  We initiate the study of computable presentations of real and complex
C*-algebras under the program of effective metric structure theory. With the
group situation as a model, we develop corresponding notions of recursive
presentations and word problems for C*-algebras, and show some analogous
results hold in this setting. Famously, every finitely generated group with a
computable presentation is computably categorical, but we provide a
counterexample in the case of C*-algebras. On the other hand, we show every
finite-dimensional C*-algebra is computably categorical.
