Vol. 34, No. 2, 1970

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Projecting onto cycles in smooth, reflexive Banach spaces

Henry Bruce Cohen and Francis E. Sullivan

Vol. 34 (1970), No. 2, 355–364

This paper deals with operator algebras generated by certain classes of norm 1 projections on smooth, reflexive Banach spaces. For a strictly increasing continuous function on the nonnegative reals, the set of “-projections” gives rise to operator algebras equal to their second commutants. The principal result is that the closed subspace generated by the set of elements Ex, where x is fixed and E runs through a Boolean algebra of -projections, is the range of a norm 1 projection that commutes with each projection in the Boolean algebra. Sufficient conditions using Clarkson type norm inequalities are given for the commutativity of the set of all -projections. Examples in Orlicz spaces are given.

Mathematical Subject Classification
Primary: 46.10
Received: 23 July 1968
Revised: 25 February 1970
Published: 1 August 1970
Henry Bruce Cohen
Francis E. Sullivan