Associativity of crossed products by partial actions, enveloping actions and partial representations

Abstract

Given a partial action <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha"> <mml:semantics> <mml:mi>α</mml:mi> <mml:annotation encoding="application/x-tex">\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a 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> on an associative algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we consider the crossed product <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A right-normal-factor-semidirect-product Subscript alpha Baseline upper G"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:msub> <mml:mo>⋊</mml:mo> <mml:mi>α</mml:mi> </mml:msub> <mml:mi>G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {A}\rtimes _\alpha G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Using the algebras of multipliers, we generalize a result of Exel (1997) on the associativity of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A right-normal-factor-semidirect-product Subscript alpha Baseline upper G"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:msub> <mml:mo>⋊</mml:mo> <mml:mi>α</mml:mi> </mml:msub> <mml:mi>G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {A}\rtimes _\alpha G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> obtained in the context of <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. In particular, we prove that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A right-normal-factor-semidirect-product Subscript alpha Baseline upper G"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:msub> <mml:mo>⋊</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>α</mml:mi> </mml:mrow> </mml:msub> <mml:mi>G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {A} \rtimes _{\alpha } G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is associative, provided that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is semiprime. We also give a criterion for the existence of a global extension of a given partial action on an algebra, and use crossed products to study relations between partial actions of groups on algebras and partial representations. As an application we endow partial group algebras with a crossed product structure.