An Algebraic Study of Averaging Operators
  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. 
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