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Kunal Mukherjee

Publications and source records attributed to Kunal Mukherjee.

51 records · Page 3Linked to original sources
arXiv2019

On Noncommutative Joinings

This paper extends the classical theory of joinings of measurable dynamical systems to the noncommutative setting from several interconnected points of view. Among these is a particularly fruitful identification of joinings with equivariant quantum channels between $W^{\ast}$-dynamical systems that provides noncommutative generalizations of many fundamental results of classical joining theory. We obtain fully general analogues of the main classical disjointness characterizations of ergodicity, primeness and mixing phenomena.

math.OA↗
arXiv2019

Noncommutative Joinings II

This paper is a continuation of the authors' previous work on noncommutative joinings, and contains a study of relative independence of W$^*$-dynamical systems. We prove that, given any separable locally compact group $G$, an ergodic W$^{*}$-dynamical $G$-system $\mathfrak{M}$ with compact subsystem $\mathfrak{N}$ is disjoint relative to $\mathfrak{N}$ from its maximal compact subsystem $\mathfrak{M}_{K}$ if and only if $\mathfrak{N}\cong\mathfrak{M}_{K}$. This generalizes recent work of Duvenhage, which established the result for $G$ abelian.

math.OA↗
arXiv2019

The Modular Symmetry of Markov Maps

A state-preserving automorphism of a von Neumann algebra induces a canonical unitary operator on the GNS Hilbert space of the state which fixes the vacuum. This unitary commutes with both the modular operator of the state and its modular conjugation. We prove an extension of this result for state-preserving unital completely positive maps.

math.OA↗
arXiv2018

Contribution of top barrier materials to high mobility in near-surface InAs quantum wells grown on GaSb(001)

Near-surface InAs two-dimensional electron gas (2DEG) systems have great potential for realizing networks of multiple Majorana zero modes towards a scalable topological quantum computer. Improving mobility in the near-surface 2DEGs is beneficial for stable topological superconducting states as well as for correlation of multiple Majorana zero modes in a complex network. Here, we investigate near-surface InAs 2DEGs (13 nm away from the surface) grown on GaSb(001) substrates, whose lattice constant is closely matched to InAs, by molecular beam epitaxy. The effect of 10-nm-thick top barrier to the mobility is studied by comparing Al$_{0.9}$Ga$_{0.1}$Sb and In$_{0.75}$Ga$_{0.25}$As as a top barrier on otherwise identical InAs quantum wells grown with identical bottom barrier and buffer layers. A 3-nm-thick capping layer on Al$_{0.9}$Ga$_{0.1}$Sb top barrier also affects the 2DEG electronic transport properties by modifying scattering from 2D remote ionized impurities at the surface. The highest transport mobility of 650,000 cm$^2$/Vs with an electron density of 3.81 $\times$ 10$^{11}$ cm$^{-2}$ was observed in an InAs 2DEG with an Al$_{0.9}$Ga$_{0.1}$Sb top barrier and an In$_{0.75}$Ga$_{0.25}$As capping layer. Analysis of Shubnikov-de Haas oscillations in the high mobility sample suggests that long-range scattering, such as remote ionized impurity scattering, is the dominant scattering mechanism in the InAs 2DEGs grown on GaSb(001) substrates. In comparison to InAs quantum wells grown on lattice-mismatched InP, the ones grown on GaSb show smoother surface morphology and higher quantum mobility. However, In$_{0.75}$Ga$_{0.25}$As top barrier in InAs quantum well grown on GaSb limits the transport mobility by charged dislocations formed in it, in addition to the major contribution to scattering from the alloy scattering.

cond-mat.mes-hall↗
arXiv2018

On compact bicrossed products

We make a comprehensive and self-contained study of compact bicrossed products arising from matched pairs of discrete groups and compact groups. We exhibit an automatic regularity property of such a matched pair and and produce an easy construction of the associated bicrossed product $\mathbb{G}$. We investigate the relative co-property $(T)$ and the relative co-Haagerup property of the pair comprising of the compact group and the bicrossed product, discuss property $(T)$ and Haagerup property of the discrete dual $\widehat{\mathbb{G}}$, and review co-amenability of $\mathbb{G}$ as well. We distinguish two such non-trivial compact bicrossed products with relative co-property $(T)$ and also provide an infinite family of pairwise non isomorphic non-trivial discrete quantum groups with property $(T)$, the existence of even one of the latter was unknown. Finally, we examine all the properties mentioned above for the crossed product quantum group given by an action by quantum automorphisms of a discrete group on a compact quantum group, and also establish the permanence of rapid decay and weak amenability and provide several explicit examples.

math.OA↗
arXiv2016

Mixing and weakly mixing abelian subalgebras of type II$_1$ factors

This paper studies weakly mixing (singular) and mixing masas in type $\rm{II}_{1}$ factors from a bimodule point of view. Several necessary and sufficient conditions to characterize the normalizing algebra of a masa are presented. We also study the structure of mixing inclusions, with special attention paid to masas of product class. A recent result of Jolissaint and Stalder concerning mixing masas arising out of inclusions of groups is revisited. One consequence of our structural results rules out the existence of certain Koopman-realizable measures, arising from semidirect products, which are absolutely continuous but not Lebesgue. We also show that there exist uncountably many pairwise non--conjugate mixing masas in the free group factors each with Pukánszky invariant $\{1,\infty\}$.

math.OA↗
arXiv2016

Generator masas in $q$-deformed Araki-Woods von Neumann algebras and factoriality

To any strongly continuous orthogonal representation of $\R$ on a real Hilbert space $\CH_\R$, Hiai constructed $q$-deformed Araki-Woods von Neumann algebras for $-1< q< 1$, which are $W^{\ast}$-algebras arising from non tracial representations of the $q$-commutation relations, the latter yielding an interpolation between the Bosonic and Fermionic statistics. We prove that if the orthogonal representation is not ergodic then these von Neumann algebras are factors whenever $dim(\CH_\R)\geq 2$ and $q\in (-1,1)$. In such case, the centralizer of the $q$-quasi free state has trivial relative commutant. In the process, we study `generator masas' in these factors and establish that they are strongly mixing. The analysis is inspired by a previous work of É. Ricard on Bo$\overset{.}{\text{z}}$ejko-Speicher's factors \cite{ER} and measure-multiplicity invariant of masas introduced by K. Dykema, A. Sinclair and R. Smith in \cite{DSS06}.

math.OA↗
arXiv2016

KMS quantum symmetric states

Let $A$ be a unital C$^*$-algebra and let $σ$ be a one-parameter automorphism group of $A$. We consider $\operatorname{QSS}_σ(A)$, the set of all quantum symmetric states on $*_1^\infty A$ that are also KMS states (for a fixed inverse temperature, for specificity taken to be $-1$) for the free product automorphism group $*_1^\inftyσ$. We characterize the elements of $\operatorname{QSS}_σ(A)$, we show that $\operatorname{QSS}_σ(A)$ is a Choquet simplex whenever it is nonempty and we characterize its extreme points.

math.OA↗
arXiv2014

The simplex of tracial quantum symmetric states

We show that the space of tracial quantum symmetric states of an arbitrary unital C*-algebra is a Choquet simplex and is a face of the tracial state space of the universal unital C*-algebra free product of A with itself infinitely many times. We also show that the extreme points of this simplex are dense, making it the Poulsen simplex when A is separable and nontrivial. In the course of the proof we characterize the centers of certain tracial amalgamated free product C*-algebras.

math.OA↗
arXiv2012

Measure-Multiplicity of the Laplacian masa

It is shown that for the Laplacian masa in the free group factors, the orthocomplement of the associated Jones' projection is an infinite direct sum of coarse bimodules.

math.OA↗
arXiv2011

Mixing subalgebras of finite von Neumann algebras

Jolissaint and Stalder introduced definitions of mixing and weak mixing for von Neumann subalgebras of finite von Neumann algebras. In this note, we study various algebraic and analytical properties of subalgebras with these mixing properties. We prove some basic results about mixing inclusions of von Neumann algebras and establish a connection between mixing properties and normalizers of von Neumann subalgebras. The special case of mixing subalgebras arising from inclusions of countable discrete groups finds applications to ergodic theory, in particular, a new generalization of a classical theorem of Halmos on the automorphisms of a compact abelian group. For a finite von Neumann algebra $M$ and von Neumann subalgebras $A$, $B$ of $M$, we introduce a notion of weak mixing of $B\subseteq M$ relative to $A$. We show that weak mixing of $B\subset M$ relative to $A$ is equivalent to the following property: if $x\in M$ and there exist a finite number of elements $x_1,...,x_n\in M$ such that $Ax\subset \sum_{i=1}^nx_iB$, then $x\in B$. We conclude the paper with an assortment of further examples of mixing subalgebras arising from the amalgamated free product and crossed product constructions.

math.OA↗
arXiv2011

Singular Masas and Measure-Multiplicity Invariant

In this paper we study relations between the \emph{left-right-measure} and properties of singular masas. Part of the analysis is mainly concerned with masas for which the \emph{left-right-measure} is the class of product measure. We provide examples of Tauer masas in the hyperfinite $\rm{II}_{1}$ factor whose \emph{left-right-measure} is the class of Lebesgue measure. We show that for each subset $S\subseteq \mathbb{N}$, there exist uncountably many pairwise non conjugate singular masas in the free group factors with \emph{Pukánszky invariant} $S\cup\{\infty\}$.

math.OA↗
arXiv2011

Hilbert von Neumann modules

We introduce a way of regarding Hilbert von Neumann modules as spaces of operators between Hilbert space, not unlike [Skei], but in an apparently much simpler manner and involving far less machinery. We verify that our definition is equivalent to that of [Skei], by verifying the `Riesz lemma' or what is called `self-duality' in [Skei]. An advantage with our approach is that we can totally side-step the need to go through $C^*$-modules and avoid the two stages of completion - first in norm, then in the strong operator topology - involved in the former approach. We establish the analogue of the Stinespring dilation theorem for Hilbert von Neumann bimodules, and we develop our version of `internal tensor products' which we refer to as Connes fusion for obvious reasons. In our discussion of examples, we examine the bimodules arising from automorphisms of von Neumann algebras, verify that fusion of bimodules corresponds to composition of automorphisms in this case, and that the isomorphism class of such a bimodule depends only on the inner conjugacy class of the automorphism. We also relate Jones' basic construction to the Stinespring dilation associated to the conditional expectation onto a finite-index inclusion (by invoking the uniqueness assertion regarding the latter).

math.QA↗
arXiv2008

Masas and Bimodule Decompositions of $\rm{II}_{1}$ Factors

The measure-multiplicity-invariant for masas in $\rm{II}_{1}$ factors was introduced in \cite{MR2261688} to distinguish masas that have the same Pukánszky invariant. In this paper we study the measure class in the measure-multiplicity-invariant. This is equivalent to studying the standard Hilbert space as an associated bimodule. We characterize the type of any masa depending on the left-right-measure using Baire category methods (selection principle of Jankov and von Neumann). We present a second proof of Chifan's result on normalisers and a measure theoretic proof of the equivalence of weak asymptotic homomorphism property (WAHP) and singularity that appeared in \cite{MR2417416}.

math.OA↗