  Several authors have recently constructed characteristic classes for classes
of infinite rank vector bundles appearing in topology and physics. These
include the tangent bundle to the space of maps between closed manifolds, the
infinite rank bundles in the families index theorem, and bundles with
pseudodifferential operators as structure group. In this paper, we construct
the corresponding K-theories for these types of bundles. We develop the
formalism of these theories and use their Chern character to detect a large
class of nontrivial elements.
