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The Ginsparg-Wilson algebra is the algebra underlying the Ginsparg-Wilsonsolution of the fermion doubling problem in lattice gauge theory. The Diracoperator of the fuzzy sphere is not afflicted with this problem. Previously wehave indicated that there is a Ginsparg-Wilson operator underlying it as wellin the absence of gauge fields and instantons. Here we develop this observationsystematically and establish a Dirac operator theory for the fuzzy sphere withor without gauge fields, and always with the Ginsparg-Wilson algebra. There isno fermion doubling in this theory. The association of the Ginsparg-Wilsonalgebra with the fuzzy sphere is surprising as the latter is not designed withthis algebra in mind. The theory reproduces the integrated U(1)_A anomaly andindex theory correctly.Comment: references added, typos corrected, section 4.2 simplified. Report.no:  SU-4252-769, DFUP-02-1

Fuzzy Ginsparg-Wilson algebra: A solution of the fermion doubling problem