In a recent work, Moslehian and Rajić have shown that the Grüss inequality holds for unital n-positive linear maps φ:A →B(H), where A is a unital C∗-algebra and H is a Hilbert space, if n ≥ 3. They also demonstrate that the inequality fails to hold, in general, if n = 1 and question whether the inequality holds if n = 2. In this article, we provide an affirmative answer to this question.

This article contains a characterization of operator systems S with the property that every positive map φ: S → Mn is decompos-able, as well as an alternate and a more direct proof of a characterization of decomposable maps due to E. Størmer.

Subsets of the set of g-tuples of matrices that are closed with respect to direct sums and compact in the free topology are characterized. They are, in a dilation theoretic sense, the hull of a single point.

Versions of well-known function theoretic operator theory results of Szegő and Widom are established for the Neil algebra. The Neil algebra is the subalgebra of the algebra of bounded analytic functions on the unit disc consisting of those functions whose derivative vanishes at the origin.

Matrix-valued polynomials in any finite number of freely noncommuting variables that enjoy certain canonical partial convexity properties are characterized, via an algebraic certificate, in terms of Linear Matrix Inequalities and Bilinear Matrix Inequalities.