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

Abstract

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 L2(G) is isomorphic to LP(G') (2 < p < oo), a statement we conjecture to be true for every pair LP(G), Lq (G') of group rings corresponding to different exponents 1 < p, q < oo. The study of ring homomorphisms between Banach algebras has a long history. Probably the first such result is due to Eidelheit [Eid] who proved that every ring isomorphism between the algebras of all bounded linear operators acting on real Banach spaces is implemented by an invertible operator and hence it is continuous and linear. For the case of complex spaces and a recent far-reaching generalization concerning not only ring but semigroup isomorphisms of standard operator algebras we refer to [Arn] and [Sem], respectively. Moreover, in the case of general semisimple Banach algebras, the most important result on the linearity of ring isomorphisms is due to Kaplansky [Kap]. Motivated by the theory of operator ranges, in our papers [Mol2], [Mol3] we studied homomorphism ranges and considered surjective and almost surjective ring homomoprhisms (continuity is never assumed) between some operator algebras and function algebras, respectively. As the main corollary in [Mol3], we obtained the existence of ring theoretical differences among some important function algebras. The result of [Mol2] is in a close relation to our present investigations. It states that there is no surjective ring homomorphism between different p-classes of an infinite-dimensional H*-algebra. These classes are common generalizations of the p-classes of compact operators and Ip spaces. Our mentioned result shows that Received by the editors March 4, 1997 and, in revised form, March 10, 1998. 1991 Mathematics Subject Classification. Primary 46K15, 47D50, 47B49; Secondary 43A15, 43A22.