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Paper IPM / M / 16074  


Abstract:  
Let \Bbb G be a locally compact quantum group and let I be closed ideal of L^{1}(\Bbb G) with y_{I} ≠ 0 for some y ∈ sp(L^{1}(\Bbb G)). In this paper, we give a characterization for compactness of \Bbb G in terms of the existence of a weakly compact left or right multiplier T on I with T(f)(y_{I}) ≠ 0 for some f ∈ I. Using this, we prove that I is an ideal in its second dual if and only if \Bbb G is compact. We also study Arens regularity of I whenever it has a bounded left approximate identity. Finally, we obtain some characterizations for amenability of \Bbb G in terms of the existence of some Imodule homomorphisms on I^{**} and on I^{*}.
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