Vern I. Paulsen

Tensor products of operator spaces

Abstract In this paper we lay the foundations for a systematic study of tensor products of subspaces of C ∗ -algebras. To accomplish this, various notions of duality are introduced and employed. Elementary proofs of the complete injectivity of the Haagerup norm, and of the extension theorem for completely bounded maps, are given. Pisiers gamma norms are examined and found to be special cases of the Haagerup norm. We identify the greatest operator space cross norm and show that the spatial tensor norm is the least operator space cross norm in an appropriate sense. Indeed most of the elementary theory of Banach space tensor norms generalizes to the category of operator spaces.

Schur Products and Matrix Completions

We prove that a necessary and sufficient condition for a given partially positive matrix to have a positive completion is that a certain Schur product map defined on a certain subspace of matrices is a positive map. By analyzing the positive elements of this subspace, we obtain new proofs of the results of H. Dym and I. Gohberg and Grone, Johnson, Sa, and Wolkowitz (Linear Algebra Appl.58 (1984), 109–124). (Linear Algebra Appl.36 (1981), 1–24). We also obtain a new proof of a result of U. Haagerup (Decomposition of completely bounded maps on operation algebras, preprint), characterizing the norm of Schur product maps, and a new Hahn-Banach type extension theorem for these maps. Finally, we obtain generalizations of many of these results to matrices of operators, which we apply to the study of representations of certain subalgebras of the n × n matrices.

Multilinear maps and tensor norms on operator systems

Abstract We extend work of Christensen and Sinclair on completely bounded multilinear forms to the case of subspaces of C ∗ algebras, and obtain a representation theorem and a Hahn-Banach extension theorem for such maps. In the second part of the paper the Haagerup norms on tensor products are investigated, and we obtain new characterizations of these quantities.

On Bohr's inequality

Bohrs inequality says that if is a bounded analytic function on the closed unit disc, then for 0 leq r ⩽ 1/3 and that 1/3 is sharp. In this paper we give an operator-theoretic proof of Bohrs inequality that is based on von Neumanns inequality. Since our proof is operator-theoretic, our methods extend to several complex variables and to non-commutative situations. We obtain Bohr type inequalities for the algebras of bounded analytic functions and the multiplier algebras of reproducing kernel Hilbert spaces on various higher-dimensional domains, for the non-commutative disc algebra An, and for the reduced (respectively full) group C*-algebra of the free group on n generators. We also include an application to Banach algebras. We prove that every Banach algebra has an equivalent norm in which it satisfies a non-unital version of von Neumanns inequality. 2000 Mathematical Subject Classification: 47A20, 47A56.

A NOTE ON JOINT HYPONORMALITY

We describe certain cones of polynomials in two variables naturally associated to the class(es) of operators T for which the tuple (T, T2, ...,Tn) is jointly (weakly) hyponormal. As an application we give an example of an operator T such that the tuple (T,T2) is jointly but not weakly hyponormal. Further, we show that there exists a polynomially hyponormal operator which is not subnormal if and only if there exists a weighted shift with the same property.

Representations of Function Algebras, Abstract Operator Spaces, and Banach Space Geometry

Let G be the closed unit ball of some norm on Cn, and let A(G) be the closure of the polynomials in the sup norm. We prove that if n ⩾ 5 then there is a contractive representation of A(G) as operators on a Hilbert space which is not completely contractive. Our technique involves introducing a numerical invariant α(X) for a normed space X which measures the difference between the minimal operator space structure which can be assigned to X, MIN(X), and the maximal structure, MAX(X). We estimate α(X) using Banach space techniques. We also prove that if X is any infinite dimensional subspace of the space of continuous functions on a compact Hausdorff space, then there exists a bounded linear map on X which is not completely bounded.

Operator system structures on ordered spaces

Given an Archimedean order unit space (V,V^+,e), we construct a minimal operator system OMIN(V) and a maximal operator system OMAX(V), which are the analogues of the minimal and maximal operator spaces of a normed space. We develop some of the key properties of these operator systems and make some progress on characterizing when an operator system S is completely boundedly isomorphic to either OMIN(S) or to OMAX(S). We then apply these concepts to the study of entanglement breaking maps. We prove that for matrix algebras a linear map is completely positive from OMIN(M_n) to OMAX(M_m) if and only if it is entanglement breaking.

Explicit construction of universal operator algebras and applications to polynomial factorization

Using the characterization of unital operator algebras developed in [6], we give explicit internal definitions of the free product and the maximal operator-algebra tensor product of operator algebras and of the group operator algebra OA(G) of a discrete semigroup G (if G is a discrete group, then OA(G) coincides with the group C*-algebra C*(G)). This approach leads to new factorization theorems for polynomials in one and two variables.

Decoherence-Insensitive Quantum Communication by Optimal

The central issue in this paper is to transmit a quantum state in such a way that after some decoherence occurs, most of the information can be restored by a suitable decoding operation. For this purpose, we incorporate redundancy by mapping a given initial quantum state to a messenger state on a larger dimensional Hilbert space via a C* -algebra embedding. Our noise model for the transmission is a phase damping channel which admits a noiseless subsystem or decoherence-free subspace. More precisely, the transmission channel is obtained from convex combinations of a set of lowest rank yes/no measurements that leave a component of the messenger state unchanged. The objective of our encoding is to distribute quantum information optimally across the noise-susceptible component of the transmission when the noiseless component is not large enough to contain all the quantum information to be transmitted. We derive simple geometric conditions for optimal encoding and construct examples of such encodings.

C^{\ast }

On presente des resultats en theorie des operateurs multivariable dont les demonstrations se relient a des techniques issues de la geometrie algebrique