General results of interpolation (eg. Nevanlinna-Pick) by elements in the noncommutative analytic Toeplitz algebra F ∞ (resp. noncommutative disc algebra An) with consequences to the interpolation by bounded operator-valued analytic functions in the unit ball of C n are obtained.Non-commutative Poisson transforms are used to provide new von Neumann type inequalities. Completely isometric representations of the quotient algebra F ∞ /J on Hilbert spaces, where J is any w * -closed, 2-sided ideal of F ∞ , are obtained and used to construct a w * -continuous, F ∞ /J-functional calculus associated to row contractions T = [T 1 , . . . , Tn] when f (T 1 , . . . , Tn) = 0 for any f ∈ J. Other properties of the dual algebra F ∞ /J are considered.Pitts [DP1] proved that F ∞ ⊂ B(F 2 ) has property A 1 (1), and Bercovici [B] proved that M k (F ∞ ) has property A 1 (1) for each k ≥ 1 (i.e., F ∞ has property A ℵ0 (1)).We refer to [Ar1], [P], and [Pi] for results on completely bounded maps and operator spaces.Let J be a WOT-closed, 2-sided ideal of F ∞ and defineRecall from [S] that a subspace N ⊂ H is semi-invariant under a semigroup of operators Σ ⊂ B(H) if for every T 1 , T 2 ∈ Σ, P N T 1 P N T 2 P N = P N T 1 T 2 P N . It is well

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.

Abstract. The framework of the paper is that of the full Fock space F 2 (H n ) and the Banach algebra F ∞ which can be viewed as non-commutative analogues of the Hardy spaces H 2 and H ∞ respectively.

Quantum effects are represented by operators on a Hilbert space satisfying 0⩽A⩽I, and sharp quantum effects are represented by projection operators. We say that an effect A is almost sharp if A=PQP for projections P and Q. We give simple characterizations of almost sharp effects. We also characterize effects that can be written as longer products of projections. For generality we first work in the formalism of von Neumann algebras. We then specialize to the full operator algebra B(H) and to finite dimensional Hilbert spaces.

We examine some structural properties of (injective and projective) tensor products of ^-spaces (projections, complemented subspaces, reflexivity, isomorphisms, etc.). We combine these results with combinatorial arguments to address the question of primarity for these spaces and their duals.Introduction.