Stable rank and approximation theorems in $H^ \infty$

Abstract:

It is conjectured that for ${H^\infty }$ the Bass stable rank $(\text {bsr})$ is $1$ and the topological stable rank $(\text {tsr})$ is $2$. ${\text {bsr}}({H^\infty })= 1$ if and only if for every $({f_1},{f_2})\; \in {H^\infty } \times {H^\infty }$ which is a corona pair (i.e., there exist ${g_{1}}$, ${g_2} \in {H^\infty }$ such that ${f_1}{g_1} + {f_2}{g_2}= 1$) there exists a $g \in {H^\infty }$ such that ${f_1} + {f_2}g \in {({H^\infty })^{ - 1}}$, the invertibles in ${H^\infty }$; however, it suffices to consider corona pairs $({f_1},{f_2})$ where ${f_1}$ is a Blaschke product. It is also shown that there exists a $g \in {H^\infty }$ such that ${f_1} + {f_2}g \in \exp ({H^\infty })$ if and only if $\log {f_1}$ can be boundedly, analytically defined on $\{ {z \in \mathbb {D}:| {f_2}(z)| < \delta } \}$, for some $\delta > 0$. ${\text {tsr}}({H^\infty })= 2$ if and only if the corona pairs are uniformly dense in ${H^\infty } \times {H^\infty }$; 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, $K= {H^2} \ominus b {H^2}$ is an ${H^\infty }$-module via the compressed Toeplitz operators ${C_f}= {P_K}{T_f}{|_K}$, for $f \in {H^\infty }$, where ${T_f}$ is the Toeplitz operator ${T_f}g= f g$, for $g \in {H^2}$. Some stable rank questions can be recast as lifting questions: for $({f_i})_1^n \subset {H^\infty }$, there exist $({g_i})_1^n$, $({h_i})_1^n \subset {H^\infty }$ such that $\sum \nolimits _{i = 1}^n {({f_i} + b {g_i}){h_i}= 1}$ if and only if the compressed Toeplitz operators $({C_{{f_i}}})_1^n$ may be lifted to Toeplitz operators $({T_{{F_i}}})_1^n$ which generate $B({H^2})$ as an ideal.