This is really just a comment, but a bit too long. Thanks Tyler. Ulrich, that is a good question. I once worked hard to understand the zeroth space of MU (and more simply BP), one prime at a time, calculating Dyer Lashof operations. I threw away my notes because the answers I was getting seemed unhelpful. I seem to recall a paper of Steve Wilson that considers the Hopf ring of BP_0, but my memory may be failing me. The problem is that these spaces are very large and seem to have nothing like the economical and intuitive description of their K-theory analogues. It could be a good problem to try to understand what these infinite loop spaces really look like and what they mean geometrically. A short modern summary of early work on GL_1(R), alias SFR, in general is in [114] on my web page. There is work in progress by John Lind that makes honest the idea that BGL_1R is a classifying space for a reasonably concrete kind of ``principal GL_1R-bundle''.

This is really just a comment, but a bit too long. Thanks Tyler. Ulrich, that is a good question. I once worked hard to understand the zeroth space of MU (and more simply BP), one prime at a time, calculating Dyer Lashof operations. I threw away my notes because the answers I was getting seemed unhelpful. I seem to recall a paper of Steve Wilson that considers the Hopf ring of BP_0, but my memory may be failing me. The problem is that these spaces are very large and seem to have nothing like the economical and intuitive description of their K-theory analogues. It could be a good problem to try to understand what these infinite loop spaces really look like and what they mean geometrically. A short modern summary of early work on GL_1(R), alias FR, in general is in [114] on my web page. There is work in progress by John Lind that makes honest the idea that BGL_1R is a classifying space for a reasonably concrete kind of ``principal GL_1R-bundle''.