  We compute the homology of the space of equivariant loops on the classifying
space of a simplicial monoid $M$ with anti-involution, provided $\pi_0 (M)$ is
central in the homology ring of $M$. The proof is similar to McDuff and Segal's
proof of the group completion theorem.
  Then we compute the homology of the $C_2$-fixed points of a Segal-type model
of the algebraic $K$-theory of an additive category with duality. As an
application we show that this fixed point space is sometimes group complete,
but not in general.
