Alvaro Arias

Noncommutative interpolation and Poisson transforms

General results of interpolation (e.g., Nevanlinna-Pick) by elements in the noncommutative analytic Toeplitz algebraF∞ (resp., noncommutative disc algebraAn) with consequences to the interpolation by bounded operator-valued analytic functions in the unit ball of ℂn are obtained. Noncommutative Poisson transforms are used to provide new von Neumann type inequalities. Completely isometric representations of the quotient algebraF∞/J on Hilbert spaces whereJ is anyw*-closed, 2-sided ideal ofF∞, are obtained and used to construct aw*-continuous,F∞/J-functional calculus associated to row contractionsT=[T1,…,Tn] whenf(T1, …, Tn)=0 for anyf∈J. Other properties of the dual algebraF∞/J are considered.

Fixed points of quantum operations

Quantum operations frequently occur in quantum measurement theory, quantum probability, quantum computation, and quantum information theory. If an operator A is invariant under a quantum operation φ, we call A a φ-fixed point. Physically, the φ-fixed points are the operators that are not disturbed by the action of φ. Our main purpose is to answer the following question. If A is a φ-fixed point, is A compatible with the operation elements of φ? We shall show in general that the answer is no and we shall give some sufficient conditions under which the answer is yes. Our results will follow from some general theorems concerning completely positive maps and injectivity of operator systems and von Neumann algebras.

FACTORIZATION AND REFLEXIVITY ON FOCK SPACES

AbstractThe framework of the paper is that of the full Fock space

Operator Hilbert spaces without the operator approximation property

\mathcal{F}^2 (\mathcal{H}_n )

A REMARK ON POSITIVE ISOTROPIC RANDOM VECTORS

and the Banach algebraF∞ which can be viewed as non-commutative analogues of the Hardy spacesH2 andH∞ respectively.An inner-outer factorization for any element in

M-complete approximate identities in operator spaces

\mathcal{F}^2 (\mathcal{H}_n )