In [12], Loy and Miller proved that a locally compact, eudoxian TR group is algebraically and order-theoretically (and hence, topologically) isomorphic to a finite product of copies of the real numbers. In [18], Wirth used their result to describe the subgroup of a locally compact TR group generated by the compact neighbourhoods of zero. The proof of Loy and Miller relied heavily on a result of Mackey (cf. [10], p. 390) and either the finite-dimensional case of the Choquet-Kendall Theorem (cf. [15], pp. 9–10) or the representation theory of Kakutani (cf. [11], Appendix). Below we use only elementary topological results and order-theoretic arguments and a theorem of Conrad [4] to characterize all non-secular, locally compact TRL groups (Theorem 3). Our proof of Theorem 3 allows us to deduce algebraically the theorems both of Loy and Miller and of Wirth, in both cases without appealing to the theorem of Conrad.
Representation theory of the algebra generated by a pair of complex structures
