The Friedrichs angle and alternating projections in Hilbert C⁎-modules
Journal of Mathematical Analysis and Applications
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.
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