Countably generated Douglas algebras

Izuchi, Keiji

doi:10.1090/S0002-9947-1987-0869406-9

Countably generated Douglas algebras
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by Keiji Izuchi

Trans. Amer. Math. Soc. 299 (1987), 171-192

DOI: https://doi.org/10.1090/S0002-9947-1987-0869406-9

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Abstract:

Under a certain assumption of $f$ and $g$ in ${L^\infty }$ which is considered by Sarason, a strong separation theorem is proved. This is available to study a Douglas algebra $[{H^\infty }, f]$ generated by ${H^\infty }$ and $f$. It is proved that (1) ball$(B/{H^\infty } + C)$ does not have exposed points for every Douglas algebra $B$, (2) Sarason’s three functions problem is solved affirmatively, (3) some characterization of $f$ for which $[{H^\infty }, f]$ is singly generated, and (4) the $M$-ideal conjecture for Douglas algebras is not true.

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