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We present a novel approach to the representation theory of finitedimensional algebras motivated by the emerging theory of graph limits.  We introduce the rank spectrum of a finite dimensional algebra $R$ over afinite field. The elements of the rank spectrum are representations of thealgebra into von Neumann regular rank algebras, and two representations areconsidered to be equivalent if they induce the same Sylvester rank functions on$R$-matrices.  Based on this approach, we can divide the finite dimensional algebras intothree types: finite, amenable and non-amenable representation types. We provethat string algebras are of amenable representation type, but the wildKronecker algebras are not. Here, the amenability of the rank algebrasassociated to the limit points in the rank spectrum plays a very importantpart.  We also show that the limit points of finite dimensional representations ofalgebras of amenable representation type can always be viewed asrepresentations of the algebra in the continuous ring invented by John vonNeumann in the 1930's.  As an application in algorithm theory, we introduce and study the notion oftesting of parameters of modules over finite dimensional algebras, that isanalogous to the property testing of bounded degree graphs introduced byGoldreich and Ron. We shall see that for string algebras all the reasonable(stable) parameters are testable.

Infinite dimensional representations of finite dimensional algebras and amenability