, Volume 48, Issue 1-3, pp 291-295

Algebra involutions on the bidual of a Banach algebra

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Let A be a Banach algebra with bounded approximate right identity. We show that a necessary condition for the bidual of A to admit an algebra involution (with respect to the first Arens product) is that A*A=A*, i.e. the dual of A has to be essential as a right A-module. In particular, for any infinite, non-discrete, locally compact Hausdorff group G, L1(G)** does not admit any algebra involution with respect to either Arens product. This implies that the main result of a paper of R.S. Doran and W. Tiller concerning L1(G)** as Banach *-algebra (see [DT]) applies only to the trivial case of finite abelian groups.

Essentially, the proof of the foregoing lemma is due to H.Rindler.