Twisted K theory with coefficients in a C* algebra and obstructions against positive scalar curvature metrics We introduce the notion of twisted Hilbert A module bundles to construct a geometric realization of twisted K theory with coefficients in a C* algebra A. Like ordinary bundles of Hilbert A modules provide a generalization of vector bundles twisted Hilbert A module bundles extend the concept of modules over bundle gerbes. Their corresponding Grothendieck group is isomorphic to K_0(C(M S)) where S denotes a bundle of C* algebras such that its structure group reduces to PU(A) = U(A) U(1). In case S is a bundle of matrix algebras the above description boils down to ordinary twisted K theory therefore K_0(C(M S)) can be understood as twisted K theory with coefficients in A ...