units in non-commutative ring spectra

Question

Let $R$ be a connective (symmetric) ring spectrum. Let $GL_1(R)$ be the space of units of $R$, i.e. the union of the components of $\Omega^{\infty}(R)$ corresponding to the units of $\pi_0(R)$. $GL_1(R)$ is a group-like monoid, therefore it makes sense to speak about $BGL_1(R)$.

In fact, if $R$ is a commutative spectrum, then $GL_1(R)$ has the structure of an infinite loop space and defines a spectrum $gl_1(R)$.

Are there any sensible methods to analyze $BGL_1(R)$ in the non-commutative case?

As an example of what I mean, let me mention that $BGL_1(R)$ maps to $K(R)$ the algebraic $K$-theory of the ring spectrum $R$, which in turn maps to $THH(R)$ - the topological Hochschild homology. Thus, information about $THH(R)$ might tell you something about $BGL_1(R)$.

Answer 1

The notation $\Omega^{\infty}$ requires care. Definitions and comparisons for symmetric, orthogonal, and EKMM spectra are given in Lind's thesis. See https://arxiv.org/abs/0908.1092. For symmetric spectra, the original source is Sagave and Schlichtkrull https://arxiv.org/abs/1103.2764. In that case, one must use fibrant approximation, and it matters what model structure one uses. When dealing with commutative ring spectra, one must use the positive stable model structure, and then one cannot just take the zeroth space. In the noncommutative case, one can use the stable model structure and then take $\Omega^{\infty}$ to mean the zeroth space of a fibrant approximation. In either case, one can use Lind's work to transfer to equivalent EKMM ring spectra, which have highly structured zeroth spaces on the point set level. Here $gl_1(R)$ is an $\mathcal{L}$-space, where $\mathcal{L}$ is the linear isometries operad. Of course, one forgets its permutations in the noncommutative case, so that it is an $A_{\infty}$ operad. One can apply a $1$-fold delooping machine to $gl_1(R)$ or one can construct an equivalent topological monoid and take its usual classifying space. That gives equivalent explicit constructions of $Bgl_1(R)$. One advantage of this approach is that one uses exactly the same direct zeroth space construction in both the commutative and noncommutative cases. This goes back to http://www.math.uchicago.edu/~may/BOOKS/e_infty.pdf; a more modern exposition is in http://www.math.uchicago.edu/~may/PAPERS/Final1.pdf. The concrete construction may help you out.