Thomas Timmermann
We introduce $C^*$-pseudo-multiplicative unitaries and (concrete) Hopf $C^*$-bimodules, which are $C^*$-algebraic variants of the pseudo-multiplicative unitaries on Hilbert spaces and the Hopf-von Neumann-bimodules studied by Enock, Lesieur, Vallin. Moreover, we associate to every regular $C^*$-pseudo-multiplicative unitary two Hopf-$C^*$-bimodules and discuss examples related to locally compact groupoids.
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We propose a weaker condition for multiplicative unitary operators related to quantum groups, than the condition of manageability introduced by S.L. Woronowicz. We prove that all the main results of the theory of manageable multiplicative unitaries remain true under this weaker condition. We also show that multiplicative unitaries arising naturally in the construction of some recent examples of non-compact quantum groups satisfy our condition, but fail to be manageable.
We use the theory of regular objects in tensor categories to clarify the passage between braided multiplicative unitaries and multiplicative unitaries with projection. The braided multiplicative unitary and its semidirect product multiplicative unitary have the same Hilbert space representations. We also show that the multiplicative unitaries associated to two regular objects for the same tensor category are equivalent and hence generate isomorphic C*-quantum groups. In particular, a C*-quantum group is determined uniquely by its tensor category of representations on Hilbert space, and any fun…
Finite quantum groupoids can be described in many equivalent ways: In terms of the weak Hopf C*-algebras of Böhm, Nill, and Szlachányi or the finite-dimensional Hopf-von Neumann bimodules of Vallin, and in terms of finite-dimensional multiplicative partial isometries or the finite-dimensional pseudo-multiplicative unitaries of Vallin. In this short note, we show that in finite dimensions, the notions of a Hopf-von Neumann bimodule and of a pseudo-multiplicative unitary coincide with the notions of a concrete Hopf-C*-bimodule and of a C*-pseudo-multiplicative unitary, respectively.
Semidirect products of C*-quantum groups: multiplicative unitaries approach
该研究针对带投影C*-量子群的分解问题，借助Radford定理的思路，在可管理乘法酉算子层面证明了带投影C*-量子群可唯一分解为普通与辫子C*-量子群的结论。
It is shown that all important features of a $\mathrm{C}^*$-algebraic quantum group $(A,Δ)$ defined by a modular multiplicative $W$ depend only on the pair $(A,Δ)$ rather than the multiplicative unitary operator $W$. The proof is based on thorough study of representations of quantum groups. As an application we present a construction and study properties of the universal dual of a quantum group defined by a modular multiplicative unitary - without assuming existence of Haar weights.
We generalise the notions of semi-regularity, regularity, and bi-regularity to unitary solutions of the braided Pentagon equation in concrete W*-categories with semi-regular/regular/bi-regular braiding, and study their properties. We show, for example, that under bi-regularity-assumptions, the leg-algebras form braided C*-bialgebras. Moreover, we "close the circle" for the representation categories: The braiding on the Yetter-Drinfeld category of a (semi-)regular multiplicative unitary in a category with (semi-)regular braiding is again (semi-)regular.
The notion of left (resp. right) regular object of a tensor C*-category equipped with a faithful tensor functor into the category of Hilbert spaces is introduced. If such a category has a left (resp. right) regular object, it can be interpreted as a category of corepresentations (resp. representations) of some multiplicative unitary. A regular object is an object of the category which is at the same time left and right regular in a coherent way. A category with a regular object is endowed with an associated standard braided symmetry. Conjugation is discussed in the context of multiplicative un…
In this paper, we will give a natural definition for morphisms between multiplicative unitaries. We will then discuss some equivalences of this definition and some interesting properties of them. Moreover, we will define normal sub-multiplicative unitaries for multiplicative unitaries of discrete type and prove an imprimitivity type theorem for discrete multiplicative unitaries.
We introduce C*-pseudo-multiplicative unitaries and concrete Hopf C*-bimodules for the study of quantum groupoids in the setting of C*-algebras. These unitaries and Hopf C*-bimodules generalize multiplicative unitaries and Hopf C*-algebras and are analogues of the pseudo-multiplicative unitaries and Hopf--von Neumann-bimod-ules studied by Enock, Lesieur and Vallin. To each C*-pseudo-multiplicative unitary, we associate two Fourier algebras with a duality pairing, a C*-tensor category of representations, and in the regular case two reduced and two universal Hopf C*-bimodules. The theory is illu…
Multiplicative Unitaries are described in terms of a pair of commuting shifts of relative depth two. They can be generated from ambidextrous Hilbert spaces in a tensor C*-category. The algebraic analogue of the Takesaki-Tatsuuma Duality Theorem characterizes abstractly C*-algebras acted on by unital endomorphisms that are intrinsically related to the regular representation of a multiplicative unitary. The relevant C*-algebras turn out to be simple and indeed separable if the corresponding multiplicative unitaries act on a separable Hilbert space. A categorical analogue provides internal charac…
We show that the deformation theory of Fréchet algebras for actions of Kählerian Lie groups developed by two of us, leads in a natural way to examples of non-compact locally compact quantum groups. This is achieved by constructing a manageable multiplicative unitary out of the Fréchet deformation of $C_0(G)$ for the action $λ\otimes ρ$ of $G\times G$ and the undeformed coproduct. We also prove that these quantum groups are isomorphic to those constructed out of the unitary dual $2$-cocycle discovered by Neshveyev and Tuset and associated with Bieliavsky's covariant $\star$-product, via the De …
By giving an interesting characterisation of amenable multiplicative unitaries in term of one dimensional representations, we show in a simple way that bicrossproducts of amenable locally compact groups is both amenable and coamenable.
Let U, V, and W be multiplicative unitaries coming from discrete Kac systems such that W is an amenable normal submultiplicative unitary of V with quotient U. We define notions for right-Hilbert bimodules of coactions of S_V and (S_V)^, their restrictions to S_W and (S_U)^, their dual coactions, and their full and reduced crossed products. If N(A) denotes the imprimitivity bimodule associated to any coaction of S_V on a C*-algebra A by Ng's imprimitivity theorem, then for any suitably nondegenerate injective coaction of S_V on a right-Hilbert A - B bimodule X we establish an isomorphism betwee…
In the setting of von Neumann algebras, measurable quantum groupoids have successfully been axiomatized and studied by Enock, Vallin, and Lesieur, whereas in the setting of $C^{*}$-algebras, a similar theory of locally compact quantum groupoids could not yet be developed. Some basic building blocks for such a theory, like analogues of a Hopf-von Neumann bimodule and of a pseudo-multiplicative unitary, were introduced in the thesis and a recent article by the author. That approach, however, is restricted to decomposable quantum groupoids which generalize $r$-discrete groupoids. Recently, we dev…
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