Positive definite functions as uniformly ergodic multipliers of the Fourier algebra

Abstract Let G be a locally compact group and let $$\phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϕ</mml:mi> </mml:math> be a positive definite function on G with $$\phi (e)=1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ϕ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>e</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . This function defines a multiplication operator $$M_\phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>ϕ</mml:mi> </mml:msub> </mml:math> on the Fourier algebra A ( G ) of G . The aim of this paper is to classify the ergodic properties of the operators $$M_\phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>ϕ</mml:mi> </mml:msub> </mml:math> , focusing on several key factors, including the subgroup $$H_\phi =\{x\in G:\phi (x)=1\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>ϕ</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo>{</mml:mo> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>G</mml:mi> <mml:mo>:</mml:mo> <mml:mi>ϕ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>}</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , the spectrum of $$M_\phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>ϕ</mml:mi> </mml:msub> </mml:math> , or how “spread-out” a power of $$M_\phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>ϕ</mml:mi> </mml:msub> </mml:math> can be. We show that the multiplication operator $$M_\phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>ϕ</mml:mi> </mml:msub> </mml:math> is uniformly mean ergodic if and only if $$H_\phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>ϕ</mml:mi> </mml:msub> </mml:math> is open and 1 is not an accumulation point of the spectrum of $$M_\phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>ϕ</mml:mi> </mml:msub> </mml:math> . Equivalently, this happens when some power of $$\phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϕ</mml:mi> </mml:math> is not far, in the multiplier norm, from a function supported on finitely many cosets of $$H_\phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>ϕ</mml:mi> </mml:msub> </mml:math> . Additionally, we show that the powers of $$M_\phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>ϕ</mml:mi> </mml:msub> </mml:math> converge in norm if, and only if, the operator is uniformly mean ergodic and $$H_\phi =\{x\in G:|\phi (x)|=1\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>ϕ</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo>{</mml:mo> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>G</mml:mi> <mml:mo>:</mml:mo> <mml:mo>|</mml:mo> <mml:mi>ϕ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>|</mml:mo> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>}</mml:mo> </mml:mrow> </mml:mrow> </mml:math> .