Abstract
Although this is a continuation of the previous two parts, it may be studied independently of my earlier work (1980, 2001) and the necessary results will be briefly restated. An extended early section motivates the problems from a finite state space to the general case via a discussion of random walks, or equivalently convolution operators and their structural analysis. This naturally leads to a study of the latter operators on certain function spaces and function algebras. It also shows a need to consider the (algebraic) structure of the state space of random walks, namely an analysis of the underlying locally compact groups and the dependence on the spectral analysis of the associated convolution operators on function spaces built on them. In the nonabelian group case (of the state space of the walks) the analysis is intimately related to amenability of the group, which is the range or state space of the random walk.
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Rao, M.M. (2004). Convolutions of Vector Fields-III: Amenability and Spectral Properties. In: Rao, M.M. (eds) Real and Stochastic Analysis. Trends in Mathematics. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-2054-1_7
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DOI: https://doi.org/10.1007/978-1-4612-2054-1_7
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