The notion of a quantum dynamical semigroup is defined using the concept of a completely positive map. An explicit form of a bounded generator of such a semigroup onB(ℋ) is derived. This is a quantum analogue of the Levy-Khinchin formula. As a result the general form of a large class of Markovian quantum-mechanical master equations is obtained.

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On the Generators of Quantum Dynamical Semigroups

Completely positive linear maps on complex matrices

This book represents the first synthesis of the considerable body of new research into positive definite matrices. These matrices play the same role in noncommutative analysis as positive real numbers do in classical analysis. They have theoretical and computational uses across a broad spectrum of disciplines, including calculus, electrical engineering, statistics, physics, numerical analysis, quantum information theory, and geometry. Through detailed explanations and an authoritative and inspiring writing style, Rajendra Bhatia carefully develops general techniques that have wide applications in the study of such matrices. Bhatia introduces several key topics in functional analysis, operator theory, harmonic analysis, and differential geometry--all built around the central theme of positive definite matrices. He discusses positive and completely positive linear maps, and presents major theorems with simple and direct proofs. He examines matrix means and their applications, and shows how to use positive definite functions to derive operator inequalities that he and others proved in recent years. He guides the reader through the differential geometry of the manifold of positive definite matrices, and explains recent work on the geometric mean of several matrices. Positive Definite Matrices is an informative and useful reference book for mathematicians and other researchers and practitioners. The numerous exercises and notes at the end of each chapter also make it the ideal textbook for graduate-level courses.

Positive Definite Matrices

We study a construction that yields a class of translation invariant states on quantum spin chains, characterized by the property that the correlations across any bond can be modeled on a finite-dimensional vector space. These states can be considered as generalized valence bond states, and they are dense in the set of all translation invariant states. We develop a complete theory of the ergodic decomposition of such states, including the decomposition into periodic “Neel ordered” states. The ergodic components have exponential decay of correlations. All states considered can be obtained as “local functions” of states of a special kind, so-called “purely generated states,” which are shown to be ground states for suitably chosen finite range VBS interactions. We show that all these generalized VBS models have a spectral gap. Our theory does not require symmetry of the state with respect to a local gauge group. In particular we illustrate our results with a one-parameter family of examples which are not isotropic except for one special case. This isotropic model coincides with the one-dimensional antiferromagnet, recently studied by Affleck, Kennedy, Lieb, and Tasaki.

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Finitely correlated states on quantum spin chains

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Subalgebras of C*-algebras

1 Fundamentals- 11 Operators and C*-algebras- 12 Two density theorems- 13 Ideals, quotients, and representations- 14 C*-algebras of compact operators- 15 CCR and GCR algebras- 16 States and the GNS construction- 17 The existence of representations- 18 Order and approximate units- 2 Multiplicity Theory- 21 From type I to multiplicity-free- 22 Commutative C*-algebras and normal operators- 23 An application: type I von Neumann algebras- 24 GCR algebras are type I- 3 Borel Structures- 31 Polish spaces- 32 Borel sets and analytic sets- 33 Borel spaces- 34 Cross sections- 4 From Commutative Algebras to GCR Algebras- 41 The spectrum of a C*-algebra- 42 Decomposable operator algebras- 43 Representations of GCR algebras

An invitation to C*-algebras

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On subalgebras of $C^*$-algebras

A d-contraction is a d-tuple (T1, . . . , Td) of mutually commuting operators acting on a common Hilbert space H such that ‖T1ξ1 + T2ξ2 + · · · + Tdξd‖ ≤ ‖ξ1‖ + ‖ξ2‖ + · · · + ‖ξd‖ for all ξ1, ξ2, . . . , ξd ∈ H. These are the higher dimensional counterparts of contractions. We show that many of the operator-theoretic aspects of function theory in the unit disk generalize to the unit ball Bd in complex d-space, including von Neumann’s inequality and the model theory of contractions. These results depend on properties of the d-shift, a distinguished d-contraction which acts on a new H2 space associated with Bd, and which is the higher dimensional counterpart of the unilateral shift. H 2 and the d-shift are highly unique. Indeed, by exploiting the noncommutative Choquet boundary of the d-shift relative to its generated C∗-algebra we find that there is more uniqueness in dimension d ≥ 2 than there is in dimension one.

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Subalgebras of C*-algebras III: Multivariable operator theory

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William Arveson

Papers

Subalgebras of C*-algebras

An invitation to C*-algebras

On subalgebras of $C^*$-algebras

Subalgebras of C*-algebras III: Multivariable operator theory

Subalgebras of c*-algebras ii