Abstract. Operator decomposable probabilities on vector spaces -generalizing (semi-)stable and self-decomposable laws -are well known. More specific concepts, multiple operator decomposable laws, generalizing the nested Urbanik classes in the case of selfdecomposability, were investigated in fundamental papers by Maejima et al. [16], [17] resp. by Maejima and R. Shah [18] for real resp. p-adic vector spaces. For locally compact groups, decomposability properties were studied by R. Shah [27], Raja [23], see also [6]. Here we are concerned with multiple decomposability on locally compact groups and generalize -as far as possible -the results in [16], [18]. In fact, as it turned out that contraction properties play an essential role, hence throughout we concentrate on contractible locally compact groups.

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