Group actions on algebras obtained by formal deformation quantization are the main topic of this
thesis. We study these actions in order to obtain an equivariant algebraic index theorem that leads to
explicit formulas in terms of equivariant characteristic classes. The Fedosov construction, as realized
in a deformed version of Gelfand's formal geometry, is used to obtain the results.
We describe the main points of Gelfand's formal geometry in the deformed case and show how it
leads to Fedosov connections and the well-known classication of formal deformation quantization in
the direction of a symplectic structure.
A group action on a deformation quantization induces an action on the underlying symplectic
manifold. We consider the lifting problem of nding group actions inducing a given action by symplectomorphisms.
We reformulate some known sucient conditions for existence of a lift and show
that they are not necessary. Given a particular lift of an action by symplectomorphisms to the deformation
quantization, we obtain a classication of all such lifts satisfying a certain technical condition.
The classication is in terms of a rst non-Abelian group cohomology. We supply tools for computing
these sets in terms of a commuting diagram with exact rows and columns. Finally we consider some
examples to formulate vanishing and non-vanishing results.
In joint work with A. Gorokhovsky and R. Nest, we prove an equivariant algebraic index theorem.
The equivariant algebraic index theorem is a formula expressing the trace on the crossed product
algebra of a deformation quantization with a group in terms of a pairing with certain equivariant
characteristic classes. The equivariant characteristic classes are viewed as classes in the periodic cyclic
cohomology of the crossed product by using the inclusion of Borel equivariant cohomology due to
Connes.