  Representations of CCR algebras in spaces of entire functions are classified
on the basis of isomorphisms between the Heisenberg CCR algebra A_H and star
algebras of holomorphic operators. To each representations of such algebras,
satisfying a regularity and a reality condition, one can associate isomorphisms
and inner products so that they become Krein star representations of A_H, with
the gauge transformations implemented by a continuous U(1) group of Krein
isometries. Conversely, any holomorphic Krein representation of A_H, having the
gauge transformations implemented as before and no null subrepresentation, is
shown to be contained in a direct sum of the above representations. The
analysis is extended to infinite dimensional CCR algebras, under a spectral
condition for the implementers of the gauge transformations.
