The Mackey bijection for complex reductive groups and continuous fields of reduced group C*-algebras
<p>The purpose of this paper is to make a further contribution to the Mackey bijection for a complex reductive group <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G">
  <mml:semantics>
    <mml:mi>G</mml:mi>
    <mml:annotation encoding="application/x-tex">G</mml:annotation>
  </mml:semantics>
</mml:math>
</inline-formula>, between the tempered dual of <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G">
  <mml:semantics>
    <mml:mi>G</mml:mi>
    <mml:annotation encoding="application/x-tex">G</mml:annotation>
  </mml:semantics>
</mml:math>
</inline-formula> and the unitary dual of the associated Cartan motion group. We shall construct an embedding of the <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript asterisk">
  <mml:semantics>
    <mml:msup>
      <mml:mi>C</mml:mi>
      <mml:mo>∗<!-- ∗ --></mml:mo>
    </mml:msup>
    <mml:annotation encoding="application/x-tex">C^*</mml:annotation>
  </mml:semantics>
</mml:math>
</inline-formula>-algebra of the motion group into the reduced <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript asterisk">
  <mml:semantics>
    <mml:msup>
      <mml:mi>C</mml:mi>
      <mml:mo>∗<!-- ∗ --></mml:mo>
    </mml:msup>
    <mml:annotation encoding="application/x-tex">C^*</mml:annotation>
  </mml:semantics>
</mml:math>
</inline-formula>-algebra of <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G">
  <mml:semantics>
    <mml:mi>G</mml:mi>
    <mml:annotation encoding="application/x-tex">G</mml:annotation>
  </mml:semantics>
</mml:math>
</inline-formula>, and use it to characterize the continuous field of reduced group <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript asterisk">
  <mml:semantics>
    <mml:msup>
      <mml:mi>C</mml:mi>
      <mml:mo>∗<!-- ∗ --></mml:mo>
    </mml:msup>
    <mml:annotation encoding="application/x-tex">C^*</mml:annotation>
  </mml:semantics>
</mml:math>
</inline-formula>-algebras that is associated to the Mackey bijection. We shall also obtain a new characterization of the Mackey bijection using the same embedding.</p>