TITLE: Why do homotopy theorists care whether or not $BP$ is $E_\infty$?
QUESTION [20 upvotes]: I have often heard that it is not known whether or not the Brown-Peterson spectrum $BP$ is an $E_\infty$-ring spectrum. Though I see that this is a somewhat natural question to ask, I have often wondered what would the practical consequences of an answer to this question would be. 
In particular, I am wondering what would a positive answer to this question mean for homotopy theory. Are there computations we would be able to do that we couldn't do before? What would be the take away if $BP$ did not admit an $E_\infty$-ring structure?
I would also appreciate any references on this problem. Thanks!

REPLY [21 votes]: I suppose I should try to answer since the question of whether or not $BP$ is an $E_{\infty}$ ring spectrum
was Problem 1 of "Problems in infinite loop space theory'', http://www.math.uchicago.edu/~may/PAPERS/16.pdf,
written in 1974, two years after $E_{\infty}$ ring spectra were first defined.   At the time, we were hoping for some good
way of starting from the relevant group law and manufacturing $BP$ from it.  The example "in nature'' that led to
the definition of $E_{\infty}$ ring spectra was $MU$, so it was impossible not to be intrigued by the question.
Some things proven since with $MU$ might be streamlined if we could start with $BP$, such as the construction
of Morava $K$-theory.   However, Basterra and Mandell proved that $BP$ is an $E_4$-spectrum, compatibly with $MU$, which is enough for such
applications.   I'm still intrigued by the question, although I admit to having no applications in mind.