In 2017, the long-standing problem of whether the Brown–Peterson spectrum $\mathrm{BP}$ admits the structure of an $E_\infty$-ring was shown in the negative.

The question dates back to 1975, formulated as Problem 1 of Peter May's “Problems in infinite loop space theory” [May75]:

Problem 1. Does the Brown–Peterson spectrum $\mathrm{BP}$ admits a model as an $E_\infty$-ring spectrum?

This was answered for the prime $2$ in Theorem 5.5.4 of [Law17] and for odd primes in Theorem 1.2 of [Sen17], which we (partly) state below:

Theorem. The $2$-local Brown–Peterson spectrum does not admit the structure of an $E_n$-algebra for any $12\leq n\leq\infty$.

Theorem. Let $p$ be an odd prime. Then the $p$-local Brown–Peterson spectrum does not admit the structure of an $E_{2\left(p^2+2\right)}$-ring .

See also this MathOverflow question.

References

[Law17] “Secondary power operations and the Brown–Peterson spectrum at the prime $2$.” arXiv:1703.00935.

[May75] “Problems in infinite loop space theory”, Conference on homotopy theory (Evanston, Ill., 1974), Notas Mat. Simpos., vol. 1, Soc. Mat. Mexicana, México, 1975, pp. 111–125. MR 761724

[Sen17] “The Brown-Peterson spectrum is not $E_{2 (p^2+2)}$ at odd primes.” arXiv:1710.09822.