Stable Rank and Approximation Theorems in H ∞

Abstract

It is conjectured that for H°° the Bass stable rank (bsr) is 1 and the topological stable rank (tsr) is 2. bsr(H°°) = 1 if and only if for every (/i > fi) G H°° x H°° which is a corona pair (i.e., there exist g, , g2 e H°°s uch that f]g1 + f2g2 = 1) there exists age H°° such that /j + f2g e (Z/00)-, the invertibles in H°° ; however, it suffices to consider corona pairs (f\ < f-¡) where /, is a Blaschke product. It is also shown that there exists age H°° such that fx + f2g 6 exp(//°°) if and only if log/j can be boundedly, analytically defined on {z e B>: \f2(z)\ < 3} , for some ô > 0. tsr(//°°) = 2 if and only if the corona pairs are uniformly dense in H°° x H°° ; however, it suffices to show that the corona pairs are uniformly dense in pairs of Blaschke products. This condition would be satisfied if the interpolating Blaschke products were uniformly dense in the Blaschke products. For b an inner function, K = H 9 bH is an H°°-module via the compressed Toeplitz operators C, = PKTj-\K, for f e H°° , where T, is the Toeplitz operator Trg = fg , for g e H. Some stable rank questions can be recast as lifting questions: for (/])" C H°° , there exist (g,)" , (h¡)" C H°°s uch that S"=i (/; + bg¡)h¡ = 1 if and only if the compressed Toeplitz operators (Cj-)" may be lifted to Toeplitz operators (7"f)" which generate B(H2) as an ideal.

On the stable rank of 𝐻^{∞}

“…Let (f\, f2) £ U2(H°°(D)), which we require to show is reducible. The hypothesis of the corollary and the proof of Theorem 1.1 in [8] show that the set {Bh: B is an interpolating Blaschke product, h £ (H°°(D))~1} is dense in H°°(D). Consequently, by Corollary 1.2 in [8] there exists interpolating Blaschke products B\, B2 and h{, h2 £ (H°°(D))~l such that (1) fiBihi + f2B2h2 = I.…”

“…We note that this is then a special case of Theorem 1. More generally, Laroco [8] has shown that sr(H°°) = 1 if log/i can be boundedly analytically defined on {z : |/2(z)| < e} for some e > 0. It is also shown in [8] (Theorem 3.6) that in proving the reducibility of a general corona pair (f , f2) we can assume f is a Blaschke product.…”

“…More generally, Laroco [8] has shown that sr(H°°) = 1 if log/i can be boundedly analytically defined on {z : |/2(z)| < e} for some e > 0. It is also shown in [8] (Theorem 3.6) that in proving the reducibility of a general corona pair (f , f2) we can assume f is a Blaschke product. Combining Theorem 1 with some of the results in [8], we have the following corollary, which was shown to me by L. Laroco.…”

“…It is also shown in [8] (Theorem 3.6) that in proving the reducibility of a general corona pair (f , f2) we can assume f is a Blaschke product. Combining Theorem 1 with some of the results in [8], we have the following corollary, which was shown to me by L. Laroco.…”

“…We now utilize the proof of Theorem 3.6 in [8]. Equation (1) (1) {zj} is an interpolating sequence in R+ ; (2) there exists n > 0 such that inffc ILy* P(zj > zk) > n > 0 ; (3) there exists a > 0 such that for all j / k, p(z¡, z¿.)…”