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 fromvone non-commutative domain ▫$\mathcal{D}$▫ in say ▫$g$▫ variables to another non-commutative domain ▫$\tilde{\mathcal{D}}$▫ in ▫$\tilde{g}$▫ variables. As a natural extension of the usual notion, an analytic free map is proper if it maps the boundary of ▫$\mathcal{D}$▫ into the boundary of ▫$\tilde{\mathcal{D}}$▫. Assuming that both domains contain 0, we show that if ▫$f \colon \mathcal{D} \to \tilde{\mathcal{D}}$▫ is a proper analytic free map, and ▫$f(0)=0$▫, then ▫$f$▫ is one-to-one. Moreover, if also ▫$g = \tilde{g}$▫, 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 ▫$\mathcal{D}$▫ and ▫$\tilde{\mathcal{D}}$▫.
COBISS.SI-ID: 15866201
We show that if the quotient of a group by its absolute centre is locally finite of exponent ▫$n$▫, then the exponent of its autocommutator subgroup is ▫$n$▫-bounded, that is, bounded by a function depending only on ▫$n$▫. If the group itself is locally finite, then its exponent is ▫$n$▫-bounded as well. Under some extra assumptions, the exponent of its automorphism group is ▫$n$▫-bounded. We determine the absolute centre and autocommutator subgroup for a large class of (infinite) abelian groups.
COBISS.SI-ID: 15996761
We prove a general archimedean positivstellensatz for hermitian operator-valued polynomials and show that it implies the multivariate Fejer-Riesz theorem of Dritschel-Rovnyak and positivstellensätze of Ambrozie-Vasilescu and Scherer-Hol. We also obtain several generalizations of these and related results. The proof of the main result depends on an extension of the abstract archimedean positivstellensatz for ▫$\ast$▫-algebras that is interesting in its own right.
COBISS.SI-ID: 15997529