IIT Madras is a public technical and research university located in Chennai, Tamil Nadu. Founded in 1959, it is recognised as an Institute of National Importance.

IIT Madras has been ranked as the top engineering institute in India for four years in a row by the National Institutional Ranking Framework of the MHRD

It currently offers undergraduate, postgraduate and research degrees across 16 disciplines in Engineering, Sciences, Humanities and Management. About 596 faculty belonging to science and engineering departments and centres of the Institute are engaged in teaching, research and industrial consultancy.

IIT Madras

In this paper, we provide a direct proof of the fact that the symmetric embedding of B(H) into any of its bimodules induced by Markov maps is compact and establishes that this property characterizes type I factors. This is an extension of a result of D. Buchholz, C. D'Antoni, and R. Longo [J. Funct. Anal. 88(2), 233-250 (1990)]. © 2021 Author(s).

World University root node

Department of Mathematics

Journal of Mathematical Physics

On symmetric embedding of type i factors

Bozejko and Speicher associated a finite von Neumann algebra MT to a self-adjoint operator T on a complex Hilbert space of the form H ⊗ H which satisfies the Yang-Baxter relation and |T| &lt; 1. We show that if dim (H) ≥ 2, then MT is a factor when T admits an eigenvector of somespecial form. © 2020 The Author(s). Published by Cambridge University Press on behalf of The Royal Society of Edinburgh.

Proceedings of the Royal Society of Edinburgh Section A: Mathematics

On the von Neumann algebras associated to Yang-Baxter operators

Operators and Matrices

On noncommutative joinings III

Journal of Mathematical Analysis and Applications

A note on coarse bimodules

The canonical unitary representation of a locally compact separable group arising from an ergodic action of the group on a von Neumann algebra with separable predual preserving a faithful normal semifinite (infinite) weight is weak mixing. On the contrary, there exists a non-ergodic automorphism of a von Neumann algebra preserving a faithful normal semifinite trace such that the spectral measure and the spectral multiplicity of the induced action are respectively the Haar measure (on the unit circle) and. Despite not even being ergodic, this automorphism has the same spectral data as that of a Bernoulli shift. © Cambridge University Press, 2017.

Ergodic Theory and Dynamical Systems

On dynamical systems preserving weights

This paper initiates the study of non-commutative dynamical systems of the form (G,Γ), where a discrete group Γ acts on a compact quantum group (CQG) G by quantum automorphisms. We obtain combinatorial conditions for such dynamical systems to be ergodic, mixing, compact, etc. and provide a wide variety of examples to illustrate these conditions. We generalize a well-known theorem of Halmos to demonstrate 'reversal of arrows' in the ergodic hierarchy relevant to the context and make a study of spectral measures for actions of (non-commutative) groups. We investigate the structure of such dynamical systems and under certain restrictions exhibit the existence and uniqueness of the maximal ergodic invariant normal subgroup of such systems. As an application, we study the size of normalizing algebras of masas arising from groups in von Neumann algebraic CQGs and show that the normalizing algebra of such masas are the von Neumann algebras generated by co-commutative CQGs. © 2017 London Mathematical Society.

Fulltext

Proceedings of the London Mathematical Society

Automorphisms of compact quantum groups:

This article 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∗-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. © The Author(s) 2017.

International Mathematics Research Notices

On noncommutative joinings

To any strongly continuous orthogonal representation of R on a real Hilbert space HR, Hiai constructed q-deformed Araki–Woods von Neumann algebras for −1&lt;q&lt;1, which are W⁎-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(HR)≥2 and q∈(−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. © 2017

Journal of Functional Analysis

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

This paper studies weakly mixing (singular) and mixing masas in type II1 factors from a bimodule point of view. Several necessary and sufficient conditions characterizing the normalizing algebra of a masa are presented. We 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,∞}. © 2016 Elsevier Inc.

Mixing and weakly mixing abelian subalgebras of type II1 factors

Analyzing endomorphisms and shift E0-semigroups of free products of von Neumann algebras, we construct uncountably many pairwise cocycle inequivalent E0-semigroups on a large class of free Araki-Woods factors. These E0-semigroups are equimodular but not modularly extendable. We also produce conjugate but not cocycle equivalent E0-semigroups on non-type I factors. © 2017 World Scientific Publishing Company.

International Journal of Mathematics

E0-semigroups of free Araki-Woods factors

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 describe the representation theory and the fusion rules of the associated bicrossed product 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 bG, and review coamenability of 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. © 2017 European Mathematical Society.

Journal of Noncommutative Geometry

On compact bicrossed products

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

KMS quantum symmetric states

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. © 2016.

The modular symmetry of Markov maps

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. © Instytut Matematyczny PAN, 2014.

Studia Mathematica

The simplex of tracial quantum symmetric states

In this paper we study relations between the left-right-measure and properties of singular masas. Part of the analysis is mainly concerned with masas for which the left-right-measure is the class of product measure. We provide examples of Tauer masas in the hyperfinite II1 factor whose left-right-measure is the class of Lebesgue measure. We show that for each subset S ⊆N, there exist uncountably many pairwise non conjugate singular masas in the free group factors with Pukanszky invariant SU ∞. © 2013 University of Houston.

Houston Journal of Mathematics

Singular masas and measure-multiplicity invariant

Weak asymptotic homomorphism property for masas in semifinite factors

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. © Glasgow Mathematical Journal Trust 2012.

Glasgow Mathematical Journal

Measure-multiplicity of the laplacian masa

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 ⊂ M relative to A. We show that weak mixing of B ⊂ M relative to a subalgebra A ⊂ B is equivalent to the following property: if x ∈ M and there exist a finite number of elements x1,..., xn ∈ M such that A x ⊂ ∑ni=1xiB, then x ∈ B. We conclude the paper with an assortment of further examples of mixing subalgebras arising from the amalgamated free product and crossed product constructions.

New York Journal of Mathematics

Mixing subalgebras of finite von Neumann algebras

Given an inclusion N⊂M of II1 factors with trivial relative commutant, this paper lists all operators x,y∈M such that the left N-module generated by x is equal to or contained in the right N-module generated by y.

Banach J. Math. Anal

Comparison of one-sided modules

The measure-multiplicity invariant of masas in II1 factors was introduced by Dykema, Smith and Sinclair 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 measure class. We present a proof of Chifan's normalizer theorem and a measure theoretic proof of the equivalence of weak asymptotic homomorphism property and singularity. © 2010. Published by Oxford University Press. All rights reserved.