The harmonic decomposition in cyclic homology

It has been shown by Cuntz and Quillen that in characteristic $0$ the kernel of the square of the “noncommutative Laplacian” on the Hochschild and cyclic complexes contains the relevant homology information. In this note, we show that the same property holds for the plain kernel of this Laplacian, as in differential geometry. Using the same ideas, we define a variant of Hochschild homology and cyclic homology and show that we recover the classical definitions in characteristic $0$.