Vol. 29, No. 3, 1969

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ISSN: 0030-8730
The Arens products and an imbedding theorem

Julien O. Hennefeld

Vol. 29 (1969), No. 3, 551–563

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.

Mathematical Subject Classification
Primary: 46.65
Received: 4 March 1968
Revised: 30 July 1968
Published: 1 June 1969
Julien O. Hennefeld