Let $k$ denote an algebraically closed field of characteristic $0$. Suppose $K=\bigoplus\_{i\geq 0}K(i)$ is a Hopf $k$-algebra which admits a connected Hopf-grading (that is, a grading which is both an algebra and coalgebra grading, with $K(0)=k$). Call such a Hopf algebra a connected Hopf-graded Hopf algebra.


Connected Hopf-graded Hopf $k$-algebras arise naturally in algebraic toplogy when studying the cohomology rings (with coeffecients in $k$) of $H$-spaces. I assume (although I'm not 100% sure), that not all such Hopf algebras arise as cohomology algebras in this way. My question is therefore the following:


Let $K$ be a connected Hopf-graded Hopf algebra. Which additional properties on $K$ guarantee that it can be viewed as the cohomology ring $K^{\*}(X;k)$ of some $H$-space $X$?



Graded Hopf algebras and H-spaces
Any Hopf algebra of the form $H^{\bullet}(X, k)$ is necessarily graded commutative, and in addition to the conditions you've given so far, the only remaining condition is a mild cardinality condition. (You do not need to assume that $k$ is algebraically closed, only that it has characteristic zero.) 


Any such Hopf algebra $K^{\bullet}$ is isomorphic, as an algebra, to the symmetric algebra (in the graded sense) of some graded vector space $V = \bigoplus\_{n \ge 1} V\_n$ over $k$. If this graded vector space has a graded predual $W = \bigoplus\_{n \ge 1} W\_n$ (so that $V\_n \cong W\_n^{\ast}$), which in particular is the case if each $V\_n$ is finite-dimensional, then this is the cohomology of the product of Eilenberg-MacLane spaces 


$$X = \prod\_{n \ge 1} K(W\_n, n)$$ 


and now the comultiplication $\psi : K^{\bullet} \to K^{\bullet} \otimes K^{\bullet}$ induces an H-space structure $X \times X \to X$ in the obvious way. Otherwise, I think some fiddling with universal coefficients shows that no candidate $X$ exists. 


This correspondence can be upgraded to an equivalence of categories. In the case $k = \mathbb{Q}$ see, for example, May and Ponto's *More Concise Algebraic Topology*, Theorem 9.1.4. 


