  A module endomorphism $f$ on an algebra $A$ is called an averaging operator
if it satisfies $f(xf(y)) = f(x)f(y)$ for any $x, y\in A$. An algebra $A$ with
an averaging operator $f$ is called an averaging algebra. Averaging operators
have been studied for over one hundred years. We study averaging operators from
an algebraic point of view. In the first part, we construct free averaging
algebras on an algebra $A$ and on a set $X$, and free objects for some
subcategories of averaging algebras. Then we study properties of these free
objects and, as an application, we discuss some decision problems of averaging
algebras. In the second part, we show how averaging operators induce Lie
algebra structures. We discuss conditions under which a Lie bracket operation
is induced by an averaging operator. Then we discuss properties of these
induced Lie algebra structures. Finally we apply the results from this
discussion in the study of averaging operators.
