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 ℱ
on the nonnegative reals, the set of “ℱ-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 ℱ-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 ℱ-projections. Examples in Orlicz spaces are
given.
