The M-ideal structure of some algebras of bounded linear operators

it is called nontrivial if {0} 6= J 6= X. This notion was introduced by Alfsen and Effros [1] and has proved useful in Banach space geometry, approximation theory and harmonic analysis; see [12] for a detailed account. A number of authors have studied the M -ideal structure in L(X), the space of bounded linear operators on a Banach space X, with special emphasis on the question whether K(X), the subspace of compact operators, is an M -ideal; see for instance [3], [7], [8], [9], [10], [11], [13], [14], [16], [17], [18], [21], [22], [24] or Chapter VI in [12]. In particular we mention the facts that, for a Hilbert space H, the M -ideals of L(H) coincide with its closed two-sided ideals [20] and that, for a subspace X of lp, K(X) is an M -ideal in L(X) if and only if X has the metric compact approximation property [7]. In this paper we show that in many cases the ideal of compact operators is the only candidate for an M -ideal in L(X). For X = lp this was done by Flinn [9]. For the function space Lp = Lp[0, 1] it has long been known that the compact operators do not form an M -ideal if p 6= 2 ([14] or, for another approach, [17]); in [12, p. 252] the problem is posed to determine the M -ideal structure of L(Lp) completely. This is done in section 2 where