Automatic surjectivity of ring homomorphisms on $H^{*}$-algebras and algebraic differences among some group algebras of compact groups

In this paper we present two automatic surjectivity results concerning ring homomorphisms between $p$-classes of an $H^{*}$-algebra which, in some sense, improve the main theorem in a recent paper by the author (Proc. Amer. Math. Soc. 124 (1996), 169-175) quite significantly. Furthermore, we apply our results to show that for arbitrary infinite compact groups $G,G'$, no quotient ring of $L^{2}(G)$ is isomorphic to $L^{p}(G')$ $(2<p\leq \infty )$, a statement we conjecture to be true for every pair $L^{p}(G), L^{q}(G')$ of group rings corresponding to different exponents $1\leq p,q\leq \infty$.