Approximate identities for ideals in L ( L p ) $L(L^p)$

IF 0.8 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society Pub Date : 2024-09-28 DOI:10.1112/blms.13161

W. B. Johnson, G. Schechtman

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Abstract

The main result is that the only non-trivial closed ideal in the Banach algebra $L (L^{p})$ of bounded linear operators on $L^{p} (0, 1)$ , $1 ⩽ p < \infty$ , that has a left approximate identity is the ideal of compact operators. The algebra $L (L^{1})$ has at least one non-trivial closed ideal that has a contractive right approximate identity as well as many, including the unique maximal ideal, that do not have a right approximate identity.

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L(L p)$ L(L^p)$理想的近似恒等式

主要结果是在Banach代数L (L p) $L(L^p)$上有界线性算子的唯一非平凡闭理想。0,1) $L^p(0,1)$, 1≤p ＆lt；∞$1\leqslant p < \infty$，它有一个左近似恒等式是紧算子的理想。代数L (l1) $L(L^1)$至少有一个具有压缩右近似恒等式的非平凡闭理想，以及许多不具有右近似恒等式的非平凡闭理想，包括唯一极大理想。

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