We study the "quantized calculus" corresponding to the algebraic ideas related to "twisted cyclic cohomology" introduced in [12]. With very similar definitions and techniques as those used in [9], we define and study "twisted entire cyclic cohomology" and the "twisted Chern character" associated with an appropriate operator theoretic data called "twisted spectral data", which consists of a spectral triple in the conventional sense of noncommutative geometry [1] and an additional positive operator having some specified properties. Furthermore, it is shown that given a spectral triple (in the conventional sense) which is equivariant under the (co-) action of a compact matrix pseudogroup, it is possible to obtain a canonical twisted spectral data and hence the corresponding (twisted) Chern character, which will be invariant (in the usual sense) under the (co-)action of the pseudogroup, in contrast to the fact that the Chern character coming from the conventional noncommutative geometry need not to be invariant. In the last section, we also try to detail out some remarks made in [3], in the context of a new definition of invariance satisfied by the conventional (untwisted) cyclic cocycles when lifted to an appropriate larger algebra.
Index map, $\sigma$ -connections, and Connes–Chern character in the setting of twisted spectral triples
