By polar decomposition, every continuous linear function $f \colon H \to K$ between Hilbert spaces can be written uniquely as $f = \widehat{f} \circ |f|$ for a positive operator $|f| \colon H \to H$ and a partial isometry $\widehat{f} \colon H \to K$ with $\ker(\widehat{f})=\ker(|f|)$. The binary operation $(f,g) \mapsto |g \circ f|$ on the set of positive operators is not associative. Is the binary operation $(f,g) \mapsto \widehat{g \circ f}$ on the set of partial isometries associative?