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Measured quantum groupoids on a finite basis and equivariant Kasparov theory. (English) Zbl 1499.19002

To every pair \(A\), \(B\) of \(C^*\)-algebras continuously acted upon by a regular measured quantum groupoid on a finite basis \(\mathcal G\), a \(\mathcal G\)-equivariant Kasparov theory group \(KK_{\mathcal G}(A,B)\) is associated, and it is shown that the Kasparov product and some other results from the equivariant Kasparov theory generalize to this setting. This is applied to monoidal equivalence, and a new proof is provided of the equivalence of the equivariant Kasparov categories \(KK_{\mathbb G_1}\) and \(KK_{\mathbb G_2}\) when \(\mathbb G_1\) and \(\mathbb G_2\) are monoidally equivalent regular locally compact quantum groups.

MSC:

19K35 Kasparov theory (\(KK\)-theory)
46L67 Quantum groups (operator algebraic aspects)
16T99 Hopf algebras, quantum groups and related topics
46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory
46L80 \(K\)-theory and operator algebras (including cyclic theory)
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[1] In virtue of Proposition 4.2.11 (1)-(4), the following definition makes sense. the pair .E j ; F j / is a G j -equivariant Kasparov A j -B j -bimodule. Let x WDOE.E j ; F j /2KK G j .A j ; B j / and J G k ;
[2] G j .x/ D OE.E k ;
[3] F k /. Let F WD F 1˚F2 2 L.E/. We denote by J G ;G j .x/ the unique element y 2 KK G .A; B/ such that b A˝A y˝B a B D OE.D; R .F //. For j D 1; 2, we have a well-defined homomorphism of abelian groups J G ;G j W KK G j .A j ; B j / ! KK G .A; B/.
[4] F /.
[5] J G j ;G ı J G ;G j D id KK G j .A j ;B j / ,
[6] J G k ;G ı J G ;G j D J G k ;G j ,
[7] J G k ;G j ı J G j ;
[8] G D J G k ;G .
[9] Proof. Formulas (1) and (2)
[10] G W KK G .A; B/ ! KK G j .A j ; B j / and J G ;G j W KK G i .A j ; B j / ! KK G .A; B/
[11] G D id KK G .A;B/ (cf. Proposition 4.2.14 (1)). Let F 2 L.E/ such that the pair .E;
[12] F / 2 KK G .A;
[13] B/. We have F D F 1˚F2 with F 1 2
[14] L.E 1 / and F 2 2 L.E 2 /. It follows from Proposition 4.2.14 (3) that J G k ;
[15] G j .OE.E j ; F j // D OE.E k ; F k /. By applying Lemma 4.2.13, we then obtain J G ;G j J G j ;G .x/ D J G ;G j .E j ; F j / D .E;
[16] F / D x:
[17] G 1 W KK G 1 .A 1 ;
[18] B 2 / is an isomorphism of abelian groups and .J G 2 ;
[19] Proof. This is an immediate consequence of Proposition 4.2.14 (2) and Theorem 4.2.15. Let us fix a third G 1 -C -algebra C 1 . Consider the induced G 2 -C -algebra C 2 WD Ind G 2 G 1 .C 1 / and the G -C -algebra C WD C 1˚C2 .
[20] J G ;G j .1 A j / D 1 A , (2) for all x 2 KK G j .A j ; C j / and y 2 KK G j .C j ; B j /,
[21] J G ;G j .x˝C j y/ D J G ;G j .x/˝C J G ;G j .y/ in KK G .A; B/:
[22] J G k ;G j .1 A j / D 1 A k , (2) for all x 2 KK G j .A j ; C j / and y 2 KK G j .C j ; B j /,
[23] J G k ;G j .x˝C j y/ D J G k ;G j .x/˝C k J G k ;G j .y/ in KK G k .A k ; B k /:
[24] Proof. This is a direct consequence of Propositions 4.2.14 (2), 4.1.4, and 4.2.17. Notations 4.2.19. We denote by KK G (resp. KK G j for j D 1; 2) the category of separable G (resp. G j )-C -algebras whose set of arrows between two G (resp. G j )-C -algebras A and B is the equivariant Kasparov group KK G .A; B/ (resp. KK G j .A; B/).
[25] G W KK G ! KK G j and J G ;G j W KK G j ! KK G are equivalences of categories inverse of each other, (2) the correspondences J G 2 ;G 1 W KK G 1 ! KK G 2 and J G 1 ;G 2 W KK G 2 ! KK G 1 are equiv-alences of categories inverse of each other.
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