Quantum Group-Twisted Tensor Products of C*-Algebras

Abstract

We put two C * -algebras together in a noncommutative tensor product using quantum group coactions on them and a bicharacter relating the two quantum groups that act. We describe this twisted tensor product in two equivalent ways, based on certain pairs of quantum group representations and based on covariant Hilbert space representations, respectively. We establish basic properties of the twisted tensor product and study some examples.2010 Mathematics Subject Classification. 81R50 (46L05 46L55).

Quantum group-twisted tensor products of C*-algebras. II

“…Let C and D be C * -algebras with a coaction of a C * -quantum group G = (A, ∆ A ). As in [12], C * -quantum groups are generated by manageable multiplicative unitaries, and Haar weights are not assumed. If G is a group, then the C * -tensor product C ⊗ D inherits a diagonal coaction.…”

Quantum Symmetries of the Twisted Tensor Products of C*-Algebras

“…By [35,Lemma 3.8], there exists a V-Heisenberg pair, i.e. a Hilbert space H and a pair of representations α ∈ Mor(C 0 (G), K(H)) and β ∈ Mor((C 0 (H), K(H)) that satisfy the following condition: 23 .…”

“…The plan of the article is as follows: in Section 2, after fixing the notations and conventions for locally compact quantum groups, we discuss the theories of twisted tensor product of C * -algebras and generalized Drinfeld doubles developed in [35] and [39], respectively. In Section 3 we study the 'universal C * -algebra' associated to the generalized Drinfeld double, and prove two results about its action on twisted tensor products, namely Lemma 3.4 and Theorem 3.5.…”

Semidirect Products of C*-Quantum Groups: Multiplicative Unitaries Approach

“…This gives Z * F = P 23 P 13 P 24 P 14 P * 14 P * 24 W C 24 W C 14 = P 23 P 13 W C 24 W C 14 . Now we use that (ι,ι) is the standard Heisenberg pair, generated by W C , and that the anti-Heisenberg pair (α,α) is constructed as in [11,Lemma 3.6]; that is, This unitary and Q witness the manageability of the braided multiplicative unitary (U,V, F). The rather technical proof of this fact is relegated to the appendix, see Lemma A.8.…”

“…The reduced bicharacter is the unique unitary W C ∈ U(Ĉ⊗C) with W C = (ι⊗ι)(W C ) or, briefly, W C = W Ĉ ιι . By construction, A ⊆ M(C) and ⊆ M(Ĉ) as C * -subalgebras of B(H).The representations (ι,ι) form a Heisenberg pair for the quantum group (C, ∆ C ) in the notation of[11]. This Heisenberg pair generates an anti-Heisenberg pair α : C → B(H),α :Ĉ → B(H) by[11, Lemma 3.6].…”