Consider $S_{2k} (\Gamma_0 (N))$ and let $S(N)$ denote the direct limit of the finite direct sums of the $S_{2k}$. Since each $S_{2k} (\Gamma_0 (N))$ is also a Hilbert space w.r.t. the Petersson inner product, $S(N)$ is as well, and we can consider the C*-algebra $S^*(N)$ of bounded operators on $S(N)$.

So has anyone actually considered this thing yet? If not, can someone comment on the possible relevance of using the Riemann-Hurwitz formula (giving the dimensions of the $S_{2k}$) to describe $S^*(N)$ as the limit of the Bratteli diagrams of the aforementioned finite direct sums? (Since $S^*(N)$ is approximately finite-dimensional, this seems like a natural thing to do...)

(NB. I asked this question on sci.math.research back in the nineties and never got an answer, and figured I'd give it another shot here given the MO userbase. Anything ridiculous here should be blamed on my younger self.)