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](https://mathoverflow.net/questions/247526/why-do-homotopy-theorists-care-whether-or-not-bp-is-e-infty).

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#References

[Law17] “Secondary power operations and the Brown–Peterson spectrum at the prime $2$.” [arXiv:1703.00935](https://arxiv.org/pdf/1703.00935.pdf).

[May75] “[Problems in infinite loop space theory](http://www.math.uchicago.edu/~may/PAPERS/16.pdf)”, 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](https://arxiv.org/pdf/1710.09822v1.pdf).