Basic Module Theory over Non-Commutative Rings with Computational Aspects of Operator Algebras

The present text surveys some relevant situations and results where basic Module Theory interacts with computational aspects of operator algebras. We tried to keep a balance between constructive and algebraic aspects.Comment: To appear in the Proceedings of the AADIOS 2012 conference, to be published in Lecture Notes in Computer Scienc

On real one-sided ideals in a free algebra

In classical and real algebraic geometry there are several notions of the radical of an ideal I. There is the vanishing radical defined as the set of all real polynomials vanishing on the real zero set of I, and the real radical defined as the smallest real ideal containing I. By the real Nullstellensatz they coincide. This paper focuses on extensions of these to the free algebra R of noncommutative real polynomials in x=(x_1,...,x_g) and x^*=(x_1^*,...,x_g^*). We work with a natural notion of the (noncommutative real) zero set V(I) of a left ideal I in the free algebra. The vanishing radical of I is the set of all noncommutative polynomials p which vanish on V(I). In this paper our quest is to find classes of left ideals I which coincide with their vanishing radical. We completely succeed for monomial ideals and homogeneous principal ideals. We also present the case of principal univariate ideals with a degree two generator and find that it is very messy. Also we give an algorithm (running under NCAlgebra) which checks if a left ideal is radical or is not, and illustrate how one uses our implementation of it.Comment: v1: 31 pages; v2: 32 page

Geometry of free loci and factorization of noncommutative polynomials

The free singularity locus of a noncommutative polynomial f is defined to be the sequence $Z_n(f)=\{X\in M_n^g : \det f(X)=0\}$ of hypersurfaces. The main theorem of this article shows that f is irreducible if and only if $Z_n(f)$ is eventually irreducible. A key step in the proof is an irreducibility result for linear pencils. Apart from its consequences to factorization in a free algebra, the paper also discusses its applications to invariant subspaces in perturbation theory and linear matrix inequalities in real algebraic geometry.Comment: v2: 32 pages, includes a table of content

Geometry of free loci and factorization of noncommutative polynomials

The free singularity locus of a noncommutative polynomial f is defined to be the sequence Zn(f)={X∈Mgn:detf(X)=0} of hypersurfaces. The main theorem of this article shows that f is irreducible if and only if Zn(f) is eventually irreducible. A key step in the proof is an irreducibility result for linear pencils. Apart from its consequences to factorization in a free algebra, the paper also discusses its applications to invariant subspaces in perturbation theory and linear matrix inequalities in real algebraic geometry

Realizations of non-commutative rational functions around a matrix centre, II: The lost-abbey conditions

In a previous paper the authors generalized classical results of minimal realizations of non-commutative (nc) rational functions, using nc Fornasini--Marchesini realizations which are centred at an arbitrary matrix point. In particular, it was proved that the domain of regularity of a nc rational function is contained in the invertibility set of a corresponding pencil of any minimal realization of the function. In this paper we prove an equality between the domain of a nc rational function and the domain of any of its minimal realizations. As for evaluations over stably finite algebras, we show that the domain of the realization w.r.t any such algebra coincides with the so called matrix domain of the function w.r.t the algebra. As a corollary we show that the domain of regularity and the stable extended domain coincide. In contrary to both the classical case and the scalar case -- where every matrix coefficients which satisfy the controllability and observability conditions can appear in a minimal realization of a nc rational function -- the matrix coefficients in our case have to satisfy certain equations, called linearized lost-abbey conditions, which are related to Taylor--Taylor expansions in nc function theory