New York Journal of Mathematics
Volume 13 (2007) 215-270


Matthew Daws

Arens-regularity of algebras arising from tensor norms

Published: July 26, 2007
Keywords: Arens products, Arens regularity, compact operators, approximable operators, nuclear operators, tensor norm, operator ideals
Subject: 47L10, 47L20, 46B22, 46B28, 46H05

We investigate the Arens products on the biduals of certain algebras of operators on nonreflexive Banach spaces. To be precise, we study the α-nuclear operators, where α is a tensor norm. This includes the approximable and nuclear operators, and we use these, together with the 2-nuclear operators, as motivating examples. The structure of the two topological centres of the bidual are studied, and typical results are that for the approximable operators, the two topological centres are always distinct, neither contains the other, and both strictly contain the original algebra. In contrast, on a nonpathological Banach space, the topological centres of the bidual of the nuclear operators coincide. Our methods allow us to also study the algebra of compact operators, even when the compacts are not equal to the approximable operators.


The financial support of the EPSRC is gratefully acknowledged.

Author information

St. John's College, Oxford, OX1 3JP