The covering dimension of a distinguished subset of the spectrum of and the algebra of real-symmetric and continuous functions on

We show that the covering dimension, dim E, of the closure E of the interval ] − 1 , 1 [ in the spectrum of H ∞ equals one. Using Suárezʼs result that dim M ( H ∞ ) = 2 , we then compute the Bass and topological stable ranks of the algebra C ( M ( H ∞ ) ) sym of real-symmetric continuous functions on M ( H ∞ ) .