## 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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