The Friedrichs angle and alternating projections in Hilbert $C^{\ast }$-modules

Abstract

Let $B$ be a $C^{\ast }$-algebra, $X$ a Hilbert $C^{\ast }$-module over $B$ and $M,N\subset X$ a pair of complemented submodules. We prove the $C^{\ast }$-module version of von Neumann's alternating projections theorem: the sequence $(P_N{P}_M{)}^n$ is Cauchy in the $\ast$-strong module topology if and only if $M\cap N$ is the complement of $\overline{M^{\perp }+N^{\perp }}$. In this case, the $\ast$-strong limit of $(P_M{P}_N{)}^n$ is the orthogonal projection onto $M\cap N$. 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 $M,N\subset X$ is well-defined and that $c(M,N)<1$ if and only if $M\cap N$ is complemented and $M+N$ is closed.

19 pages. We added Lemma 3.10, sharpened Proposition 3.12, and discuss vector bundles in Remark 3.14

References in corpus (1)

• Halmos' two projections theorem for Hilbert $C^{\ast }$-module operators and the Friedrichs