On the stable rank and reducibility in algebras of real symmetric functions

Abstract

Abstract. Let A R (D) denote the set of functions belonging to the disc algebra having real Fourier coefficients. We show that A R (D) has Bass and topological stable ranks equal to 2, which settles the conjecture made by Brett Wick in [18]. We also give a necessary and sufficient condition for reducibility in some real algebras of functions on symmetric domains with holes, which is a generalization of the main theorem in [18]. A sufficient topological condition on the symmetric open set D is given for the corresponding real algebra A R (D) to have Bass stable rank equal to 1.

Approximation by invertible elements and the generalized $E$-stable rank for $A({\boldsymbol D})_{\mathsf R}$ and $C({\boldsymbol D})_{\mathrm{sym}}$

“…Hence f 2 + ag = 0 on E for some a ∈ A(D) R . Since the real symmetric polynomials are dense in A(D) R (see for example [12] or [22]), there exists p ∈ A(D) R ∩ C [z] such that pf + ag = 0 on E. By moving a little bit the zeros of p (non real zeros in pairs), we may assume that p and g have no zeros in common in D.…”

“…Our goal in section two is to determine the generalized E-stable rank for A(D) R and to characterize those functions zero free on E that can be uniformly approximated on E by elements invertible in A(D) R . Only recently it has been shown (see [22] and [14]) that the Bass and topological stable rank of A(D) R coincide:…”

“…We refer the reader to [3], [4], [5], [6], [10], [13], [14], [19], [20], [21], [22], [23], [26] for a glimpse on several aspects and computations of stable ranks.…”

Function theory in real Hardy spaces

“…We point to the work of Mikkola and Sasane [9], Mortini and Wick [10], [11], Rupp and Sasane [13], and Wick [16], [17] as sources of further motivation and additional references. These papers deal with problems in systems theory, stable ranks for Banach algebras, and corona problems.…”

A distinguished real Banach algebra

“…Recently, there have been some papers studying concepts from Control theory and K-theory within this algebra (simultaneous stabilization, stable ranks,...) (see [6,7,9]). …”