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

topology and its applications

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 $M(H^\infty)$