Cayley transform, outer exponential and spinor norm
The authors generalize a construction of Connes by defining for an -bundle over smooth manifold and a reduced cyclic cohomology class a sequence of de Rham cohomology classes . Here is a convenient algebra, defined by the authors, and is a locally trivial bundle with standard fibre a right finitely generated projective -module and bounded -modules homomorphisms as transition functions.
Summary: Arrays of numbers may be written not only on a line (= ``a vector'') or in the plain (= ``a matrix'') but also on a circle (= ``a circular vector''), on a torus (= ``a toroidal matrix'') etc. In the latter case, the immanent index-rotation ambiguity converts the standard ``scalar'' product into a binary operation with several interesting properties.