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 $M(H^{\infty })$

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 Suárez's result that $\dim M(H^{\infty })=2$, we then compute the Bass and topological stable ranks of the algebra $C(M(H^{\infty }){)}_{\mathrm{sym}}$ of real-symmetric continuous functions on $M(H^{\infty })$.

18 pages