Completely isometric representations of 𝑀_{𝑐𝑏}𝐴(𝐺) and 𝑈𝐶𝐵(𝐺̂)*

Neufang, Matthias; Ruan, Zhong-Jin; Spronk, Nico

doi:10.1090/S0002-9947-07-03940-2

Completely isometric representations of $M_{cb}A(G)$ and $UCB(\hat G)^*$
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by Matthias Neufang, Zhong-Jin Ruan and Nico Spronk PDF

Trans. Amer. Math. Soc. 360 (2008), 1133-1161 Request permission

Abstract:

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 $M_{cb}A(G)$, which is dual to the representation of the measure algebra $M(G)$, on $\mathcal {B}(L_2(G))$. The image algebras of $M(G)$ and $M_{cb}A(G)$ in $\mathcal {CB}^{\sigma } (\mathcal {B}(L_2(G)))$ 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 $UCB(\hat G)^*$ on $\mathcal {B}(L_2(G))$, which can be regarded as a duality result of Neufang’s completely isometric representation theorem for $LUC(G)^*$.

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