Projective limits and balanced topological groups

The purpose of these notes is to provide background material for a Workshop in Topological Groups given at the Summer Topology Conference at the C.W. Post campus of Long Island University during the period August 4–7, 1999. We will be looking at the two concepts of ‘balanced groups’ and ‘functionally balanced groups’. In 1992 a paper was published by the author in conjunction with S. Rothman, H. Strassberg, and T.S. Wu where it was shown that the two concepts are equivalent for locally compact groups. anced group” is one in which the left and right uniform structures are equivalent and a “functionally balanced group” is one on which the classes of left uniformly continuous bounded real valued functions and those that are right uniformly continuous coincide. In our paper we were able to use projective limits of Lie groups to establish the desired result. This endeavor together with a classical theorem of Graev suggested that this result might be extended to those groups that are projective limits of metric groups. tes consist of four sections: 1. tive limits of topological groups. onally balanced groups. tive limits of groups. ript, the work of Protasov and Saryev and the state of the problem. n 1 is a quick introduction with proofs to the concept of a projective limit of topological groups. Section 2 introduces the concept of a functionally balanced group. It describes an important characterization of such groups due to Protasov and Saryev and an important corollary of this characterization. Section 3 has a complete proof of Graevʹs motivating theorem and obtains a number of theorems, which to my knowledge are new, concerning projective limits of metric groups. Finally, Section 4 contains an account of a miniconference held at C.W. Post after the workshop ended and includes a proof of the Protasov–Saryev characterization which has not been available previously in English and also a result of Protasov that reduces the whole problem of ‘functional balance’ versus ‘balance’ to the solution of a simply stated question.