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Zbl 1180.46040
Alaminos, J.; Extremera, J.; Villena, A.R.
Uniqueness of rotation invariant norms.
(English)
[J] Banach J. Math. Anal. 3, No. 1, 85-98, electronic only (2009). ISSN 1735-8787/e

For each $N=1,2,\dots$, let $S^N$ denote the $N$-dimensional Euclidean sphere endowed with the Lebesgue measure. Given a set $J$ of continuous linear operators on a Banach space $X$, we say that $J$ determines the norm topology of $X$ if any complete norm $|\cdot|$ on $X$ such that the operator $T$ maps $(X,|\cdot|)$ continuously to $(X,|\cdot|)$ for each $T\in J$, is equivalent to the given norm $\|\cdot\|$ on $X$. It was shown by {\it K.\,Jarosz} [Uniqueness of translation invariant norms'', J.~Funct.\ Anal.\ 174, No.\,2, 417--429 (2000; Zbl 0981.46006)] that the set of operators on $L^p(S^1)$, with $1<p<\infty$, corresponding to all rotations on $S^1$, determines the norm topology of $L^p(S^1)$. {\it A.\,R.\thinspace Villena} established in [Invariant functionals and the uniqueness of invariant norms'', J.~Funct.\ Anal.\ 215, No.\,2, 366--398 (2004; Zbl 1067.46041)] a similar result for $L^p(S^N)$, $N\ge 2$. In the present, interesting paper, the authors study the question: how many rotations are necessary to determine the topology of $L^p(S^N)$ with $1<p<\infty$ and $N\ge 1$? One of the main results shows that, if $N\ge 2$, then there exist finitely many rotations of $S^N$ such that the set consisting of the corresponding rotation operators on $L^p(S^N)$ determine the norm topology of the space for $1<p\le \infty$. On the other hand, it is shown that the norm topology of $L^2 (S^1)$ cannot be determined by the set of operators corresponding to rotations by elements of any thin set of rotations of $S^1$.
[Anthony To-Ming Lau (Alberta)]
MSC 2000:
*46H40 Automatic continuity
43A15 Lp-spaces and other function spaces on groups, etc.
43A20 L1 algebras on groups, etc.

Keywords: invariant norm; automatic continuity; Dirichlet set; Kazhdan's property; rotation of sphere

Citations: Zbl 0981.46006; Zbl 1067.46041

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