In this paper we apply the theory of quasi-free states of CAR algebras and their Bogolubov automorphisms to give an alternative C*algebraic construction of tthe determinant and pfaffian line bundles discussed by Pressley-Segal and by Borthwick. The basic property of the pfaffian of being the holomorphic square root of the determinant bundle (after restriction to the isotropic grassmannian) is derived from a Fock-anti-Fock correspondence and an application of the Powers-Stormer purification procedure. A Borel-Weil type description of the infinite dimensional Spin^c-representation is discussed, via a Shale-Stinespring implementation of Bogolubov transformations.