Joint continuity of multiplication on the dual of the left uniformly continuous functions

摘要

Let G be a locally compact group and LUC(G) the C*-algebra of the bounded left uniformly continuous functions on G. The spectrum G LUC of LUC(G) is the universal semigroup compactification of G with respect to the joint continuity property: the multiplication on G×G LUC is jointly continuous. The paper studies the joint weak* continuity of multiplication on LUC(G)* and, in particular, the question how the joint continuity property of G LUC can be related to a property concerning the whole algebra LUC(G)*. The group G is naturally replaced by the measure algebra M(G), and LUC(G)* can be identified with M(G LUC), the space of regular Borel measures on G LUC. It is shown that the joint weak* continuity can fail even on bounded sets of M(G)×M(G LUC), but, on the other hand, the multiplication on M(G)×M(G LUC) is positive continuous in the sense of Jewett. Keywords Left uniformly continuous function - Convolution - Joint continuity property - Positive continuous - Topological centre Communicated by Jimmie D. Lawson.