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- W2172003246 abstract "Abstract For a Banach space $mathfrak{X}$, let $mathcal{B}(mathfrak{X})$ denote the Banach algebra of all continuous linear operators on $mathfrak{X}$. First, we study the lattice of closed ideals in $mathcal{B}(mathfrak{J}_p)$, where $1 lt p t infty$ and $mathfrak{J}_p$ is the $p$th James space. Our main result is that the ideal of weakly compact operators is the unique maximal ideal in $mathcal{B}(mathfrak{J}_p)$. Applications of this result include the following. (i) The Brown–McCoy radical of $mathcal{B}(mathfrak{X})$, which by definition is the intersection of all maximal ideals in $mathcal{B}(mathfrak{X})$, cannot be turned into an operator ideal. This implies that there is no ‘Brown–McCoy’ analogue of Pietsch’s construction of the operator ideal of inessential operators from the Jacobson radical of $mathcal{B}(mathfrak{X})/mathcal{A}(mathfrak{X})$. (ii) For each natural number $n$ and each $n$-tuple $(m_1,dots,m_n)$ in ${k^2mid kinmathbb{N}}cup{infty}$, there is a Banach space $mathfrak{X}$ such that $mathcal{B}(mathfrak{X})$ has exactly $n$ maximal ideals, and these maximal ideals have codimensions $m_1,dots,m_n$ in $mathcal{B}(mathfrak{X})$, respectively; the Banach space $mathfrak{X}$ is a finite direct sum of James spaces and $ell_p$-spaces. Second, building on the work of Gowers and Maurey, we obtain further examples of Banach spaces $mathfrak{X}$ such that all the maximal ideals in $mathcal{B}(mathfrak{X})$ can be classified. We show that the ideal of strictly singular operators is the unique maximal ideal in $mathcal{B}(mathfrak{X})$ for each hereditarily indecomposable Banach space $mathfrak{X}$, and we prove that there are $2^{2^{aleph_0}}$ distinct maximal ideals in $mathcal{B}(mathfrak{G})$, where $mathfrak{G}$ is the Banach space constructed by Gowers to solve Banach’s hyperplane problem. AMS 2000 Mathematics subject classification: Primary 47D30; 47D50; 46H10; 16D30" @default.
- W2172003246 created "2016-06-24" @default.
- W2172003246 creator A5048851482 @default.
- W2172003246 date "2002-10-01" @default.
- W2172003246 modified "2023-10-16" @default.
- W2172003246 title "MAXIMAL IDEALS IN THE ALGEBRA OF OPERATORS ON CERTAIN BANACH SPACES" @default.
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- W2172003246 doi "https://doi.org/10.1017/s0013091500001097" @default.
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