Papers by Mithun Mukherjee
We describe here the higher rank numerical range, as defined by Choi, Kribs and Życzkowski, of a ... more We describe here the higher rank numerical range, as defined by Choi, Kribs and Życzkowski, of a normal operator on an infinite dimensional Hilbert space in terms of its spectral measure. This generalizes a result of Avendaño for self-adjoint operators. An analogous description of the numerical range of a normal operator by Durszt is derived for the higher rank numerical range as an immediate consequence. It has several interesting applications. We show using Durszt’s example that there exists a normal contraction T for which the intersection of the higher rank numerical ranges of all unitary dilations of T contains the higher rank numerical range of T as a proper subset. Finally, we strengthen and generalize a result of Wu by providing a necessary and sufficient condition for the higher rank numerical range of a normal contraction being equal to the intersection of the higher rank numerical ranges of all possible unitary dilations of it.
D. Bures defined a metric $\beta $ on states of a $C^*$-algebra and this concept has been general... more D. Bures defined a metric $\beta $ on states of a $C^*$-algebra and this concept has been generalized to unital completely positive maps $\phi : \mathcal A \to \mathcal B$, where $\mathcal B$ is either an injective $C^*$-algebra or a von Neumann algebra. We introduce a new distance $\gamma $ for the same classes of unital completely positive maps. We use in our definition the distance between representations on the same Hilbert $C^*$-module in contrast to the Bures metric which uses one representation and distinct vectors. This metric can be expressed in terms of a class of completely positive maps on free products of $C^*$-algebras and in this setting $\gamma $ looks like Wasserstein metric on probability measures. Surprisingly, when the range algebra $\mathcal B$ is injective, $\gamma $ and $\beta $ are related by the following explicit formula: $\beta ^2= 2-\sqrt{4- \gamma ^2} .$ A deep result of Choi and Li on constrained dilation is the main tool in proving this formula.
Structure theorem of the generator of a norm continuous completely positive semigroup: an alternative proof using Bures distance
Positivity
Here we present an alternative proof using Bures distance that the generator L of a norm continuo... more Here we present an alternative proof using Bures distance that the generator L of a norm continuous completely positive semigroup acting on a $$C^*$$C∗-algebra $${\mathcal {B}}\subset \mathcal B(H)$$B⊂B(H) has the form $$ L(b) = \Psi (b) + k^*b+bk$$L(b)=Ψ(b)+k∗b+bk, $$b\in {\mathcal {B}}$$b∈B for some completely positive map $$\Psi :{\mathcal {B}}\rightarrow {\mathcal {B}}(H)$$Ψ:B→B(H) and $$k\in {\mathcal {B}}(H)$$k∈B(H).
Transactions of the American Mathematical Society
We introduce the notion of additive units, or 'addits', of a pointed Arveson system, and demonstr... more We introduce the notion of additive units, or 'addits', of a pointed Arveson system, and demonstrate their usefulness through several applications. By a pointed Arveson system we mean a spatial Arveson system with a fixed normalised reference unit. We show that the addits form a Hilbert space whose codimension-one subspace of 'roots' is isomorphic to the index space of the Arveson system, and that the addits generate the type I part of the Arveson system. Consequently the isomorphism class of the Hilbert space of addits is independent of the reference unit. The addits of a pointed inclusion system are shown to be in natural correspondence with the addits of the generated pointed product system. The theory of amalgamated products is developed using addits and roots, and an explicit formula for the amalgamation of pointed Arveson systems is given, providing a new proof of its independence of the particular reference units. (This independence justifies the terminology 'spatial product' of spatial Arveson systems). Finally a cluster construction for inclusion subsystems of an Arveson system is introduced and we demonstrate its correspondence with the action of the Cantor-Bendixson derivative in the context of the random closed set approach to product systems due to Tsirelson and Liebscher.
Annals of Functional Analysis, 2015
It is known that the spatial product of two product systems is intrinsic. Here we extend this res... more It is known that the spatial product of two product systems is intrinsic. Here we extend this result by analysing subsystems of the tensor product of product systems. A relation with cluster systems in the sense of [4] is established. In a special case, we show that the amalgamated product of product systems through strictly contractive units is independent of the choices of the units. The amalgamated product in this case is isomorphic to the tensor product of the spatial product of the two and the type I product system of index one.
Annals of Functional Analysis
It is known that the spatial product of two product systems is intrinsic. Here we extend this res... more It is known that the spatial product of two product systems is intrinsic. Here we extend this result by analysing subsystems of the tensor product of product systems. A relation with cluster systems in the sense of [4] is established. In a special case, we show that the amalgamated product of product systems through strictly contractive units is independent of the choices of the units. The amalgamated product in this case is isomorphic to the tensor product of the spatial product of the two and the type I product system of index one.
We introduce the notion of additive units and roots of a unit in a spatial product system. The se... more We introduce the notion of additive units and roots of a unit in a spatial product system. The set of all roots of any unit forms a Hilbert space and its dimension is the same as the index of the product system. We show that a unit and all of its roots generate the type I part of the product system. Using properties of roots, we also provide an alternative proof of the Powers' problem that the cocycle conjugacy class of Powers sum is independent of the choice of intertwining isometries. In the last section, we introduce the notion of cluster of a product subsystem and establish its connection with random sets in the sense of Tsirelson ([27]) and Liebscher ([11]).
Integrators of matrices
Linear Algebra and its Applications, 2007
emis.ams.org
The notion of amalgamation of product systems has been introduced in [7] which generalizes the co... more The notion of amalgamation of product systems has been introduced in [7] which generalizes the concept of Skeide product, introduced by Skeide, of two product systems via a pair of normalized units. In this paper we show that amalgamation leads to a setup where a product system is generated by two subsystems and conversely whenever a product system is generated by two subsystems, it could be realized as an amalgamated product. We parameterize all contractive morphism from a Type I product system to another Type I product system and compute index of amalgamated product through contractive morphisms.
Arxiv preprint arXiv:0907.0095, 2009
Here we generalize the concept of spatial tensor product, introduced by Skeide, of two product sy... more Here we generalize the concept of spatial tensor product, introduced by Skeide, of two product systems via a pair of normalized units. This new notion is called amalgamated tensor product of product systems, and now the amalgamation can be done using a contractive morphism. Index of amalgamation product (when done through units) adds up for normalized units but for non-normalized units, the index is one more than the sum. We define inclusion systems and use it as a tool for index computations. It is expected that this notion will have other uses.
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Papers by Mithun Mukherjee