TITLE: Where does the definition of "Tower of Algebras" come from?
QUESTION [5 upvotes]: A tower of algebras is a sequence of algebras
$$A_0 \hookrightarrow A_1 \hookrightarrow \cdots \hookrightarrow A_n \hookrightarrow \cdots$$ 
with embeddings $A_n \otimes A_m \hookrightarrow A_{n+m}$ satisfying an associativity condition.
I found this definition recently in papers in representation theory such as:
"Algebraic structures on Grothendieck groups of a tower of algebras"
by Huilan Li and Nantel Bergeron
http://arxiv.org/pdf/math/0612170
"Combinatorial Hopf algebras and Towers of Algebras - Dimension, Quantization, and Functoriality"
by Nantel Bergeron, Thomas Lam and Huilan Li
http://arxiv.org/pdf/0903.1381
"Representation theories of some towers of algebras related to the symmetric groups and their Hecke algebras"
by F. Hivert and N. Thiéry
http://garsia.math.yorku.ca/fpsac06/papers/75.pdf
Also, I found this definition in a homotopy theoretic (and operadic) paper:
"On Quillen homology and a homotopy completion tower for algebras over operads"
by J. Harper and K. Hess.

My question is: where does the definition of tower of algebras come from and what was the original motivation to introduce this object?

REPLY [4 votes]: I don't know what the original motivation was (or where the original definition was). But one reasonable conceptual explanation for the definition is this: let $\mathcal C$ be a category with objects $\mathbb N$, endomorphisms $A_n$, and other morphism spaces 0. Then the embeddings $A_m \otimes A_n \to A_{m+n}$ (and the associativity conditions) are what is needed to provide $\mathcal C$ with a strict monoidal structure, where $m \otimes n := m + n$. Then one reason towers of algebras tend to be interesting is that monoidal categories tend to be interesting.