Single generator problem

Tanaka, Jun-ichi

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

Single generator problem
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Trans. Amer. Math. Soc. 348 (1996), 4113-4129 Request permission

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.

References

Raúl E. Curto, Paul S. Muhly, and Takahiko Nakazi, Ergodic $H^1$ is not a dual space, Hokkaido Math. J. 16 (1987), no. 1, 75–79. MR 878844, DOI 10.14492/hokmj/1381517832
Manfred Denker and Ernst Eberlein, Ergodic flows are strictly ergodic, Advances in Math. 13 (1974), 437–473. MR 352403, DOI 10.1016/0001-8708(74)90075-9
Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
Frank Forelli, Analytic and quasi-invariant measures, Acta Math. 118 (1967), 33–59. MR 209771, DOI 10.1007/BF02392475
T. W. Gamelin, $H^{p}$ spaces and extremal functions in $H^{1}$, Trans. Amer. Math. Soc. 124 (1966), 158–167. MR 213877, DOI 10.1090/S0002-9947-1966-0213877-4
Theodore W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1969. MR 0410387
John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
Toshihiro Hamachi, Yukimasa Oka, and Motosige Osikawa, A classification of ergodic non-singular transformation groups, Mem. Fac. Sci. Kyushu Univ. Ser. A 28 (1974), 113–133. MR 432859, DOI 10.2206/kyushumfs.28.113
H. Helson, Analyticity on compact abelian groups, Algebras in analysis (Proc. Instructional Conf. and NATO Advanced Study Inst., Birmingham, 1973) Academic Press, London, 1975, pp. 1–62. MR 0427959
Henry Helson, Compact groups with ordered duals. VII, J. London Math. Soc. (2) 20 (1979), no. 3, 509–515. MR 561142, DOI 10.1112/jlms/s2-20.3.509
Henry Helson, The spectral theorem, Lecture Notes in Mathematics, vol. 1227, Springer-Verlag, Berlin, 1986. MR 873504, DOI 10.1007/BFb0101629
Calvin C. Moore and Klaus Schmidt, Coboundaries and homomorphisms for nonsingular actions and a problem of H. Helson, Proc. London Math. Soc. (3) 40 (1980), no. 3, 443–475. MR 572015, DOI 10.1112/plms/s3-40.3.443
Paul S. Muhly, Function algebras and flows, Acta Sci. Math. (Szeged) 35 (1973), 111–121. MR 331068
Paul S. Muhly, Function algebras and flows. III, Math. Z. 136 (1974), 253–260. MR 493357, DOI 10.1007/BF01214129
Paul S. Muhly, Isometries of ergodic Hardy spaces, Israel J. Math. 36 (1980), no. 1, 50–74. MR 589657, DOI 10.1007/BF02761230
William Parry, Compact abelian group extensions of discrete dynamical systems, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 13 (1969), 95–113. MR 260976, DOI 10.1007/BF00537014
Jun-ichi Tanaka, Blaschke cocycles and generators, Pacific J. Math. 142 (1990), no. 2, 357–378. MR 1042051, DOI 10.2140/pjm.1990.142.357
Jun-ichi Tanaka, Flows in fibers, Trans. Amer. Math. Soc. 343 (1994), no. 2, 779–804. MR 1202421, DOI 10.1090/S0002-9947-1994-1202421-8
T. P. Srinivasan and Ju-kwei Wang, Weak ${}^{\ast }$-Dirichlet algebras, Function Algebras (Proc. Internat. Sympos. on Function Algebras, Tulane Univ., 1965) Scott-Foresman, Chicago, Ill., 1966, pp. 216–249. MR 0198282
Russell C. Walker, The Stone-Čech compactification, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 83, Springer-Verlag, New York-Berlin, 1974. MR 0380698, DOI 10.1007/978-3-642-61935-9