Ideals in \(L(L_1)\) (Q2290816)

The purpose of the paper under review is to investigate closed (two-sided) ideals in the Banach algebra \(L(X)\) of bounded linear operators on a Banach space $X$, with special attention given to the function space \(L_p:=L_p(0,1)\) when \(p=1\). It is known that the compact operators are the only nontrivial closed ideal in \(L(X)\) when \(X\) is one of the classical sequence spaces \(\ell_p\), \(1 \leq p < \infty\), or \(c_0\), or the Hilbert function space \(L_2\). When \(1 < p \neq 2 < \infty\), it was proved recently that there is a continuum of closed ideals in \(L(L_p)\) [\textit{T. Schlumprecht} and \textit{A. Zsák}, J. Reine Angew. Math. 735, 225--247 (2018; Zbl 1464.47048)]. The situation for \(X=L_1\) is quite different, previously no one has proved that the Banach algebra \(L(L_1)\) contains a closed ideal distinct from the following five: the compact operators; the strictly singular operators; the operators on \(L_1\) that factor through \(\ell_1\); the Dunford-Pettis operators; and the \(L_1\)-singular operators on \(L_1\). For an interesting and fairly complete exposition on the closed ideals in \(L(L_p)\), see the introduction of the paper under review. Recall some notions: an operator \(T:X \to Y\) between Banach spaces is \(Z\)-singular for an Banach space \(Z\) if \(TS\) is not an (into)isomorphism for any operator \(S:Z \to X\); an operator is strictly singular if it is \(Z\)-singular for every infinite-dimensional \(Z\); an ideal in \(L(X)\) is called small if it is contained in the ideal of strictly singular operators, otherwise is called large. The main result of the paper is a wise construction of a continuum of small closed ideals in \(L(L_1)\). This solves a long standing question from \textit{A. Pietsch} [Operator ideals, Berlin: VEB Deutscher Verlag der Wissenschaften (1978; Zbl 0399.47039)]. Also, the authors mention that it is not known if there are more than a continuum of closed ideals in \(L(L_1)\) and pose the question if there are more than three large ideals in \(L(L_1)\). As consequence of the main result, it is proved that \(L(C(K))\) has a continuum of small closed ideals for every compact uncountable metric space \(K\). Finally, by using a duality argument, the authors prove that \(L(\ell_\infty)\) has a continuum of small closed ideals and also show that distinct small closed ideals in \(L(L_1)\) produce distinct small closed ideals in \(L(L_\infty)\).