Sheep anti Human Protein C polyclonal antibody [HRP]

We establish some geometric results on K-theory with coefficients in Z/kZ. The first one is a new proof of the Atiyah-Patodi-Singer mod k index theorem (Math Proc Camb Philos Soc 79:71-99, 1976) in the case of Dirac operators, i.e. in a geometric situation. The second one is a Grothendieck-Riemann-Roch theorem for Z/k Z-K-theory.

Let M be a factor acting on a complex, separable Hilbert space H. An operator a is an element of M is said to be irreducible in M if W*(a), the von Neumann subalgebra generated by a in M, is an irreducible subfactor of M, i.e., W*(a)' boolean AND M = CI. In this note, we prove that each operator a is an element of M is a sum of two irreducible operators in M, which can be viewed as a natural generalization of a theorem in [Proc. Amer. Math. Soc. 21 (1969), pp. 251-252], with a completely different proof.