I'll discuss the Banach algebra structure of the spaces of bounded linear operators on ell_p and L_p:=L_p(0,1).  

The main new results are

1. The only non trivial closed ideal in L(L_p), for 1 <= 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".