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Let X be a separable Banach
space, B(X) be the algebra of all bounded linear operators on X, and 𝒞 be the
algebra of all compact linear operators. This paper centers around the general
question of giving a construction of B(X) as a Banach algebra starting from
𝒞.
It is a result of Schatten and von Neumann that if H is a Hilbert space, then
there is an isometric imbedding of B(H) onto 𝒞∗∗, where 𝒞∗∗ denotes the second dual
of 𝒞. Moreover, each of the two Arens products on 𝒞∗∗ coincides with the
multiplication induced on 𝒞∗∗ by operator multiplication on B(H). The proofs of
these results make strong use of the Hilbert space structure.
In this paper we generalize these results to a large class of uniformly
convex spaces. Moreover, we show that even when B(X) is not equal to
𝒞∗∗ it is still possible to construct B(X) as a Banach algebra starting from
𝒞.
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