at.algebraic topology - Values of the multiplicative group over a ring spectrum

It is slightly complicated.

One has a number of adjunctions:
$$
begin{eqnarray*}
mathbb{G}_m(B) &=& Hom_{A-alg}(A[mathbb{Z}],B) \
&simeq& Hom_{E_infty-rings}(mathbb{S}[mathbb{Z}],B)\
&simeq& Hom_{E_infty-spaces}(mathbb{Z},GL_1(B))\
&simeq& Hom_{spectra}(Hmathbb{Z},gl_1(B)).
end{eqnarray*}
$$
(Note these adjunctions are weak equivalences of spaces, and the last two adjunctions require a fair amount of theory to make rigorous.)

The problem is that it is usually quite difficult to compute the maps out of the Eilenberg-Mac Lane spectrum $Hmathbb{Z}$ unless the target is also an Eilenberg-Mac Lane space. In the case where the algebra $B$ comes from a simplicial commutative ring, this is true and so one at least knows that the set of homotopy classes of maps $[Hmathbb{Z}, gl_1(B)]$ surjects onto $pi_0(B)^times$. Even for complex K-theory, the calculation is somewhat involved (but doable), but the only method that I can immediately think of involves the Bousfield-Kuhn functor.