Let $G$ be a finite $p$-group. We prove that whenever the commuting probability of $G$ is greater than $(2p^2 + p - 2)/p^5$, the unramified Brauer group of the field of $G$-invariant functions is trivial. Equivalently, all relations between commutators in $G$ are consequences of some universal ones. The bound is best possible, and gives a global lower bound of $1/4$ for all finite groups. The result is attained by describing the structure of groups whose Bogomolov multipliers are nontrivial, and Bogomolov multipliers of all of their proper subgroups and quotients are trivial. Applications include a classification of $p$-groups of minimal order that have nontrivial Bogomolov multipliers and are of nilpotency class $2$, a nonprobabilistic criterion for the vanishing of the Bogomolov multiplier, and establishing a sequence of Bogomolov's absolute $\gamma$-minimal factors which are $2$-groups of arbitrarily large nilpotency class, thus providing counterexamples to some of Bogomolov's claims. In relation to this, we fill a gap in the proof of triviality of Bogomolov multipliers of finite simple groups.
COBISS.SI-ID: 17284185
The study of matrix inequalities in a dimension-free setting is in the realm of free real algebraic geometry. In this paper we investigate constrained trace and eigenvalue optimization of noncommutative polynomials. We present Lasserre’s relaxation scheme for trace optimization based on semidefinite programming (SDP) and demonstrate its convergence properties. Finite convergence of this relaxation scheme is governed by flatness, i.e., a rank-preserving property for associated dual SDPs. If flatness is observed, then optimizers can be extracted using the Gelfand–Naimark–Segal construction and the Artin–Wedderburn theory verifying exactness of the relaxation. To enforce flatness we employ a noncommutative version of the randomization technique championed by Nie. The implementation of these procedures in our computer algebra system NCSOStools is presented and several examples are given to illustrate our results.
COBISS.SI-ID: 2048288770
Finsler's Lemma charactrizes all pairs of symmetric $n \times n$ real matrices $A$ and $B$ which satisfy the property that $v^T A v ) 0$ for every nonzero $v \in \mathbb{R}^n$ such that $v^T B v = 0$. We extend this characterization to all symmetric matrices of real multivariate polynomials, but we need an additional assumption that $B$ is negative semidefinite outside some ball. We also give two applications of this result to Noncommutative Real Algebraic Geometry which for $n=1$ reduce to the usual characterizations of positive polynomials on varieties and on compact sets.
COBISS.SI-ID: 17193817
We construct an exotic one-parameter semigroup of endomorphisms of a symmetric cone $C$, whose generator is not the sum of a Lie group generator and an endomorphism of $C$. The question is motivated by the theory of affine processes on symmetric cones, which plays an important role in mathematical finance. On the other hand, theoretical question that we solve in this paper seems to have been implicitly open even much longer then this motivation suggests.
COBISS.SI-ID: 17257561
Let $\mathcal{H}$ be a complex Hilbert space, let $\mathcal{D} \to \mathcal{B(H)}$ ( be a discrete masa (maximal abelian selfadjoint algebra) and let $\mathcal{A}$ be a linear subspace (or a singleton subset) of $\mathcal{B(H)}$ not necessarily having any nontrivial intersection with $\mathcal{D}$. Assume that the commutator $AD - DA$ is quasinilpotent for every $A \in \mathcal{A}$ and every $D \in \mathcal{D}$. We prove that $\mathcal{A}$ and $\mathcal{D}$ are simultaneously triangularizable. If $\mathcal{D}$ is a continuous masa, there exist compact operators satisfying this condition that fail to have a multiplicity-free triangularization together with $\mathcal{D}$. However, we prove an analogous result in the case where $\mathcal{A}$ is a finite-dimensional space of operators of finite rank.
COBISS.SI-ID: 17441881