {\em affine morphisms at zero level} in the affine category over the planar algebra associated to the subfactor is isomorphic to the fusion algebra of the subfactor as a *-algebra. This identification paves the way to analyze the structure of affine $P$-modules with weight zero for any subfactor planar algebra $P$ (possibly having infinite depth). Further, for irreducible depth two subfactor planar algebras, we establish an additive equivalence between the category of affine $P$-modules and the center of the category of $N$-$N$-bimodules generated by $L^2(M)$; this partially verifies a conjecture of Jones and Walker.

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