Injectivity of the modules Lp (G)

01/17/2012 - 4:00pm
01/17/2012 - 5:00pm
H.G. Dales

Let A be a Banach algebra, and let E be a Banach A-bimodule. A basic question is to determine when E is injective (in a sense that I shall explain). A fundamental result of Helemski is that E is injective whenever A is an amenable Banach algebra and E is a dual module. The following has been open for a long time: Suppose that all (or some specific) dual Banach A- bimodules are injective. Does it always follow that A is amenable? Let G be a locally compact group, and let L1(G) be the standard group algebra. Then Lp(G) is a Banach L1(G)-bimodule for each p ≥ 1, and it is a dual module when p > 1. Suppose that G is an amenable group. Then L1(G) is an amenable Banach algebra, and so Lp(G) is injective for each p > 1. We can now show that, conversely, G is amenable whenever Lp(G) is injective for some p > 1. Our approach is to use the theory of ‘multi-norms’, which I shall go over briefly. The talk is based on the following paper: H. G. Dales, M. Daws, H. L. Pham, and P. Ramsden, Multi-norms and the injectivity of Lp(G), J. London Math. Soc., to appear.

Davidson Lecture Hall, Claremont McKenna College
Analysis Seminar-Dales.pdf46.79 KB

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