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 $M(H^\infty)$

TL;DR

This paper determines the covering dimension of a specific subset of the spectrum of $H^ty$, showing it is one, and computes the Bass and topological stable ranks of real-symmetric continuous functions on the spectrum.

Contribution

It establishes the dimension of a distinguished subset of the spectrum and calculates the algebraic invariants of real-symmetric continuous functions on it.

Findings

The closure of the interval $]-1,1[$ in the spectrum has covering dimension one.

The spectrum $M(H^ty)$ has dimension two.

The Bass and topological stable ranks of the algebra of real-symmetric continuous functions are computed.

Abstract

We show that the covering dimension, $\dim E$, of the closure $E$ of the interval $]-1,1[$ in the spectrum of $H^\infty$ equals one. Using Su\'arez's result that $\dim M(H^\infty)=2$, we then compute the Bass and topological stable ranks of the algebra $C(M(H^\infty))_{\rm\scriptscriptstyle sym}$ of real-symmetric continuous functions on $M(H^\infty)$.