ALGEBRA ENVIRONMENTS I. GEOMETRIC AND TOPOLOGICAL STRUCTURES.

Algebra environments capture properties of non-commutative conditional expectations in a general algebraic setting. Their study relies on algebraic geometry, topology, and differential geometry techniques. The structure algebraic and Banach manifolds of algebra environments and their Zariski and smooth tangent vector bundles are particular objects of interest. A description of derivations on algebra environments compatible with geometric structures is an additional issue. Grassmann and flag manifolds of unital involutive algebras and spaces of projective compact group representations in C^* -algebras are analyzed as structure manifolds of associated algebra environments. [ABSTRACT FROM AUTHOR]