Let G be a locally compact group. It is shown that there exists a natural completely isometric representation of the completely bounded Fourier multiplier algebra McbA(G), which is dual to the representation of the measure algebra M(G), on B(L2(G)). The image algebras of M(G) and M cbA(G) in CBσ(B(L2(G))) are intrinsically characterized, and some commutant theorems are proved. It is also shown that for any amenable group G, there is a natural completely isometric representation of UCB(Ĝ)ß on B(L2(G)), which can be regarded as a duality result of Neufang's completely isometric representation theorem for LUC(G)ß.

dx.doi.org/10.1090/S0002-9947-07-03940-2
Transactions of the American Mathematical Society
School of Mathematics and Statistics

Neufang, M, Ruan, Z.-J. (Zhong-Jin), & Spronk, N. (Nico). (2008). Completely isometric representations of McbA(G) and UCB(Ĝ)ß. Transactions of the American Mathematical Society, 360(3), 1133–1161. doi:10.1090/S0002-9947-07-03940-2