  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.
