Matrix convex hulls of free semialgebraic sets (Q2790584)

The authors study a noncommutative (free) analog of real algebraic geometry. In particular, they consider matrix convex sets and their projections. A free semialgebraic set which is convex and bounded and open can be represented as the solution of a linear matrix inequality (LMI), so there should be not too many convex free semialgebraic sets. Also, in contrast to the Tarski-Seidenberg theorem in real algebraic geometry over a real closed field, the projection of a free convex semialgebraic set need not be free semialgebraic.NEWLINENEWLINEThis motivates the authors to the study matrix convex hull of free semialgebraic sets. They present a construction of a sequence of LMI domains in increasingly many variables whose projections are successively finer outer approximations of the matrix hull of a free semialgebraic set. It is based on free analogs of moments and Hankel matrices. The virtues of this construction are demonstrated.