Let $S$ and $\rm ku$ denote the sphere spectrum and the connective K-theory spectrum, respectively.


Then is the morphism ${\rm ku} \to F(F({\rm ku},S),S)$ an equivalence?
Here $F$ denotes the mapping spectrum.



Is ku reflexive as a spectrum?
Writing $D(-) = F(-, S)$, I believe that there is an equivalence $$D ku \simeq \prod\_{n=0}^\infty \Sigma^{-2n-1} H(\widehat{\mathbb Z} / \mathbb Z),$$ where $\widehat{\mathbb Z}$ is the profinite completion of the integers. This would split as a similar product after dualizing a second time, so can't be equivalent to $ku$.


The claimed equivalence is a combination of the following facts:


1. $D H \mathbb Q \simeq \Sigma^{-1} H \widehat{\mathbb Z} / \mathbb Z$: Writing $H \mathbb Q = \operatorname{colim}( \cdots \to S \xrightarrow n S \to \cdots)$, we find $D H \mathbb Q = \lim( \cdots \leftarrow S \xleftarrow n S \leftarrow \cdots)$, and the conclusion follows from the Milnor sequence and the finite torsion exponents of the higher homotopy groups of spheres.
2. $D H A \simeq 0$ for $A$ torsion abelian: Since $S$ is harmonic, $D H \mathbb F\_p \simeq 0$, and $A$ torsion abelian has a filtration whose quotients are prime torsion.
3. $D H \mathbb Q \simeq D H \mathbb Z$: There's a fiber sequence $$D H \mathbb Z \leftarrow D H \mathbb Q \leftarrow D H(\mathbb Q / \mathbb Z),$$ and the last spectrum is null by the above.
4. For $X$ coconnective, $D X \simeq D (\mathbb Q \otimes X)$: $X$ can be written as the colimit of a filtration whose quotients are Eilenberg–Mac Lanes, to which one applies the previous facts.
5. $L\_{H \mathbb Z} KU \simeq \mathbb Q \otimes KU$: Using Snaith's theorem $KU \simeq \Sigma^\infty\_+ \mathbb CP^\infty[\beta^{-1}]$, one calculates the integral homology of $KU$ to be rational.
6. $D KU \simeq \prod\_{n=-\infty}^\infty \Sigma^{-2n-1} H(\widehat{\mathbb Z} / \mathbb Z)$: Since $S$ is $H\mathbb Z$–local, $DKU \simeq D(L\_{H\mathbb Z} KU) \simeq D(\mathbb Q \otimes KU)$. Rational spectra are Eilenberg–Mac Lane, hence determined by their homotopy groups, so one combines Bott periodicity and the first fact.


The main claim then follows from the fiber sequence $$D ku \leftarrow D KU \leftarrow D(\tau\_{< 0} KU).$$ We've analyzed the middle term, and the last term starts off coconnective so (after applying the above facts) turns out to be the connective part of $D KU$, which finally determines the first term to be the remaining coconnective part.


