Single generator problem

Tanaka, Jun-ichi

doi:10.1090/S0002-9947-96-01566-8

Single generator problem
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by Jun-ichi Tanaka

Trans. Amer. Math. Soc. 348 (1996), 4113-4129

DOI: https://doi.org/10.1090/S0002-9947-96-01566-8

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Abstract:

Using the Stone-Čech compactification $\beta \mathbf Z$ of integers, we introduce a free extension of an almost periodic flow. Together with some properties of outer functions, we see that, in a certain class of ergodic Hardy spaces $H^p(\mu )$, $1\le p\le \infty$, the corresponding subspaces $H_0^p(\mu )$ are all singly generated. This shows the existence of maximal weak-$^*$ Dirichlet algebras, different from $H^\infty$ of the disc, for which the single generator problem is settled.

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