We show that a representation of a Banach algebra A on a Banach space X can be extended to a canonical representation of $$A^{**}$$ A ∗ ∗ on X if and only if certain orbit maps $$A\rightarrow X$$ A → X are weakly compact. When this is the case, we show that the essential space of the representation is complemented if A has a bounded left approximate identity. This provides a tool to disregard the difference between degenerate and nondegenerate representations. Our results have interesting consequences both in $$C^*$$ C ∗ -algebras and in abstract harmonic analysis. For example, a $$C^*$$ C ∗ -algebra A has an isometric representation on an $$L^p$$ L p -space, for $$p\in [1,\infty ){\setminus }\{2\}$$ p ∈ [ 1 , ∞ ) \ { 2 } , if and only if A is commutative. Moreover, the $$L^p$$ L p -operator algebra of a locally compact group is universal with respect to arbitrary representations on $$L^p$$ L p -spaces.

中文翻译：

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将 Banach 代数的表示扩展到它们的双对数

我们表明，当且仅当某些轨道映射 $$A\rightarrow 时，Banach 代数 A 在 Banach 空间 X 上的表示可以扩展为 X 上 $$A^{**}$$ A ∗ ∗ 的规范表示X$$ A → X 是弱紧致的。在这种情况下，我们证明如果 A 具有有界左近似恒等式，则表示的基本空间是互补的。这提供了一种忽略退化和非退化表示之间差异的工具。我们的结果在 $$C^*$$ C ∗ 代数和抽象调和分析中都有有趣的结果。例如，$$C^*$$ C ∗ -代数 A 在 $$L^p$$ L p 空间上具有等距表示，对于 $$p\in [1,\infty ){\setminus } \{2\}$$ p ∈ [ 1 , ∞ ) \ { 2 } ，当且仅当 A 是可交换的。而且，