On Completions and Dense Subspaces of Strongly Facially Symmetric Spaces

Abstract

We study the behavior of strongly facially symmetric spaces under completion and passage to norm-dense subspaces. We introduce a natural face-density condition guaranteeing that the completion of a normed SFS-space remains strongly facially symmetric. We prove that a norm-dense subspace of a neutral strongly facially symmetric space inherits the neutral SFS-structure whenever it is invariant under the ambient generalized Peirce projections. Several examples and counterexamples are presented, including intermediate subspaces of the trace class and the dense subspace $C[0,1]\subset L_1[0,1]$.