  We study string topology for classifying spaces of connected compact Lie
groups, drawing connections with Hochschild cohomology and equivariant homotopy
theory. First, for a compact Lie group $G$, we show that the string topology
prospectrum $LBG^{-TBG}$ is equivalent to the homotopy fixed-point prospectrum
for the conjugation action of $G$ on itself, $G^{hG}$. Dually, we identify
$LBG^{-ad}$ with the homotopy orbit spectrum $(DG)_{hG}$, and study ring and
co-ring structures on these spectra. Finally, we show that in homology, these
products may be identified with the Gerstenhaber cup product in the Hochschild
cohomology of $C^*(BG)$ and $C_*(G)$, respectively. These, in turn, are
isomorphic via Koszul duality.
