  We define a genuine $\mathbb{Z}/2$-equivariant real algebraic $K$-theory
spectrum $KR(A)$, for every genuine $\mathbb{Z}/2$-equivariant spectrum $A$
equipped with a compatible multiplicative structure. This construction extends
the real $K$-theory of Hesselholt-Madsen for discrete rings and the Hermitian
$K$-theory of Burghelea-Fiedorowicz for simplicial rings. We construct a
natural trace map of $\mathbb{Z}/2$-spectra $tr\colon KR(A)\to THR(A)$ to the
real topological Hochschild homology spectrum, which extends the $K$-theoretic
trace of B\"okstedt-Hsiang-Madsen.
  The trace provides a splitting of the real $K$-theory of the spherical
group-ring. We use this splitting on the geometric fixed points of $KR$, which
we regard as an $L$-theory of genuinely equivariant ring spectra, to
reformulate the Novikov conjecture on the homotopy invariance of the higher
signatures purely in terms of the module structure of the rational $L$-theory
of the "Burnside group-ring".
