The notion was introduced by Cuntz and Quillen for the applications to cyclic homology.
[1] A quasi-free algebra generalizes a free algebra, as well as the coordinate ring of a smooth affine complex curve.
Because of the latter generalization, a quasi-free algebra can be thought of as signifying smoothness on a noncommutative space.
denotes the differential envelope of A; i.e., the universal differential-graded algebra generated by A.
Given an A-bimodule E, a right connection on E is a linear map that satisfies
[9] One of basic properties of a quasi-free algebra is that the algebra is left and right hereditary (i.e., a submodule of a projective left or right module is projective or equivalently the left or right global dimension is at most one).
[10] This puts a strong restriction for algebras to be quasi-free.
In particular, a polynomial ring over a field is quasi-free if and only if the number of variables is at most one.
An analog of the tubular neighborhood theorem, called the formal tubular neighborhood theorem, holds for quasi-free algebras.