Amenable Banach Algebras by Volker Runde

Author:Volker Runde
Language: eng
Format: epub
ISBN: 9781071603512
Publisher: Springer New York

(iii)

is a normal, dual Banach M(G)-bimodule.

Proof.

To prove (i), let , and note that

As by Proposition D.4.2(iii), the norm continuity of thus yields that . To show that , let . Since —identified with its canonical image in —is dense in , there is a net in such that . Passing to a subnet, we may suppose that converges to some or tends to (which, of course, can only occur if G is not compact). In the first case, we have

and in the second one

In either case, we holds, which proves (i).

For the proof of (ii), let , and let . With standing for the bimodule action of M(G) on defined in (5.11) and (5.12), it is routinely checked that . Let . As

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