Deformation estimates and applications to Toeplitz algebras

Abstract We discuss explicit approximations of uniformly continuous functions by Lipschitz continuous functions. Such constructions are relevant in the analysis of Toeplitz operators. As model cases, we treat $${\mathbb {C}}^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> with the Euclidean metric and the complex unit ball $${\mathbb {B}}^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>B</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> equipped with the Bergman metric in details. In both cases a flow of approximating real analytic Lipschitz functions with a control of the Lipschitz constants is defined. We provide two applications of these results to semi-commutator estimates in Berezin Toeplitz quantization as well as to the analysis of certain $$C^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> algebras generated by Toeplitz operators on the complex unit ball $${\mathbb {B}}^2.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>B</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>.</mml:mo> </mml:mrow> </mml:math>