1.

Pure states, positive matrix polynomials and sums of hermitian squares

Let $ M $ be an archimedean quadratic module of real $ t \times t $ matrix polynomials in $ n $ variables, and let $ S \subseteq \Bbb R^{n}$ be the set of all points where each element of $ M $ is positive semidefinite. Our key finding is a natural bijection between the set of pure states of $ M $ and $ S \times \Bbb P^{t-1} (\Bbb R)$. This leads us to conceptual proofs of positivity certificates for matrix polynomials, including the recent seminal result of Hol and Scherer: If a symmetric matrix polynomial is positive definite on $S$, then it belongs to $ M $.

COBISS.SI-ID: 15816793

2.

The Procesi-Schacher conjecture and Hilbert's 17th problem for algebras with involution

In 1976 Procesi and Schacher developed an Artin-Schreier type theory for central simple algebras with involution and conjectured that in such an algebra a totally positive element is always a sum of hermitian squares. In this paper elementary counterexamples to this conjecture are constructed and cases are studied where the conjecture does hold. Also, a Positivstellensatz is established for noncommutative polynomials, positive semidefinite on all tuples of matrices of a fixed size.

COBISS.SI-ID: 15580249

3.

A method for computing lowest eigenvalues of symmetric polynomial differential operators by semidefinite programming

A method for computing global infima of real multivariate polynomials based on semidefinite programming was developed by Shor. The present article aims to extend a variant of their method to noncommutative symmetric polynomials. We extend the method from polynomials to polynomial differential operators.

COBISS.SI-ID: 15609177

4.

On the spectral radius of positive operators on Banach sequence spaces

We study under what conditions on a Banach lattice, every positive weak Dunford-Pettis operator is weakly compact.

COBISS.SI-ID: 15607641

5.

Powerful actions and non-abelian tensor products of powerful p-groups

In 1987 Lubotzky and Mann introduced the concept of a powerful $p$-group. In this paper we generalise this definition as follows. Suppose $M$ and $N$ are finite $p$-groups with $N$ acting on $M$. Then, we say $N$ acts powerfully on $M$ if $p$ is odd and the induced action of $N$ upon $M/M^p$ is trivial. The paper contains fundamental properties of powerful actions. Finally, as an application,we study the non-Abelian tensor product of powerful $p$-groups.

COBISS.SI-ID: 15596121