A noncommutative Sierpinski gasket

A quantized version of the Sierpinski gasket is proposed, on purely topological grounds, as a C*-algebra A infinity with a suitable form of self-similarity. Several properties of A infinity are studied, in particular its nuclearity, the structure of ideals as well as the description of irreducible representations and extremal traces. A harmonic structure is introduced, giving rise to a self-similar Dirichlet form E. A spectral triple is also constructed, extending the one already known for the classical gasket, from which E can be reconstructed. Moreover we show that A infinity is a compact quantum metric space.