Hochschild homology criteria for trivial algebra structures

We prove two similar results by quite different methods. The first one deals with augmented artinian algebras over a field: we characterize the trivial algebra structure on the augmentation ideal in terms of the maximality of the dimensions of the Hochschild homology (or cyclic homology) groups. For the second result, let X be a 1-connected finite CW-complex. We characterize the trivial algebra structure on the cohomology algebra of X with coefficients in a fixed field in terms of the maximality of the Betti numbers of the free loop space.