  Functional integrals are defined in terms of locally compact topological
groups and their associated Banach-valued Haar integrals. This approach
generalizes the functional integral scheme of Cartier and DeWitt-Morette. The
definition allows a construction of functional Mellin transforms. In turn, the
functional Mellin transforms can be used to define functional traces,
logarithms, and determinants. The associated functional integrals are useful
tools for probing function spaces in general and $C^\ast$-algebras in
particular. Several interesting aspects are explored.
