Cyclic cohomology for graded $C^{\ast ,r}$-algebras and its pairings with van Daele $K$-theory

We consider cycles for graded $C^{\ast ,r}$-algebras (Real $C^{\ast }$-algebras) which are compatible with the $\ast$-structure and the real structure. Their characters are cyclic cocycles. We define a Connes type pairing between such characters and elements of the van Daele $K$-groups of the $C^{\ast ,r}$-algebra and its real subalgebra. This pairing vanishes on elements of finite order. We define a second type of pairing between characters and $K$-group elements which is derived from a unital inclusion of $C^{\ast }$-algebras. It is potentially non-trivial on elements of order two and torsion valued. Such torsion valued pairings yield topological invariants for insulators. The two-dimensional Kane-Mele and the three-dimensional Fu-Kane-Mele strong invariant are special cases of torsion valued pairings. We compute the pairings for a simple class of periodic models and establish structural results for two dimensional aperiodic models with odd time reversal invariance.