We realize Kellendonk's C*-algebra of an aperiodic tiling as the tight C*-algebra of the inverse semigroup associated to the tiling, thus providing further evidence that the tight C*-algebra is a good candidate to be the natural associative algebra to go along with an inverse semigroup.

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Semigroup Forum

Exel, R. (Ruy), Gonçalves, D. (Daniel), & Starling, C. (2012). The tiling C*-algebra viewed as a tight inverse semigroup algebra. Semigroup Forum, 84(2), 229–240. doi:10.1007/s00233-011-9359-x