K. Sumesh

Bio: K. Sumesh is an academic researcher from Indian Statistical Institute. The author has contributed to research in topics: Dilation (operator theory) & Physics. The author has an hindex of 4, co-authored 11 publications receiving 80 citations. Previous affiliations of K. Sumesh include Indian Institute of Technology Madras.

Stinespring's Theorem for maps on Hilbert C ∗ -modules

Abstract: Stinespring's representation theorem is a fundamental theorem in the theory of completely positive maps. It is a structure theorem for completely positive maps from a C*- algebra into the C*- algebra of bounded operators on a Hilbert space. This theorem provides a representation for completely positive maps, showing that they are simple modifications of *- homomorphisms. One may consider it as a natural generalization of the well-known Gelfand-Naimark-Segal theorem for states on C*-algebras. Resently, a theorem which looks like Stinesprings theorem was presented by Mohammad B. Asadi in for a class of unital maps on Hilbert C*-modules. This result can also be proved by removing a techical condition of Asadis theorem. The assumption of unitality on maps under consideration can also be remove. This result looks even more like Stinesprings theorem.

Bures Distance For Completely Positive Maps

Abstract: Bures had defined a metric on the set of normal states on a von Neumann algebra using GNS representations of states. This notion has been extended to completely positive maps between C*-algebras by Kretschmann, Schlingemann and Werner. We present a Hilbert C*-module version of this theory. We show that we do get a metric when the completely positive maps under consideration map to a von Neumann algebra. Further, we include several examples and counter examples. We also prove a rigidity theorem, showing that representation modules of completely positive maps which are close to the identity map contain a copy of the original algebra.

On a generalization of Ando’s dilation theorem

Abstract: We introduce the notion of Q-commuting operators which includes commuting operators. We prove a generalized version of the commutant lifting theorem and Ando’s dilation theorem in the context of Q-commuting operators.

On a tensor-analogue of the Schur product

Abstract: We consider the tensorial Schur product\(R \circ ^\otimes S = [r_{ij} \otimes s_{ij}]\) for \(R \in M_n(\mathcal {A}), S\in M_n(\mathcal {B}),\) with \(\mathcal {A}, \mathcal {B}~\text{ unital }~ C^*\)-algebras, verify that such a ‘tensorial Schur product’ of positive operators is again positive, and then use this fact to prove (an apparently marginally more general version of) the classical result of Choi that a linear map \(\phi :M_n \rightarrow M_d\) is completely positive if and only if \([\phi (E_{ij})] \in M_n(M_d)^+\), where of course \(\{E_{ij}:1 \le i,j \le n\}\) denotes the usual system of matrix units in \(M_n (:= M_n(\mathbb C))\). We also discuss some other corollaries of the main result.

Covariant version of the Stinespring type theorem for Hilbert C*-modules

Abstract: In this paper, we prove a covariant version of the Stinespring theorem for Hilbert C*-modules. Also, we show that there is a bijective correspondence between operator valued completely positive maps, (u′, u)-covariant with respect to the dynamical system (G, η, X) on Hilbert C*-modules and (u′, u)-covariant operator valued completely positive maps on the crossed product G ×ηX of X by η.

Covariant version of the stinespring type theorem for hilbert c -modules

Abstract: We prove a covariant version of the Stinespring theorem for Hilbert C � -modules. A completely positive linear map from a C � -algebra A to another C � -algebra B is a map ' : A ! B with the property that ('(aij)) n=1 is a positive element in the C � -algebra Mn(B) of all nn matrices with elements in B for all positive matrices (aij) n=1 in Mn(A) and for all positive integersn. The study of completely positive maps is motivated by the applications of the theory of completely positive maps to quantum information theory (operator valued completely positive maps on C � -algebras are used as mathematical model for quantum operations) and quantum probability.

Quantum Markov chains: A unification approach

Abstract: In this paper, we study a unified approach for quantum Markov chains (QMCs). A new quantum Markov property that generalizes the old one, is discussed. We introduce Markov states and chains on gener...

CP-Semigroups and Dilations, Subproduct Systems and Superproduct Systems: The Multi-Parameter Case and Beyond

Abstract: These notes are the output of a decade of research on how the results about dilations of one-parameter CP-semigroups with the help of product systems, can be put forward to d-parameter semigroups - and beyond. While preliminary work on the two- and d-parameter case is based on the approach via the Arveson-Stinespring correspondence of a CP-map by Muhly and Solel (and limited to von Neumann algebras), here we explore consequently the approach via Paschke's GNS-correspondence of a CP-map by Bhat and Skeide. (A comparison is postponed to Appendix A(iv).) The generalizations are multi-fold, the difficulties often enormous. In fact, our only true if-and-only-if theorem, is the following: A Markov semigroup over (the opposite of) an Ore monoid admits a full (strict or normal) dilation if and only if its GNS-subproduct system embeds into a product system. Already earlier, it has been observed that the GNS- (respectively, the Arveson-Stinespring) correspondences form a subproduct system, and that the main difficulty is to embed that into a product system. Here we add, that every dilation comes along with a superproduct system (a product system if the dilation is full). The latter may or may not contain the GNS-subproduct system; it does, if the dilation is strong - but not only. Apart from the many positive results pushing forward the theory to large extent, we provide plenty of counter examples for almost every desirable statement we could not prove. Still, a small number of open problems remains. The most prominent: Does there exist a CP-semigroup that admits a dilation, but no strong dilation? Another one: Does there exist a Markov semigroup that admits a (necessarily strong) dilation, but no full dilation?