Résumé : It is well known that the mod $p$ homology of an infinite loop spaces has the structure of a so called AR-allowable Hopf algebra, i.e., a Hopf algebra on which the Steenrod algebra and the Dyer-Lashof algebra act satisfying some compatibility conditions. The work of McClure et. al. (as well as more recent works of Bousfield) gives its mod $p$ $K$-theory counter part. In this talk we discuss its $BP$ cohomology counter part. As a concrete example, we show thata $BP^*(S^3)$ doesn't admit such a structure.