A completely entangled subspace of maximal dimension

A completely entangled subspace of a tensor product of Hilbert spaces is a subspace with no non-trivial product vector. K. R. Parthasarathy determined the maximum dimension possible for such a subspace. Here we present a simple explicit example of one such space. We determine the set of product vectors in its orthogonal complement and see that it spans whole of the orthogonal complement. This way we are able to determine the minimum dimension possible for an unextendible product basis (UPB) consisting of product vectors which are linearly independent but not necessarily mutually orthogonal.Comment: 8 page

Nilpotent Completely Positive Maps

We study the structure of nilpotent completely positive maps in terms of Choi-Kraus coefficients. We prove several inequalities, including certain majorization type inequalities for dimensions of kernels of powers of nilpotent completely positive maps.Comment: 10 page

Pure Semigroups of Isometries on Hilbert C*-Modules

We show that pure strongly continuous semigroups of adjointable isometries on a Hilbert C*-module are standard right shifts. By counter examples, we illustrate that the analogy of this result with the classical result on Hilbert spaces by Sz.-Nagy, cannot be improved further to understand arbitrary isometry semigroups of isometries in the classical way.Comment: 18 pages; correction of an awful lot of typos; avoiding in some places a conflict with the known terminology 'reducing subspace

Domination for quantum Markov semigroups (Research on structure of operators by order and related topics)

A well-known theorem of W. Arveson states that a completely positive (CP) map dominated (difference is CP) by a given CP map is described through a positive contraction in the commutant of the homomorphism of Stinespring representation. This contraction plays the role of Radon-Nikodym derivative of measure theory. We look at implications of this to one parameter semigroups of unital completely positive maps, known as quantum Markov semigroups. We focus our attention to quantum Markov semigroups of B(H), the algebra of all bounded operators on a Hilbert space. These semigroups can be dilated to semigroups of unital endomorphisms (E₀-semigroups) of B(K) for some other Hilbert space K. Here we see that dominated CP semigroups dilate to dominated semigroups of the dilation. Moreover at the level of Eo-semigroups the dominated CP semigroups are described through positive contractive local cocycles, which are really families of contractions of 'Radon-Nikodym derivatives' mentioned above. We analyze the structure for quantum semigroups with bounded generators

Standard noncommuting and commuting dilations of commuting tuples

We introduce a notion called `maximal commuting piece' for tuples of Hilbert space operators. Given a commuting tuple of operators forming a row contraction there are two commonly used dilations in multivariable operator theory. Firstly there is the minimal isometric dilation consisting of isometries with orthogonal ranges and hence it is a noncommuting tuple. There is also a commuting dilation related with a standard commuting tuple on Boson Fock space. We show that this commuting dilation is the maximal commuting piece of the minimal isometric dilation. We use this result to classify all representations of Cuntz algebra O_n coming from dilations of commuting tuples.Comment: 18 pages, Latex, 1 commuting diagra

A model theory for q-commuting contractive tuples

A contractive tuple is a tuple (T1, . . . , Td) of operators on a common Hilbert space such that (0.1) T1T∗1 + + TdT∗d ≤ 1. It is said to be q-commuting if TjTi = qijTiTj for all 1 ≤ i < j ≤ d, where qij , 1 ≤ i < j ≤ d are complex numbers. These are higher-dimensional and non-commutative generalizations of a contraction. A particular example of this is the q-commuting shift. In this note, we investigate model theory for q-commuting contractive tuples using representations of the q-commuting shift