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

Abstract

We determine the generalized E-stable ranks for the real algebra, C(D) sym , of all complex valued continuous functions on the closed unit disk, symmetric to the real axis, and its subalgebra A(D) R of holomorphic functions. A characterization of those invertible functions in C(E) is given that can be uniformly approximated on E by invertibles in A(D) R . Finally, we compute the Bass and topological stable rank of C(K) sym for real symmetric compact planar sets K.

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 $M(H^\infty)$

“…Proof. This follows as in [16] by using that bsr C(E, R) = 2 and bsr C(M + , C) = 2. For the reader's convenience we present those parts that need replacing D by M(H ∞ ), and D + by M + .…”

“…In recent years the real counterparts to the classical complex function algebras A(D), A(K), H ∞ (D) have gained a certain interest due to their appearance in control theory. These are, for example, the algebras A(K) sym = {f ∈ C(K), f holomorphic in K • and f (z) = f (z) for all z ∈ K}, where K is a real-symmetric compact set in C (that is K satisfies z ∈ K ⇐⇒ z ∈ K), A R (D) = {f ∈ A(D) : f real valued on [−1, 1]}, and [16,20,27,28,33,34]). If D is the closed unit disk, then of course A(D) sym = A R (D).…”

The Bass stable rank for the real Banach algebra A(K)sym

“…For matter of comparison, let us recall the situation in C(K) sym , that was done for the stable rank one part in [24, Theorem 6.5] and for the stable rank two part in [13,Theorem 3.4]. Note that the proof of the sufficiency of (1) below was based on the fact we confirmed above that bsr(A(K) sym ) = 1 whenever K ∩ R is totally disconnected.…”

“…For quite recent papers see Mikkola and Sasane [11], Mortini and Rupp [13], Mortini and Wick [15,16], Rupp and Sasane [23,24] and Wick [27].…”

The absolute stable rank of C(X,τ)

“…The Bass and topological stable ranks for this algebra have recently been determined (see [4]). In view of our main Theorem 4.1 above, we may also deduce the absolute stable rank of C (K ) sym .…”