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Is there a good notion of the spectrum of units $R^\ast$ in a (possibly non-connective) $E_\infty$-ring spectrum $R$?

A standard definition (see section 1.2 in http://arxiv.org/abs/0810.4535) seems to output a connective spectrum $gl_1(R)$: the underlying space is the pullback of $\Omega^\infty(R) \to \pi_0(R)$ over the units $\pi_0(R)^\times \subset \pi_0(R)$. While this is completely reasonable if $R$ is itself connective, is there a better operation when one does not assume $R$ to be connective?

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    $\begingroup$ My understanding is that constructing a sane notion of spectra of units sensitive to coconnective information is an interesting open problem. As spaces of units were initially developed to understand twists of spectra parametrized over a space, and as all spaces are themselves connective, this insensitivity wasn't initially considered to be an issue. Steffen Sagave has proposed a model for periodic $E_\infty$ ring spectra; maybe you'd enjoy reading about that. arxiv.org/abs/1111.6731 $\endgroup$
    – Eric Peterson
    Commented Apr 4, 2014 at 7:14
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    $\begingroup$ Whoa. Units were initially developed to study obstructions to orientability of bundles and fibrations, among other (and deeper) things, such as understanding F/Top as BO-{\otimes} as an infinite loop space away from 2. It was maybe 30 years later that we understood orientations in terms of understanding twists of parametrized spectra. $\endgroup$
    – Peter May
    Commented Apr 5, 2014 at 2:14

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