TITLE: How often does suspension define an action of Z/2 on a category of module spectra?
QUESTION [13 upvotes]: Let R be the 2-periodic complex K-theory spectrum, or any other naturally occuring 2-periodic E-infty ring spectrum.  The suspend-once functor gives an autoequivalence of the category of R-module spectra, and since R is 2-periodic applying it twice is isomorphic to the identity functor.  In some coarse sense this defines an action of Z/2 on the category of R-modules.  
There is a better, more complicated and more correct notion of Z/2-action on a collection of module spectra.  Module spectra form an infty-category, and we can ask if Z/2 acts on this infty-category.  One way to put it: the suspend-once functor and the homotopy between suspend-twice and the identity give us a map from RP^2 into the classifying space of automorphisms of this infty-category.  Does this extend to a map from BZ/2 = RP^infty?

REPLY [12 votes]: As Jacob says, to produce a map $RP^\infty\to B\mathrm{Pic}(KU)$, we should use the fact that there is an infinite loop splitting $\mathrm{Pic}(KU)=\mathrm{Cliff}_{\mathbb{C}} \times Y$, where $Y$ is the 3-connected cover.
So, when I got to do the exercise of computing homotopy classes of maps $RP^\infty\to B\mathrm{Cliff}_{\mathbb{C}}$ (using knowledge of the $\infty$-loop k-invariants, as decribed in the other answers),  I get $[RP^\infty, B\mathrm{Cliff}_{\mathbb{C}}] = \mathbb{Z}/8$: each stage in the postnikov tower of $B\mathrm{Cliff}$ adds a factor of $2$, and all the extensions are non-trivial.  In particular, the elements which are congruent to $1$ mod 2 in $\mathbb{Z}/8$ are non-trivial on the bottom cell.  So, if I've done the calculation right (and that's a big if), there are four distinct ways to make $\mathbb{Z}/2$ act so that the generator acts by suspension.
I find it interesting that I get $\mathbb{Z}/8$, which reminds me of real Bott periodicity.  In fact, the calculations make it look like
$$\mathrm{map}_*(RP^\infty, B\mathrm{Cliff}_{\mathbb{C}}) \approx \mathrm{Cliff}_{\mathbb{R}}$$
as infinite loop spaces. 
Is there a way to take a real Clifford algebra $A$, and functorially produce from it a monoidal $\mathbb{Z}/2$-action on (the symmetric monodial Morita $2$-groupoid of) complex Clifford algebras?