If $N\ge 2$, then there exist finitely many rotations of the sphere
$\mathbb{S}^N$ such that the set of the corresponding rotation
operators on $L^p(\mathbb{S}^N)$ determines the norm topology for
$1 \leq p \leq\infty, p \neq 1$. For $N=1$ the situation is different: the norm
topology of $L^2(\mathbb{S}^1)$ cannot be determined by the set of
operators corresponding to the rotations by elements of any `thin'
set of rotations of $\mathbb{S}^1$.
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References
J. Alaminos, J. Extremera and A.R. Villena, Applications of Kazhdan's property (T) to automatic continuity, J. Math. Anal. Appl., 307 (2005), 167--180.
N. Bourbaki, Éléments de mathématique. Intégration, Hermann, Paris, 1963.
Mathematical Reviews (MathSciNet):
MR179291
J.B. Conway, A Course in Functional Analysis, Graduate Texts in Mathematics, 96, Springer-Verlag, New York, 1985.
Mathematical Reviews (MathSciNet):
MR768926
V.G. Drinfeld, Finitely additive measure on $S^2$ and $S^3$, invariant with respect to rotations, Funct. Anal. Appl., 18 (1984), 245--246.
Mathematical Reviews (MathSciNet):
MR757256
G.B. Folland, A Course in Abstract Harmonic Analysis, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1995.
P. de la Harpe and A. Valette, La propriété (T) pour les groupes localement compacts, Astérisque 175, Soc. Math. de France, 1989.
B. Host, J.F. Méla, and F. Parreau, Non singular transformations and spectral analysis of measures, Bull. Soc. Math. France, 119 (1991), 33--90.
K. Jarosz, Uniqueness of translation invariant norms, J. Funct. Anal., 174 (2000), 417--429.
T.W. Korner, Some results on Kronecker, Dirichlet and Helson sets, Ann. Inst. Fourier (Grenoble), 20 fasc. 2 (1970), 219--324.
Mathematical Reviews (MathSciNet):
MR284771
G.A. Margulis, Some remarks on invariant means, Mh. Math., 90 (1980), 233--235.
Mathematical Reviews (MathSciNet):
MR596890
A.M. Sinclair, Automatic Continuity of Linear Operators, Cambridge Univ. Press, Cambridge, UK, 1976.
Mathematical Reviews (MathSciNet):
MR487371
D. Sullivan, For $n>3$ there is only finitely-additive rotationally invariant measure on the $n$-sphere defined on all Lebesgue measurable sets, Bull. Amer. Math. Soc., 4 (1981), 121--123.
Mathematical Reviews (MathSciNet):
MR590825
A.R. Villena, Invariant functionals and the uniqueness of invariant norms, J. Funct. Anal., 215 (2004), 365--398.
A. Wilansky, Functional Analysis, Blaisdell Publishing Co., Ginn and Co., New York-Toronto-London, 1964.
Mathematical Reviews (MathSciNet):
MR170186
A. Zygmund, Trigonometric Series. Vol. I, II, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2002.
