74 research outputs found
Matrix Convex Hulls of Free Semialgebraic Sets
This article resides in the realm of the noncommutative (free) analog of real
algebraic geometry - the study of polynomial inequalities and equations over
the real numbers - with a focus on matrix convex sets and their projections
. A free semialgebraic set which is convex as well as bounded and open
can be represented as the solution set of a Linear Matrix Inequality (LMI), a
result which suggests that convex free semialgebraic sets are rare. Further,
Tarski's transfer principle fails in the free setting: The projection of a free
convex semialgebraic set need not be free semialgebraic. Both of these results,
and the importance of convex approximations in the optimization community,
provide impetus and motivation for the study of the free (matrix) convex hull
of free semialgebraic sets.
This article presents the construction of a sequence of LMI domains
in increasingly many variables whose projections are
successively finer outer approximations of the matrix convex hull of a free
semialgebraic set . It is based on free analogs of
moments and Hankel matrices. Such an approximation scheme is possibly the best
that can be done in general. Indeed, natural noncommutative transcriptions of
formulas for certain well known classical (commutative) convex hulls does not
produce the convex hulls in the free case. This failure is illustrated on one
of the simplest free nonconvex .
A basic question is which free sets are the projection of a free
semialgebraic set ? Techniques and results of this paper bear upon this
question which is open even for convex sets.Comment: 41 pages; includes table of contents; supplementary material (a
Mathematica notebook) can be found at
http://www.math.auckland.ac.nz/~igorklep/publ.htm
The Tracial Hahn-Banach Theorem, Polar Duals, Matrix Convex Sets, and Projections of Free Spectrahedra
This article investigates matrix convex sets and introduces their tracial
analogs which we call contractively tracial convex sets. In both contexts
completely positive (cp) maps play a central role: unital cp maps in the case
of matrix convex sets and trace preserving cp (CPTP) maps in the case of
contractively tracial convex sets. CPTP maps, also known as quantum channels,
are fundamental objects in quantum information theory.
Free convexity is intimately connected with Linear Matrix Inequalities (LMIs)
L(x) = A_0 + A_1 x_1 + ... + A_g x_g > 0 and their matrix convex solution sets
{ X : L(X) is positive semidefinite }, called free spectrahedra. The
Effros-Winkler Hahn-Banach Separation Theorem for matrix convex sets states
that matrix convex sets are solution sets of LMIs with operator coefficients.
Motivated in part by cp interpolation problems, we develop the foundations of
convex analysis and duality in the tracial setting, including tracial analogs
of the Effros-Winkler Theorem.
The projection of a free spectrahedron in g+h variables to g variables is a
matrix convex set called a free spectrahedrop. As a class, free spectrahedrops
are more general than free spectrahedra, but at the same time more tractable
than general matrix convex sets. Moreover, many matrix convex sets can be
approximated from above by free spectrahedrops. Here a number of fundamental
results for spectrahedrops and their polar duals are established. For example,
the free polar dual of a free spectrahedrop is again a free spectrahedrop. We
also give a Positivstellensatz for free polynomials that are positive on a free
spectrahedrop.Comment: v2: 56 pages, reworked abstract and intro to emphasize the convex
duality aspects; v1: 60 pages; includes an index and table of content
Free bianalytic maps between spectrahedra and spectraballs in a generic setting
Given a tuple of matrices, the collection of
those tuples of matrices (of the same size) such that is called a spectraball . Likewise,
given a tuple of matrices the collection of
tuples of matrices (of the same size) such that is a free spectrahedron
. Assuming and are irreducible, plus an additional mild
hypothesis, there is a free bianalytic map
normalized by and if and only if
and spans an algebra. Moreover is unique, rational and has an elegant
algebraic representation.Comment: 19 page
Proper Analytic Free Maps
This paper concerns analytic free maps. These maps are free analogs of
classical analytic functions in several complex variables, and are defined in
terms of non-commuting variables amongst which there are no relations - they
are free variables. Analytic free maps include vector-valued polynomials in
free (non-commuting) variables and form a canonical class of mappings from one
non-commutative domain D in say g variables to another non-commutative domain
D' in g' variables. As a natural extension of the usual notion, an analytic
free map is proper if it maps the boundary of D into the boundary of D'.
Assuming that both domains contain 0, we show that if f:D->D' is a proper
analytic free map, and f(0)=0, then f is one-to-one. Moreover, if also g=g',
then f is invertible and f^(-1) is also an analytic free map. These conclusions
on the map f are the strongest possible without additional assumptions on the
domains D and D'.Comment: 17 pages, final version. To appear in the Journal of Functional
Analysi
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