Contributions to a noncommutative real algebraic geometry : doctoral thesis

Zalar, Aljaž (author), Cimprič, Jaka (mentor), Klep, Igor (comentor)

We study Positivstellensätze from noncommutative real algebraic geometry. Of these, we focus on two specific ones. A version of the matrix Fejér-Riesz theorem characterizes positive semidefinite matrix polynomials on the real line. This characterization has already been extended from the real line to a disjoint union of finitely many closed intervals in the case of scalar polynomials and to a single closed interval in the case of matrix polynomials. Our first interest in this thesis is to figure out what can be said in the case of matrix polynomials and a disjoint union of finitely many closed intervals. Algebraic certificates of positivity for noncommutative matrix polynomials on matrix convex sets, such as the solution set of a linear matrix inequality (LMI), have recently attracted much attention among real algebraic geometers. In the case of LMIs many certificates are known. Since every closed matrix convex set containing the origin is the solution set of a linear operator inequality (LOI), this attracts the second interest of this thesis which is to extend the certificates from matrix to operator polynomials. Our main result referring to the first problem is a denominator-free characterization in the case of a compact union, called a Compact Positivstellensatz. The technique in the proof is the adaptation of Schur complements and eliminating the denominators with the help of known results for scalar polynomials. We also construct counterexamples for the extension of the characterization to almost all non-compact unions. By developing the connections between matrix polynomials and Laurent matrix polynomials we obtain the matrix Positivstellensatz on a disjoint union of finitely many closed arcs in the unit complex circle and finally, using this result we come to a Non-compact Positivstellensatz for a non-compact union of finitely many closed intervals in the real line using only simple denominators. Referring to the second problem our first result is an algebraic characterization for the domination of the solution sets of monic LOIs, called a Linear Positivstellensatz. The techniques used are complete positivity and the theory of operator algebras. We provide examples which show that the monicity assumption is necessary. As a consequence we also obtain the description of the polar dual of the LOI. Next we focus on the question of the equality of the solution sets of two LOIs which turns out to be a harder one. We present the answer for LOIs with compact operator coeficients, called a Linear Gleichstellensatz. Namely, under some minimality assumption, the LOIs are unitarily equivalent. The idea is to understand the unital $C^\ast$-algebras generated by the coefficients and $\ast$-isomorphisms between them. We show by examples that the answer does not extend to arbitrary LOIs. Finally, we establish a Convex Positivstellensatz which characterizes matrix polynomials positive semidefinite on the solution set of a LOI and show that in the univariate case it extends to operator polynomials.

• Language: English
• Keywords: real algebraic geometry, positive polynomials, quadratic module, Positivstellensatz, free positivity, linear operator inequality, spectrahedron, noncommutative polynomials
• Work type: Doctoral dissertation
• Typology: 2.08 - Doctoral Dissertation
• Organization: FMF - Faculty of Mathematics and Physics
• Place of publishing: Ljubljana
• Publisher: [A. Zalar]
• Year: 2017
• Number of pages: XIV, 124 str.
• PID: 20.500.12556/RUL-97408
• UDC: 517.9(043.3)
• COBISS.SI-ID: 17986137
• Publication date in RUL: 24.10.2017

ZALAR, Aljaž, 2017, Contributions to a noncommutative real algebraic geometry : doctoral thesis [online]. Doctoral dissertation. Ljubljana : A. Zalar. [Accessed 23 June 2025]. Retrieved from: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=eng&id=97408

Secondary language

• Language: Slovenian
• Title: Prispevki k nekomutativni realni algebraični geometriji

Abstract

Študiramo Positivstellensätze iz nekomutativne realne algebraične geometrije. Med njimi se osredotočimo na posebna primera. Verzija matričnega Fejér-Rieszovega izreka karakterizira pozitivno semidefinitne matrične polinome na realni osi. Ta karakterizacija je že bila razširjena iz realne osi na disjunktno unijo končno mnogo zaprtih intervalov v primeru skalarnih polinomov in na primer enega samega zaprtega intervala v primeru matričnih polinomov. Prvi cilj tega dela je ugotoviti, kaj se da povedati za matrične polinome in disjunktno unijo končno mnogo zaprtih intervalov. Algebraična zagotovila za pozitivnost nekomutativnih matričnih polinomov na matrično konveksnih množicah, kot je množica rešitev linearne matrične neenakosti, so nedavno pritegnila pozornost med realnimi algebraičnimi geometri in veliko je že bilo narejenega. Ker je vsaka zaprta matrična konveksna množica, ki vsebuje izhodišče, množica rešitev linearne operatorske neenakosti (LON), to motivira drugi cilj tega dela, ki je razširitev zagotovil za pozitivnost iz matričnih na operatorske polinome. Naš glavni rezultat pri študiju prvega problema je karakterizacija brez imenovalcev v primeru kompaktnih unij, ki se imenuje Kompaktni Positivstellensatz. Tehnika v dokazu je predelava Schurovih komplementov in odprava imenovalcev z uporabo znanih rezultatov za skalarne polinome. Konstruiramo tudi protiprimere za razširitev te karakterizacije na skoraj vse nekompaktne unije. S študijem povezav med matričnimi polinomi in Laurentovimi matričnimi polinomi izpeljemo matrični Positivstellensatz na disjunktni uniji končno mnogo zaprtih lokov na enotski kompleksni krožnici. Z uporabo tega rezultata nato izpeljemo Nekompaktni Positivstellensatz za nekompaktno unijo zaprtih intervalov na realni osi, v katerem nastopajo le enostavni imenovalci. Naš prvi rezultat pri študiju drugega problema je algebraično zagotovilo za dominacijo množic rešitev eničnih LONov, ki se imenuje Linearni Positivstellensatz. Glavni uporabljeni tehniki sta popolna pozitivnost in teorija operatorskih algeber. Predstavimo tudi primere, ki pokažejo, da je predpostavka eničnosti potrebna. Opišemo tudi polaro LONa. Nato se osredotočimo na vprašanje enakosti množic rešitev dveh LONov, kar se izkaže za težji problem. Predstavimo odgovor za LONe s kompaktnimi operatorskimi koeficienti, ki se imenuje Linearni Gleichstellensatz. Ta pove, da sta pri predpostavki minimialnosti LONa unitarno ekvivalentna. Ideja je razumeti unitalne $C^\ast$-algebre, generirane s koeficienti LONa, in $\ast$-homomorfizme med njimi. S primeri pokažemo, da se izrek ne razširi na poljubne LONe. Na koncu izpeljemo Konveksni Positivstellensatz, ki karakterizira nekomutativne matrične polinome, pozitivno semidefinitne na množici rešitev LONa. V primeru ene spremenljivke pa ga razširimo na operatorske polinome.

• Keywords: realna algebraična geometrija, pozitivni polinomi, kvadratni moduli, prosta pozitivnost, linearna operatorska neenakost, spektraeder, nekomutativni polinomi