The Friedrichs angle and alternating projections in Hilbert C⁎-modules

Journal of Mathematical Analysis and Applications, vol. 516, pp. 126474

Abstract

Let B be a C⁎-algebra, X a Hilbert C⁎-module over B and M,N⊂X a pair of complemented submodules. We prove the C⁎-module version of von Neumann's alternating projections theorem: the sequence (PNPM)n is Cauchy in the ⁎-strong module topology if and only if M∩N is the complement of M⊥+N⊥‾. In this case, the ⁎-strong limit of (PMPN)n is the orthogonal projection onto M∩N. We use this result and the local-global principle to show that the cosine of the Friedrichs angle c(M,N) between any pair of complemented submodules M,N⊂X is well-defined and that c(M,N)<1 if and only if M∩N is complemented and M+N is closed.

Authors 2

1. Bram Mesland corresponding Aachen Mathematisch Instituut

   Leiden University

   Affiliation as printed

   Mathematisch Instituut, Universiteit Leiden, Netherlands

2. Adam Rennie

   University of Wollongong

   Affiliation as printed

   School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, Australia