A Non-commutative Real Nullstellensatz Corresponds to a Non-commutative Real Ideal; Algorithms

This article takes up the challenge of extending the classical Real Nullstellensatz of Dubois and Risler to left ideals in a *-algebra A. After introducing the notions of non-commutative zero sets and real ideals, we develop three themes related to our basic question: does an element p of A having zero set containing the intersection of zero sets of elements from a finite set S of A belong to the smallest real ideal containing S? Firstly, we construct some general theory which shows that if a canonical topological closure of certain objects are permitted, then the answer is yes, while at the purely algebraic level it is no. Secondly for every finite subset S of the free *-algebra R of polynomials in g indeterminates and their formal adjoints, we give an implementable algorithm which computes the smallest real ideal containing S and prove that the algorithm succeeds in a finite number of steps. Lastly we provide examples of noncommutative real ideals for which a purely algebraic non-commutative real Nullstellensatz holds. For instance, this includes the real (left) ideals generated by a finite sets S in the *-algebra of n by n matrices whose entries are polynomials in one-variable. Further, explicit sufficient conditions on a left ideal in R are given which cover all the examples of such ideals of which we are aware and significantly more.Comment: Improved results compared to earlier version

Some analysable instances of mu-synthesis

I describe a verifiable criterion for the solvability of the 2 by 2 spectral Nevanlinna-Pick problem with two interpolation points, and likewise for three other special cases of the mu-synthesis problem. The problem is to construct an analytic 2 by 2 matrix function F on the unit disc subject to a finite number of interpolation constraints and a bound on the cost function sup {mu(F(z)): |z| < 1}, where mu is an instance of the structured singular value of a 2 by 2 matrix.Comment: 19 pages. The paper is based on a talk given at the Helton 65th birthday conference, San Diego, November 2010, and summarizes a number of results, mainly taken from papers by Jim Agler and me. To appear in the conference proceeding

Forced Out : The Agony of the Refugee in Our Time

This book weaves together autobiographical statements by refugees, eyewitness accounts of this human tragedy, photographs, and facts. Kismaric, Shawcross, and Brown paint rather bleak pictures in their articles on the world-wide plight of political refugees. 100 bibl. ref