TITLE: Self-similar matrices?
QUESTION [5 upvotes]: Does anyone know anything about self-similar (infinite) matrices, with more or less fractal(-like) structure and admitting meaningful matrix-algebra operations?

REPLY [2 votes]: You should learn about the hyperfinite II_1 factor, which is the limit of the inclusions
M_1 --> M_2 --> M_4 --> M_8 --> ....

(here M_k is the k by k matrices over $\mathbb{C}$) where each inclusion is given by tensoring with the identity matrix in M_2. Every 'finite' element is "self-similar" in a sense.