Density of Toeplitz operators in rotation-invariant Toeplitz algebras

Abstract We use results and techniques from Werner’s “quantum harmonic analysis” to show that G -invariant Toeplitz operators on $$\mathcal {F}^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>F</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math> are norm dense in G -invariant Toeplitz algebras for all subgroups G of the affine unitary group $$U_n\ltimes \mathbb {C}^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>⋉</mml:mo> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:math> . Additionally, we prove that the quasi-radial Toeplitz operators are dense in the quasi-radial Toeplitz algebra over the Bergman space $$\mathcal {A}^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math> and prove that G -invariant Toeplitz operators are SOT dense in the algebra of all G -invariant bounded operators on $$\mathcal {F}^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>F</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math> .