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.
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