Introverted algebras with mean value

Let A be an introverted algebra with mean value. We prove that its spectrum \Delta (A) is a compact topological semigroup, and that the kernel K(\Delta (A)) of \Delta (A) is a compact topological group over which the mean value on A can be identified as the Haar integral. Based on these facts and also on the fact that K(\Delta (A)) is an ideal of \Delta (A), we define the convolution over \Delta (A). We then use it to derive some new convergence results involving the convolution product of sequences. These convergence results provide us with an efficient method for studying the asymptotics of nonlocal problems. The obtained results systematically establish the connection between the abstract harmonic analysis and the homogenization theory. To illustrate this, we work out some homogenization problems in connection with nonlocal partial differential equations.