Infinite loop of a p-completed specta vs p-completion of infinite loop of the spectra Assume that we have a connective spectrum $X$, and denote the $p$-completion of this spectrum in the sense of Bousfield by $X^{\wedge}_p$ (which is given by the function spectrum $F(S^{-1}\mathbb{Z}/p^{\infty},X)$). Let $(\Omega^{\infty}_0X)^{\wedge}_p$ be the profinite $p$-completion of the zeroth component of the infinite loop space of $X$. Is that true that $\Omega^{\infty}_0X^{\wedge}_p\simeq (\Omega^{\infty}_0X)^{\wedge}_p$, meaning that they are weakly equivalent as topological spaces ?
Remark. I was thinking about  two specific spectra $X$ and $Y$ and a map between them which I spare you with their descriptions that I know the map induces equivalence between $p$-completions $X^{\wedge}_p\simeq Y^{\wedge}_p$. I'd like to show $H_*(\Omega^{\infty}_0X,\mathbb{Z}/p)\cong H_*(\Omega^{\infty}_0Y,\mathbb{Z}/p)$. This made me to ask the previous question.
 A: There is a standard natural short exact sequence for the calculation of $\pi_n(X^{\wedge}_p)$ from
$\pi_n(X)$, and it applies whether $X$ is a spectrum or a space.  (I'm assuming you mean
Bousfield completion at $p$ for both).  For spaces that is in Bousfield-Kan, and a more recent treatment
is in More Concise Algebraic Topology [MP], by Kate Ponto and myself.  See [Theorem 11.1.2, MP].
The homotopy groups of $\Omega^{\infty}_0 (X^{\wedge}_p)$ are the positive degree homotopy groups of
$X^{\wedge}_p$ and are $p$-complete in the sense defined in [MP].  Therefore $\Omega^{\infty}_0 (X^{\wedge}_p)$ 
is $p$-complete by [Theorem 11.1.1, MP].  By the universal property of completion at $p$, there is a map 
$(\Omega^{\infty}_0 X)^{\wedge}_p \to \Omega^{\infty}_0 (X^{\wedge}_p)$ under $\Omega^{\infty}_0X$ . It induces isomorphisms on 
homotopy groups, as you want. 
A: Suppose $X$ is of finite type (i.e., $\pi _n(X)$ finitely generated for all $n$) and $0$-connected.
Then one can use Milnor-type exact sequence to show that
$\pi _*(X_p^{\wedge})\cong \pi _*(X)_p^{\wedge}$
On the other hand,  since $X$ is of finite type, so is $\Omega ^{\infty }X$, and
according to Bousfield, Kan, {\it Homotopy Limits, Completions, and Localizations}
(Splinger Lecture Notes in Mathematics 304}, Part I, Chapter VI example 5.2, we 
have 
$\pi _*((\Omega ^{\infty }X)_p^{\wedge})\cong (\pi _*(\Omega ^{\infty }X))_p^{\wedge}$
(recall that we have finite type hypothesis, so tensoring with $p$-adics amounts to completing at $p$) where $(\Omega ^{\infty }X)_p^{\wedge}$ denotes
the Bousfield-Kan $p$-completion of the space $\Omega ^{\infty }X$.  It is also known (loc.cit) that under this assumption, Bousfield-Kan completion agrees with QUillen's or Sullivan's profinite $p$-completion.
Combining all these and the isomorphism
$\pi _*(Y)\cong \pi _*(\Omega ^{\infty}Y)\mbox{ for $*\geq 0$)}$ for any spectra $Y$, we see that $(\Omega ^{\infty }X)_p^{\wedge}$ and
$\Omega ^{\infty }(X_p^{\wedge})$ are weakly equivalent.  
(if you prefer, one can say that the natural map $\Omega ^{\infty }X
\rightarrow \Omega ^{\infty }(X_p^{\wedge})$ factors through $(\Omega ^{\infty }X)_p^{\wedge}$ and the resulting map is a weak equivalence.
