A distinguished real Banach algebra

Abstract

Abstract. -We present a new and elementary approach to characterize the maximal ideals and their associated multiplicative linear functionals for a classical real Banach algebra of analytic functions. 20.4.2009Let C s (T) be the set of all (complex-valued) continuous functions on the unit circle T = {z ∈ C : |z| = 1} that are symmetric; this means that f (e −it ) = f (e it ). Associated with C s (T) is the subset, A s , of those functions in C s (T) that admit a holomorphic extension to the interior D of the unit disk.The holomorphic extension to the disk is given by the Poisson-integralNote that for z = re it this coincides with the convolutionof f with the Poisson-kernel P r (t) := 1−r 2 1+r 2 −2r cos t . Using Fourier representation, we see that the Fourier series of an element f ∈ A s formally can be written as F[f ] = ∞ n=0f n e int , where the Fourier coefficients, given as usual by the formulaf n := 1 2π 2π 0 f (e it )e −int dt, are real numbers (see [3]). It is standard knowledge that the Taylor-MacLaurin series for P [f ] can be written as P [f ](z) = ∞ n=0f n z n . With respect to the usual pointwise operations of addition, multiplication and scalar-multiplication by reals, C s (T) and A s become real algebras.2000 Mathematics Subject Classification. -46J15.

The covering dimension of a distinguished subset of the spectrum $M(H^\infty)$ of $H^\infty$ and the algebra of real-symmetric and continuous functions on $M(H^\infty)$

“…In view of Corollary 1.3 and [12,Theorem 1.3.20] we have the following result on the structure of the maximal ideals of C(M(H ∞ )) sym and their associated multiplicative linear functionals (see also [13] for the case of the algebra A R (D)).…”

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.…”

“…Suppose that f ∈ A(D) R has constant sign on E ∩ [−1, 1], say f > 0 there, and that f does not vanish on E. Due to symmetry, f has no zeros on K, either. Since the polynomials with real coefficients are dense in A(D) R (see, e.g., [12]), we may assume without loss of generality that f is a polynomial that has no zeros on K and is positive on…”

“…In this paper we look upon A(D) R as a function algebra defined on D (see [12]). 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 .…”