Duality, extreme points and hulls for noncommutative partial convexity
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| Format: | Preprint |
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2024
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| author | Klep, Igor McCullough, Scott Štrekelj, Tea |
| author_facet | Klep, Igor McCullough, Scott Štrekelj, Tea |
| contents | This article studies generalizations of (matrix) convexity, including partial convexity and biconvexity, under the umbrella of $Γ$-convexity. Here $Γ$ is a tuple of free symmetric polynomials determining the geometry of a $Γ$-convex set. The paper introduces the notions of $Γ$-operator systems and $Γ$-ucp maps and establishes a Webster-Winkler type categorical duality between $Γ$-operator systems and $Γ$-convex sets. Next, a notion of an extreme point for $Γ$-convex sets is defined, paralleling the concept of a free extreme point for a matrix convex set. To ensure the existence of such points, the matricial sets considered are extended to include an operator level. It is shown that the $Γ$-extreme points of an operator $Γ$-convex set $K$ are in correspondence with the free extreme points of the operator convex hull of $Γ(K).$ From this result, a Krein-Milman theorem for $Γ$-convex sets follows. Finally, relying on the results of Helton and the first two authors, a construction of an approximation scheme for the $Γ$-convex hull of the matricial positivity domain {(also known as a free semialgebraic set)} $D_p$ of a free symmetric polynomial $p$ is given. The approximation consists of a decreasing family of $Γ$-analogs of free spectrahedra, whose projections, under mild assumptions, in the limit yield the $Γ$-convex hull of $D_p.$ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_13267 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Duality, extreme points and hulls for noncommutative partial convexity Klep, Igor McCullough, Scott Štrekelj, Tea Operator Algebras Functional Analysis 46L07, 13J30, 46N10, 47L07, 52A30 This article studies generalizations of (matrix) convexity, including partial convexity and biconvexity, under the umbrella of $Γ$-convexity. Here $Γ$ is a tuple of free symmetric polynomials determining the geometry of a $Γ$-convex set. The paper introduces the notions of $Γ$-operator systems and $Γ$-ucp maps and establishes a Webster-Winkler type categorical duality between $Γ$-operator systems and $Γ$-convex sets. Next, a notion of an extreme point for $Γ$-convex sets is defined, paralleling the concept of a free extreme point for a matrix convex set. To ensure the existence of such points, the matricial sets considered are extended to include an operator level. It is shown that the $Γ$-extreme points of an operator $Γ$-convex set $K$ are in correspondence with the free extreme points of the operator convex hull of $Γ(K).$ From this result, a Krein-Milman theorem for $Γ$-convex sets follows. Finally, relying on the results of Helton and the first two authors, a construction of an approximation scheme for the $Γ$-convex hull of the matricial positivity domain {(also known as a free semialgebraic set)} $D_p$ of a free symmetric polynomial $p$ is given. The approximation consists of a decreasing family of $Γ$-analogs of free spectrahedra, whose projections, under mild assumptions, in the limit yield the $Γ$-convex hull of $D_p.$ |
| title | Duality, extreme points and hulls for noncommutative partial convexity |
| topic | Operator Algebras Functional Analysis 46L07, 13J30, 46N10, 47L07, 52A30 |
| url | https://arxiv.org/abs/2412.13267 |