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

TL;DR

This paper extends von Neumann's alternating projections theorem to Hilbert $C^{*}$-modules, characterizing the convergence of projection sequences and defining the Friedrichs angle between submodules.

Contribution

It introduces a $C^{*}$-module version of von Neumann's theorem and establishes conditions for the Friedrichs angle to be well-defined and less than one.

Findings

The sequence $(P_N P_M)^n$ converges iff $M igcap N$ is complemented.

The $*$-strong limit of the sequence is the orthogonal projection onto $M igcap N$.

The Friedrichs angle $c(M,N)$ is well-defined and less than 1 iff $M igcap N$ is complemented and $M+N$ is closed.

Abstract

Let $B$ be a $C^{*}$-algebra, $X$ a Hilbert $C^{*}$-module over $B$ and $M,N\subset X$ a pair of complemented submodules. We prove the $C^{*}$-module version of von Neumann's alternating projections theorem: the sequence $(P_{N}P_{M})^{n}$ is Cauchy in the $*$-strong module topology if and only if $M\cap N$ is the complement of $\overline{M^{\perp}+N^{\perp}}$. In this case, the $*$-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.