Completely Compact Herz–Schur Multipliers of Dynamical Systems
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Abstract
Abstract We prove that if G is a discrete group and $$(A,G,\alpha )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mi>α</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is a C*-dynamical system such that the reduced crossed product $$A\rtimes _{r,\alpha } G$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:msub> <mml:mo>⋊</mml:mo> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>,</mml:mo> <mml:mi>α</mml:mi> </mml:mrow> </mml:msub> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> possesses property (SOAP) then every completely compact Herz–Schur $$(A,G,\alpha )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mi>α</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -multiplier can be approximated in the completely bounded norm by Herz–Schur $$(A,G,\alpha )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mi>α</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -multipliers of finite rank. As a consequence, if G has the approximation property (AP) then the completely compact Herz–Schur multipliers of A ( G ) coincide with the closure of A ( G ) in the completely bounded multiplier norm. We study the class of invariant completely compact Herz–Schur multipliers of $$A\rtimes _{r,\alpha } G$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:msub> <mml:mo>⋊</mml:mo> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>,</mml:mo> <mml:mi>α</mml:mi> </mml:mrow> </mml:msub> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> and provide a description of this class in the case of the irrational rotation algebra.
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|---|---|---|
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