Stable Relations and their Loci in Operator Algebra Variables

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AbstractLoringThe abstract properties of curves in a unit cube defined by polynomials are well understood. In particular, points that approximately are zeros of the polynomials are close to exact zeros. The same set of polynomials can become "unstable" when applied to matrices (and even more so when applied to operator algebra variables). That is, there can be contractive matrices that, when substituted into the polynomials give results near zero, and yet there is no actual matrix-root close to the approximate solution. We call such approximate solutions phantom solutions. These phantom approximate solutions can be misleading in the context of computer models. There is a notion of stability for a relation that does not require the relation to be polynomial. Much of this project will study relations that remain stable when applied to operator algebra variables. These are rare, but tend to be related to interesting invariants of operator algebras. Stability results limited to the realm of matrices will also be explored, with the hope of finding some applications to numerical analysis.Polynomial relations have stability properties that we rely on every day. Take the example of a laser cutting some flat material, where the cut-points are determined as the zero-set of a set of equations. If the x and y coordinates of the laser come close to satisfying these equations, we go ahead and fire the laser, for we are sure that close to this approximate solution there is an exact solution, and thus we are cutting close to where we should. In computer modeling, one's model is often based on more that just numerical variables. In particular, much of numerical analysis involves matrix-valued computations. It turns out that equations that are stable for ordinary values become unstable for matrix values. That is, there can be approximate solutions to equations that are not close to any true solutions. We call such things phantom approximate solutions. This project will study these phantom approximate solutions and ways to detect and avoid them.