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An averaging operator on an associative algebra $A$ is an algebraicabstraction of the time average operator on the space of real-valued functionsdefined in time-space. In this paper, we consider relative averaging operatorson a bimodule $M$ over an associative algebra $A$. A relative averagingoperator induces a diassociative algebra structure on the space $M$. The fulldata consisting of an associative algebra, a bimodule and a relative averagingoperator is called a relative averaging algebra. We define bimodules over arelative averaging algebra that fits with the representations of diassociativealgebras. We construct a graded Lie algebra and a $L_\infty$-algebra that arerespectively controlling algebraic structures for a given relative averagingoperator and relative averaging algebra. We also define cohomologies ofrelative averaging operators and relative averaging algebras and find a longexact sequence connecting various cohomology groups. As applications, we studydeformations and abelian extensions of relative averaging algebras. Finally, wedefine homotopy relative averaging algebras and show that they induce homotopydiassociative algebras.

Controlling structures, deformations and homotopy theory for averaging algebras