On Spectral Invariance of Non–Commutative Tori

Around 1980 Connes extended the notions of geometry to the noncommutative setting. Since then non-commutative geometry has turned into a very active area of mathematical research. As a first non-trivial example of a noncommutative manifold Connes discussed subalgebras of rotation algebras, the socalled non-commutative tori. In the last two decades researchers have unrevealed the relevance of non-commutative tori in a variety of mathematical and physical fields. In a recent paper we have pointed out that non-commutative tori appear very naturally in Gabor analysis. In the present paper we show that Janssen’s result on good window classes in Gabor analysis has already been proved in a completely different context and in a very disguised form by Connes in 1980. Our treatment relies on non-commutative analogs of Wiener’s lemma for certain subalgebras of rotation algebras by Gröchenig and Leinert.