Date of Submission


Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name



Theoretical Statistics and Mathematics Unit (TSMU-Bangalore)


Bhat, B. V. Rajarama (TSMU-Bangalore; ISI)

Abstract (Summary of the Work)

Completely positive (CP-) maps are special kinds of positivity preserving maps on C ∗ -algebras. W.F. Stinespring [Sti55] obtained a structure theorem for CP-maps showing that they are closely connected with ∗-homomorphisms. W. Arveson and other operator algebraists quickly realized the importance of these maps. Presently the role of the theory of CP-maps in our understanding of C ∗ -algebras and von Neumann algebras is well recognised. It has been argued by physicists that CPmaps are physically more meaningful than just positive maps due to their stability under ampliations. From quantum probabilistic point of view CP-maps are quantum analogues of stochastic or sub-stochastic transition probability maps. Therefore one begins with such maps in order to construct quantum Markov processes. Recently there has been lot of interest in quantum computation and quantum information theory and here trace preserving, unital CP-maps play the role of quantum channels. This justifies detailed study of CP-maps and related concepts.Often it is the structure theorems that makes a theory worth studying. GNStheorem and Stinespring’s theorem are the basic structure theorems for CP-maps. Our main tool to study CP-maps is the theory of Hilbert C ∗ -modules. They are objects similar to Hilbert spaces. Close connections between CP-maps and Hilbert C ∗ -modules are well-known ([Kas80, Mur97, Pas73]).Given CP-maps ϕ1 and ϕ2 between unital C ∗ -algebras A and B, by a common representation module for them we mean a Hilbert A-B-module E where they can be represented, that is, there exists xi ∈ E such that ϕi(·) = hxi ,(·)xii. We define β as the infimum of the norm differences x1 − x2 taken over all common representation modules E and representing vectors xi ∈ E, and call it Bures distance. We show the existence of a sort of universal module where we can take infimum to compute the Bures distance, and thereby prove that β is a metric when the CP-maps under consideration map to a von Neumann algebra or to an injective C ∗ -algebra. However, β is not a metric when the range algebra is a general C ∗ -algebra. The definition of Bures distance is abstract and does not give us indications as to how to compute it for concrete examples. We show that Bures distance can be computed using intertwiners between two (minimal) GNS-constructions of CP-maps. We also prove a rigidity theorem, showing that GNS-representation modules ([Pas73]) of CP-maps which are close to the identity map contain a copy of the original algebra.If ϕ : A → B is a linear map, then by a ϕ-map we mean a linear map T : E → F from a Hilbert A-module E into a Hilbert B-module F such that T(x1), T(x2)i = ϕ(x1, x2) for all xi ∈ E, that is, T preserves the inner product up to the linear map ϕ. We prove that if E is full and if ϕ is bounded linear, then ϕ will be automatically CP. Moreover,


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Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.


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