## Title:
Closed Ideals in the Banach Algebra of Bounded Linear Operators on L
_{p}(0,1).
## Abstract:
I'll discuss the Banach algebra structure of the spaces of bounded linear operators on l
_{p} and L_{p}$\equiv $L_{p}(0,1). The main new results are1. The only non trivial closed ideal in $L(L$ _{p}), $1\le \; p\; <\; \infty $, that has a left approximate identity is the ideal of compact operators
(joint with N. C. Phillips and G. Schechtman).2. There are infinitely many; in fact, a continuum; of closed ideals in $L(L$ _{1}) (joint with G. Pisier and G. Schechtman).The second result answers a question from the 1978 book of A. Pietsch, "Operator ideals". |

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