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Let $E$ be an $E_1$ ring spectrum. Then I believe the center of $E$ is an $E_2$ ring spectrum over which $E$ is an $E_1$ algebra, given by the endomorphisms of $E$ as a bimodule over itself.

Question: What is there to say about the center of Morava $K$ theory? For instance what are its homotopy groups?

By the Hopkins-Mahowald theorem this spectrum is not $p^1$ torsion for finite nonzero height.

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    $\begingroup$ It depends on the $E_1$-form of Morava $K$-theory you consider. For example, if it's an Azumaya form, the center is $E$-theory $\endgroup$
    – Maxime Ramzi
    Commented Apr 6 at 19:45
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    $\begingroup$ There is work of Angeltveit that computes the topological Hochschild coholomogy of K(n) (for certain versions of K(n), as Maxime points out) to indeed be Morava E-theory, just in case you want some kind of reference (even if vague). I don't recall which paper specifically. Perhaps this is discussed in Hopkins-Lurie though. $\endgroup$
    – Jonathan Beardsley
    Commented Apr 7 at 23:05

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As others pointed out in the comments, the topological Hochschild cohomology of $K(n)$ (i.e., the center of $K(n)$) was calculated by Angeltveit. Here is the paper: https://doi.org/10.2140/gt.2008.12.987 Angeltveit's answer is that the center of $K(n)$ is the Morava $E$-theory spectrum $E_n$ or some extension thereof, depending on the $A_{\infty}$-structure on $K(n)$ that you began with.

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    $\begingroup$ Note also that there is a subtlety with what exactly one calls Morava $K$-theory : the $2$-periodic version or the $2(p^n-1)$-periodic version. "How many forms are there ?" and "what is their center ?" will depend on this $\endgroup$
    – Maxime Ramzi
    Commented Apr 8 at 23:47

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