Abstract
A quantized version of the Sierpinski gasket is proposed, on purely topological grounds, as a C⁎-algebra A∞ with a suitable form of self-similarity. Several properties of A∞ 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∞ is a compact quantum metric space.
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