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We prove a 
$\unicode[STIX]{x1D6E4}$
-equivariant version of the algebraic index theorem, where 
$\unicode[STIX]{x1D6E4}$
is a discrete group of automorphisms of a formal deformation of a symplectic manifold. The particular cases of this result are the algebraic version of the transversal index theorem related to the theorem of A. Connes and H. Moscovici for hypo-elliptic operators and the index theorem for the extension of the algebra of pseudodifferential operators by a group of diffeomorphisms of the underlying manifold due to A. Savin, B. Sternin, E. Schrohe and D. Perrot.
We define the filtrated
$\text{K}$
-theory of a
${{\text{C}}^{*}}$
-algebra over a finite topological space
$X$
and explain how to construct a spectral sequence that computes the bivariant Kasparov theory over
$X$
in terms of filtrated
$\text{K}$
-theory.
For finite spaces with a totally ordered lattice of open subsets, this spectral sequence becomes an exact sequence as in the Universal Coefficient Theorem, with the same consequences for classification.
We also exhibit an example where filtrated
$\text{K}$
-theory is not yet a complete invariant. We describe two
${{\text{C}}^{*}}$
-algebras over a space
$X$
with four points that have isomorphic filtrated
$\text{K}$
-theory without being
$\text{KK}\left( X \right)$
-equivalent. For this space
$X$
, we enrich filtrated
$\text{K}$
-theory by another
$\text{K}$
-theory functor to a complete invariant up to
$\text{KK}\left( X \right)$
-equivalence that satisfies a Universal Coefficient Theorem.
In this paper we shall compute the cyclic cohomology of a non-commutative torus, i.e., a certain algebra 
associated with an antisymmetric bicharacter of a finite rank free abelian group G.
The main result is
1.1
where
The method of computation generalises the computation of the cyclic cohomology of the irrational rotation algebras given by Connes in [3]. (Our method works equally well also in the rational case, which was dealt with by a different method by Connes in [3].)
We first describe the Hochschild cohomology of in an explicit way, and then combine this description with the exact sequence of [3]:
1.2
