Uniqueness of norm on L(G) and C(G) when G is a compact group |
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Authors: | J Extremera |
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Institution: | Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain |
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Abstract: | Let G be a compact group. If the trivial representation of G is not weakly contained in the left regular representation of G on L02(G) and X is either Lp(G) for 1<p?∞ or C(G), then we show that every complete norm |·| on X that makes translations from (X,|·|) into itself continuous is equivalent to ||·||p or ||·||∞ respectively. If 1<p?∞ and every left invariant linear functional on Lp(G) is a constant multiple of the Haar integral, then we show that every complete norm |·| on Lp(G) that makes translations from (Lp(G),|·|) into itself continuous and that makes the map t?Lt from G into bounded is equivalent to ||·||p. |
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Keywords: | Uniqueness of norm Mean-zero weak containment property Translation invariant functionals |
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