Non-degeneracy results for (multi-)pushouts of compact groups

The fact that embeddings of W∗-algebras are equalizers implies the analogous results for both compact groups and C∗-algebras, due respectively to Poguntke and Hofmann-Neeb. We also prove a number of results on pushouts (and more generally, amalgamated free products) in the category of compact groups. Call a family of compact-group embeddings H≤Gi algebraically sound if the corresponding group-theoretic pushout embeds in its Bohr compactification. We (a) show that a family of normal embeddings is algebraically sound in the sense that Gi admit embeddings Gi≤G into a compact group which agree on H; (b) give equivalent characterizations of coherently embeddable families of normal embeddings in representation-theoretic terms, via Clifford theory; (c) characterize those compact connected Lie groups H for which all finite families of normal embeddings H⊴Gi are coherently embeddable (not having central 2-tori is a sufficient, but not necessary condition), and (d) show that families of split embeddings of compact groups are always algebraically sound.