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 $\mathbb{C}$). \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.
