Toeplitz Algebras and Extensions of Irrational Rotation Algebras

Park, Efton

Canadian mathematical bulletin, Tome 48 (2005) no. 4, pp. 607-613

Résumé

For a given irrational number $\theta$ , we define Toeplitz operators with symbols in the irrational rotation algebra ${{\mathcal{A}}_{\theta }}$ , and we show that the ${{C}^{*}}$ algebra $\mathfrak{J}\left( {{\mathcal{A}}_{\theta }} \right)$ generated by these Toeplitz operators is an extension of ${{\mathcal{A}}_{\theta }}$ by the algebra of compact operators. We then use these extensions to explicitly exhibit generators of the group $K{{K}^{1}}\left( {{\mathcal{A}}_{\theta }},\mathbb{C} \right)$ . We also prove an index theorem for $\mathfrak{J}\left( {{\mathcal{A}}_{\theta }} \right)$ that generalizes the standard index theorem for Toeplitz operators on the circle.

Détail

DOI : 10.4153/CMB-2005-056-2

Mots-clés : 47B35, 46L80, Toeplitz operators, irrational rotation algebras, index theory