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We consider cycles for graded $C^{*,r}$-algebras (Real $C^{*}$-algebras)which are compatible with the $*$-structure and the real structure. Theircharacters are cyclic cocycles. We define a Connes type pairing between suchcharacters and elements of the van Daele $K$-groups of the $C^{*,r}$-algebraand its real subalgebra. This pairing vanishes on elements of finite order. Wedefine a second type of pairing between characters and $K$-group elements whichis derived from a unital inclusion of $C^{*}$-algebras. It is potentiallynon-trivial on elements of order two and torsion valued. Such torsion valuedpairings yield topological invariants for insulators. The two-dimensionalKane-Mele and the three-dimensional Fu-Kane-Mele strong invariant are specialcases of torsion valued pairings. We compute the pairings for a simple class ofperiodic models and establish structural results for two dimensional aperiodicmodels with odd time reversal invariance.

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