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Spaces of operators as continuous function spaces. (English) Zbl 0922.46014

Let \(X\) and \(Y\) be Banach spaces, \({\mathcal L}(X,Y)\) and \({\mathcal K}(X,Y)\) denote spaces of bounded and compact linear operators, respectively, \(X_1\) denote the closed unit ball of \(X\) and \(\partial_eX_1\) the set of extreme points of \(X_1\). In this short note, it is shown that if \({\mathcal K}(X,Y)\) is isometric to \(C(\Omega)\) for a compact space \(\Omega\) then \(X^*\) is isometric to \(C(\Omega_1)\) and \(Y\) is isometric to \(C(\Omega_2)\) for some compact sets \(\Omega_1\) and \(\Omega_2\). In the case of bounded operators it is shown that if \(\partial_eX_1\) is nonempty, then \({\mathcal L}(X,Y)\) isometric to a \(C(\Omega)\) will imply that \(X\) is isometric to an abstract \(L\)-space and \(Y\) is isometric to a \(C(\Omega_1)\). The following two results are proved in this note:
Theorem. Let \(X\) and \(Y\) be Banach spaces. Suppose \({\mathcal K}(X,Y)\) is isometric to \(C(\Omega)\) for some compact set \(\Omega\). Then there are compact sets \(\Omega_1\) and \(\Omega_2\) such that \(\Omega\) is homeomorphic to \(\Omega_1\times \Omega_2\) and \(X^*\) is isometric to \(C(\Omega_1)\) and \(Y\) is isomorphic to \(C(\Omega_2)\).
Proposition. Suppose \(X\) is a Banach space such that \(\partial_eX_1\) is nonempty. If \({\mathcal L}(X,Y)\) is isometric to a \(C(\Omega)\) space then \(X\) is isometric to an \(L\)-space and \(Y\) is isometric to \(C(\Omega_1)\) for some compact set \(\Omega_1\).

MSC:

46B20 Geometry and structure of normed linear spaces
47L05 Linear spaces of operators
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