B. V. Rajarama Bhat

TENSOR PRODUCT SYSTEMS OF HILBERT MODULES AND DILATIONS OF COMPLETELY POSITIVE SEMIGROUPS

In this paper we study the problem of dilating unital completely positive (CP) semigroups (quantum dynamical semigroups) to weak Markov flows and then to semigroups of endomorphisms (E0-semigroups) using the language of Hilbert modules. This is a very effective, representation free approach to dilation. In this way we are able to identify the right algebra (maximal in some sense) for endomorphisms to act. We are led inevitably to the notion of tensor product systems of Hilbert modules and units for them, generalizing Arvesons notions for Hilbert spaces. In the course of our investigations we are not only able to give new natural and transparent proofs of well-known facts for semigroups on , but also extend the results immediately to much more general setups. For instance, Arveson classifies E0-semigroups on up to cocycle conjugacy by product systems of Hilbert spaces.5 We find that conservative CP-semigroups on arbitrary unital C*-algebras are classified up to cocycle conjugacy by product systems of Hilbert modules. Looking at other generalizations, it turns out that the role played by E0-semigroups on in dilation theory for CP-semigroups on is now played by E0-semigroups on , the full algebra of adjointable operators on a Hilbert module E. We have CP-semigroup versions of many results proved by Paschke27 for CP maps.

On greatest common divisor matrices and their applications

Abstract Let S = { x 1 , x 2 ,… x n } be a set of distinct positive integers. The n × n matrix [ S ] = (( s ij )), where s ij = ( x i , x j ), the greatest common divisor of x i and x j , is called the greatest common divisor (GCD) matrix on S . We study the structure of a GCD matrix and obtain interesting relations between its determinant. Eulers totient function, and Moebius function. We also determine some arithmetic progressions related to GCD matrices. Then we generalize the results to general partially ordered sets and show a variety of applications.

Kolmogorov’s existence theorem for Markov processes inC* algebras

Given a family of transition probability functions between measure spaces and an initial distribution Kolmogorov’s existence theorem associates a unique Markov process on the product space. Here a canonical non-commutative analogue of this result is established for families of completely positive maps betweenC* algebras satisfying the Chapman-Kolmogorov equations. This could be the starting point for a theory of quantum Markov processes.

Inclusion systems and amalgamated products of product systems

Here we generalize the concept of Skeide product, introduced by Skeide, of two product systems via a pair of normalized units. This new notion is called amalgamated product of product systems, and now the amalgamation can be done using contractive morphisms. 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.

A COMPLETELY ENTANGLED SUBSPACE OF MAXIMAL DIMENSION

Consider a tensor product of finite-dimensional Hilbert spaces with dimension , 1 ≤ i ≤ k. Then the maximum dimension possible for a subspace of with no non-zero product vector is known to be d1 d2…dk - (d1 + d2 + … + dk + k - 1. We obtain an explicit example of a subspace of this kind. We determine the set of product vectors in its orthogonal complement and show that it has the minimum dimension possible for an unextendible product basis of not necessarily orthogonal product vectors.

The Schur-Horn theorem for operators with finite spectrum

The carpenter problem in the context of II1 factors, formulated by Kadison asks: Let A ⊂ M be a masa in a type II1 factor and let E be the normal conditional expectation from M onto A. Then, is it true that for every positive contraction A in A, there is a projection P in M such that E(P ) = A? In this note, we show that this is true if A has finite spectrum. We will then use this result to prove an exact Schur-Horn theorem for (positive)operators with finite spectrum and an approximate Schur-Horn theorem for general (positive)operators.

Minimal isometric dilations of operator cocycles

We obtain existence, uniqueness results for minimal isometric dilations of contractive cocycles of semigroups of unital *-endomorphisms ofB(H. This generalizes the result of Sz. Nagy on minimal isometric dilations of semigroups of contractive operators on a Hilbert space. In a similar fashion we explore results analogus to Sarasons characterization that subspaces to which compressions of semigroups are again semigroups are semi-invariant subspaces, in the context of cocycles and quantum dynamical semigroups.

Bures Distance For Completely Positive Maps

D. 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

Maximal Commutative Subalgebras Invariant for CP-Maps: (Counter-)Examples ∗

C^*

Minimal Cuntz-Krieger dilations and representations of Cuntz-Krieger algebras

-algebras by D. Kretschmann, D. Schlingemann and R. F. Werner. We present a Hilbert