  In this paper we propose and partially carry out a program to use $K$-theory
to refine the topological realization problem of unstable algebras over the
Steenrod algebra. In particular, we establish a suitable form of algebraic
models for $K$-theory of spaces, called $\psi^p$-algebras, which give rise to
unstable algebras by taking associated graded algebras mod $p$. The
aforementioned problem is then split into (i) the \emph{algebraic} problem of
realizing unstable algebras as mod $p$ associated graded of $\psi^p$-algebras
and (ii) the \emph{topological} problem of realizing $\psi^p$-algebras as
$K$-theory of spaces. Regarding the algebraic problem, a theorem shows that
every connected and even unstable algebra can be realized. We tackle the
topological problem by obtaining a $K$-theoretic analogue of a theorem of Kuhn
and Schwartz on the so-called Realization Conjecture.

Algebraic Topology