Uniqueness of dilation invariant norms

Moreno, E.; Villena, A.

doi:10.1090/S0002-9939-04-07327-7

Uniqueness of dilation invariant norms
HTML articles powered by AMS MathViewer

by E. Moreno and A. R. Villena PDF

Proc. Amer. Math. Soc. 132 (2004), 2067-2073 Request permission

Abstract:

Let $\delta _a$ be a nontrivial dilation. We show that every complete norm $\|\cdot \|$ on $L^1(\mathbb {R}^N)$ that makes $\delta _a$ from $(L^1(\mathbb {R}^N),\|\cdot \|)$ into itself continuous is equivalent to $\|\cdot \|_1$. $\delta _a$ also determines the norm of both $C_0(\mathbb {R}^N)$ and $L^p(\mathbb {R}^N)$ with $1<p<\infty$ in a weaker sense. Furthermore, we show that even all the dilations do not determine the norm on $L^\infty (\mathbb {R}^N)$.

References

H. G. Dales, Banach algebras and automatic continuity, London Mathematical Society Monographs. New Series, vol. 24, The Clarendon Press, Oxford University Press, New York, 2000. Oxford Science Publications. MR 1816726
J. Extremera, J. F. Mena, and A. R. Villena, Uniqueness of the topology on $L^1(G)$, Studia Math. 150 (2002), no. 2, 163–173. MR 1892727, DOI 10.4064/sm150-2-5
J. Extremera, J. F. Mena, and A. R. Villena, Uniqueness of norm on $L^p(G)$ and $C(G)$ when $G$ is a compact group. J. Funct. Anal. 197 (2003), 212-227.
J. Extremera and A. R. Villena, Uniqueness of norm on $L^1(G)$ when $G$ is a Moore group, preprint, 2002.
Krzysztof Jarosz, Uniqueness of translation invariant norms, J. Funct. Anal. 174 (2000), no. 2, 417–429. MR 1768981, DOI 10.1006/jfan.2000.3593
B. E. Johnson, The uniqueness of the (complete) norm topology, Bull. Amer. Math. Soc. 73 (1967), 537–539. MR 211260, DOI 10.1090/S0002-9904-1967-11735-X
B. E. Johnson, Continuity of homomorphisms of Banach $G$-modules, Pacific J. Math. 120 (1985), no. 1, 111–121. MR 808931, DOI 10.2140/pjm.1985.120.111
A. R. Villena, Uniqueness of the topology on spaces of vector-valued functions, J. London Math. Soc. (2) 64 (2001), no. 2, 445–456. MR 1853463, DOI 10.1112/S0024610701002423
Gordon S. Woodward, Translation-invariant linear forms on $C_{0}(G)$, $C(G)$, $L^{p}(G)$ for noncompact groups, J. Functional Analysis 16 (1974), 205–220. MR 0344801, DOI 10.1016/0022-1236(74)90064-0