Jones index theory by Hilbert C*-bimodules and K-theory

Abstract

In this paper we introduce the notion of Hilbert <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper C Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">C</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>∗</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">{\mathrm {C}}^{*}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-bimodules, replacing the associativity condition of two-sided inner products in Rieffel’s imprimitivity bimodules by a Pimsner-Popa type inequality. We prove Schur’s Lemma and Frobenius reciprocity in this setting. We define minimality of Hilbert <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper C Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">C</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>∗</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">{\mathrm {C}}^{*}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-bimodules and show that tensor products of minimal bimodules are also minimal. For an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> bimodule which is compatible with a trace on a unital <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper C Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">C</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>∗</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">{\mathrm {C}}^{*}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, its dimension (square root of Jones index) depends only on its <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K upper K"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mi>K</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">KK</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-class. Finally, we show that the dimension map transforms the Kasparov products in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K upper K left-parenthesis upper A comma upper A right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mi>K</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">KK(A,A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to the product of positive real numbers, and determine the subring of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K upper K left-parenthesis upper A comma upper A right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mi>K</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">KK(A,A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> generated by the Hilbert <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper C Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">C</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>∗</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">{\mathrm {C}}^{*}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-bimodules for a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper C Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">C</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>∗</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">{\mathrm {C}}^{*}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-algebra generated by Jones projections.