  The rank-three tensor models, which have a rank-three tensor as their only
dynamical variable, may be interpreted as models of dynamical fuzzy spaces. In
this interpretation, the generalized Hermiticity condition on the rank-three
tensor leads to a cyclic property of the algebra of functions on fuzzy spaces.
The fuzzy spaces with the cyclic property are shown to have various physically
interesting characteristics. (i) Although the function algebras of the kind are
nonassociative in general, various properties analogous to quantum mechanics
hold on the fuzzy spaces. (ii) The symmetry of the rank-three tensor models can
be shown to be represented systematically by n-ary transformations on the fuzzy
spaces. The transformations contain, for instance, diffeomorphism on fuzzy
spaces. (iii) There exists a systematic procedure of truncating the function
algebras of the kind, and it can be used to consider subspaces,
compactifications, lattice theories, and coarsegraining procedures of fuzzy
spaces in physical applications.
