Multiplicative Unitaries Research Articles

Braided Quantum Groups and Their Bosonizations in the C*-Algebraic Framework

Abstract We present a general theory of braided quantum groups in the ${\textrm {C}^*}$-algebraic framework using the language of multiplicative unitaries. Starting with a manageable multiplicative unitary in the representation category of the quantum codouble of a regular quantum group $\mathbb {G}$ we construct a braided ${\textrm {C}^*}$-quantum group over $\mathbb {G}$ as a ${\textrm {C}^*}$-bialgebra in the monoidal category of the $\mathbb {G}$-Yetter–Drinfeld ${\textrm {C}^*}$-algebras. Furthermore, we establish a one-to-one correspondence between braided ${\textrm {C}^*}$-quantum groups and ${\textrm {C}^*}$-quantum groups with a projection. Consequently, we generalise the bosonization construction for braided Hopf-algebras of Radford and Majid to braided ${\textrm {C}^*}$-quantum groups. Several examples are discussed. In particular, we show that the complex quantum plane admits a braided ${\textrm {C}^*}$-quantum group structure over the circle group ${\mathbb {T}}$ and identify its bosonization with the simplified quantum $\textrm {E}(2)$ group.

Co-actions, Isometries, and isomorphism classes of Hilbert modules

We show that a A-linear map of Hilbert A-modules is induced by a unitary Hilbert module operator if and only if it extends to an ordinary unitary on appropriately defined enveloping Hilbert spaces. Applications to the theory of multiplicative unitaries let us to compute the equivalence classes of Hilbert modules over a class of C*-algebraic quantum groups. We, thus, develop a theory that, for example, could be used to show non-existence of certain co-actions. In particular, we show that the Cuntz semigroup functor takes a co-action to a multiplicative action.

From Hopf algebras to topological quantum groups. A short history, various aspects and some problems

Quantum groups have been studied within several areas of mathematics and mathematical physics. This has led to different approaches, each of them with their own techniques and conventions. Starting with Hopf algebras, where there is a general consensus, moving in the direction of topological quantum groups, where there is no such consensus, it is easy to get lost. Not only many difficulties have to be overcome, but also several choices must be made. The way this is done is often confusing. Some choices even turn out to be rather annoying. As an introductory lecture at the conference on Topological quantum groups and Hopf algebras in 2016, we have explained these choices, difficulties and annoyances, encountered on the road from Hopf algebras to topological quantum groups. In these notes, we discuss more aspects of the development of locally compact quantum groups. We not only explain some of these difficulties in greater detail, but we also give background information about the different steps, combined with some historical comments. We start with finite quantum groups and continue with discrete quantum groups, compact quantum groups and algebraic quantum groups. Multiplicative unitaries are an important side track before we finally arrive at locally compact quantum groups. Along the way, we also formulate some interesting remaining problems in the theory.

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

C*-quantum groups with projection are the noncommutative analogues of semidirect products of groups. Radford's Theorem about Hopf algebras with projection suggests that any C*quantum group with projection decomposes uniquely into an ordinary C*-quantum group and a "braided" C*-quantum group. We establish this on the level of manageable multiplicative unitaries.

The Drinfeld double for $C^*$-algebraic quantum groups

In this article, we establish the duality between the generalised Drinfeld double and generalised quantum codouble within the framework of modular or manageable (not necessarily regular) multiplicative unitaries, and discuss several properties.

A Hilbert space approach to approximate diagonals for locally compact quantum groups

For a locally compact quantum group $\mathbb{G}$, the quantum group algebra $L^1(\mathbb{G})$ is operator amenable if and only if it has an operator bounded approximate diagonal. It is known that if $L^1(\mathbb{G})$ is operator biflat and has a bounded approximate identity then it is operator amenable. In this paper, we consider nets in $L^2(\mathbb{G})$ which suffice to show these two conditions and combine them to make an approximate diagonal of the form $\omega_{{W'}^*\xi\otimes\eta}$ where $W$ is the multiplicative unitary and $\xi\otimes\eta$ are simple tensors in $L^2(\mathbb{G})\otimes L^2(\mathbb{G})$. Indeed, if $L^1(\mathbb{G})$ and $L^1(\hat{\mathbb{G}})$ both have a bounded approximate identity and either of the corresponding nets in $L^2(\mathbb{G})$ satisfies a condition generalizing quasicentrality then this construction generates an operator bounded approximate diagonal. In the classical group case, this provides a new method for constructing approximate diagonals emphasizing the relation between the operator amenability of the group algebra $L^1(G)$ and the Fourier algebra $A(G)$.

COMPLETELY POSITIVE MULTIPLIERS OF QUANTUM GROUPS

We show that any completely positive multiplier of the convolution algebra of the dual of an operator algebraic quantum group 𝔾 (either a locally compact quantum group, or a quantum group coming from a modular or manageable multiplicative unitary) is induced in a canonical fashion by a unitary corepresentation of 𝔾. It follows that there is an order bijection between the completely positive multipliers of L1(𝔾) and the positive functionals on the universal quantum group [Formula: see text]. We provide a direct link between the Junge, Neufang, Ruan representation result and the representing element of a multiplier, and use this to show that their representation map is always weak*–weak*-continuous.

Noncommutative Topological Entropy of Endomorphisms of Cuntz Algebras II

A study of noncommutative topological entropy of gauge invariant endomorphisms of Cuntz algebras began in our earlier work with J.\,Zacharias is continued and extended to endomorphisms which are not necessarily of permutation type. In particular it is shown that if \mathsf{H} is an N -dimensional Hilbert space, V is an irreducible multiplicative unitary on \mathsf{H} \otimes \mathsf{H} and F: \mathsf{H} \otimes \mathsf{H} \to \mathsf{H} \otimes \mathsf{H} is the tensor flip, then the Voiculescu entropy of the Longo's canonical endomorphism \rho_{VF} \in {\textrm{End}}(\mathcal{O}_N) is equal to \log N .

C*-pseudo-multiplicative unitaries, Hopf C*-bimodules and their Fourier algebras

AbstractWe 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-bimodules studied by Enock, Lesieur and Vallin. To each C*-pseudo-multiplicative unitary, we associate two Fourier algebras with a duality pairing and in the regular case two Hopf C*-bimodules. The theory is illustrated by examples related to locally compact Hausdorff groupoids. In particular, we obtain a continuous Fourier algebra for a locally compact Hausdorff groupoid.

Pseudo-multiplicative unitaries on C*-modules and Hopf C*-families I

Pseudo-multiplicative unitaries on C*-modules generalize the multiplicative unitaries of Baaj and Skandalis [1], and are analogues of the pseudo-multiplicative unitaries on Hilbert spaces studied by Enock, Lesieur, Vallin [5], [10], [21]. We introduce Hopf C*-families on C*-bimodules and associate to special classes of pseudo-multiplicative unitaries two Hopf C*-families. Furthermore, we discuss dual pairings and counits on these Hopf C*-families, étalé and proper pseudo-multiplicative unitaries, and two classes of examples. In a later article, we will study regularity conditions on pseudo-multiplicative unitaries, coactions of Hopf C*-families on C*-algebras, and duality.

From multiplicative unitaries to quantum groups II

It is shown that all important features of a 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.

Quantum Bohr compactification

We introduce a non commutative analog of the Bohr compactification. Starting from a general quantum group G we define a compact quantum group bG which has a universal property such as the universal property of the classical Bohr compactification for topological groups. We study the object bG in special cases when G is a classical locally compact group, dual of a classical group, discrete or compact quantum group as well a quantum group arising from a manageable multiplicative unitary. We also use our construction to give new examples of compact quantum groups.

The regular representation and regular covariant representation of crossed products of Woronowicz C*-algebras

In this paper, it is shown that the regular representation and regular covariant representation of the crossed products A ×α G correspond to the twisted multiplicative unitary operators, where A is a Woronowicz C*-algebra acted upon by a discrete group G. Meanwhile, it is also shown that the regular covariant C*-algebra is the Woronowicz C*-algebra which corresponds to a multiplicative unitary. Finally, an explicit description of the multiplicative unitary operator for C(SUq(2)) × α ℤ is given in terms of those of the Woronowicz C*-algebra C(SUq(2)) and the discrete group G.

QUANTUM GROUPOIDS OF COMPACT TYPE

To any groupoid, equipped with a Haar system, Jean-Michel Vallin had associated several objects (pseudo-multiplicative unitary, Hopf-bimodule) in order to generalize, up to the groupoid case, the classical notions of multiplicative unitary and Hopf–von Neumann algebra, which were intensely used to construct quantum groups in the operator algebra setting. In two former articles (one in collaboration with Jean-Michel Vallin), starting from a depth-2 inclusion of von Neumann algebras, we have constructed such objects, which allowed us to study two ‘quantum groupoids’ dual to each other. We are now investigating in greater details the notion of pseudo-multiplicative unitary, following the general strategy developed by Baaj and Skandalis for multiplicative unitaries. AMS 2000 Mathematics subject classification: Primary 46L89; 22A22; 81R50

Twisted product structure and representation theory of the quantum group U q(2)

We present a construction of a compact quantum group, based on the method invented by Woronowicz, which uses a twisted determinant. In our case the result turns out to be the quantum group U q (2). Our construction in a natural way shows that this group is a twisted product of its two quantum subgroups SU q (2) and U (1). This allows us to obtain description of irreducible unitary representations of U q (2). We also give the construction of the multiplicative unitary (Kac-Takesaki) operator for this quantum group.

Unitaires multiplicatifs commutatifs

In a previous article, we proved that every commutative multiplicative unitary is a multiple of the multiplicative unitary associated to a locally compact group. In the present Note we give a simpler proof of this generalization of a theorem of Weil. To cite this article: S. Baaj, G. Skandalis, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

REGULAR OBJECTS, MULTIPLICATIVE UNITARIES AND CONJUGATION

The notion of left (respectively 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 (respectively right) regular object, it can be interpreted as a category of corepresentations (respectively 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 unitaries and their associated Hopf C*-algebras. It is shown that the conjugate of a left regular object is a right regular object in the same category. Furthermore the representation category of a locally compact quantum group has a conjugation. The associated multiplicative unitary is a regular object in that category.

QUANTUM 'ax + b' GROUP

'ax + b' is the group of affine transformations of the real line R. In quantum version ab = q2ba, where q2 = e-i ℏ is a number of modulus 1. The main problem of constructing quantum deformation of this group on the C*-level consists in non-selfadjointness of Δ(b) = a ⊗ b + b ⊗ I. This problem is overcome by introducing (in addition to a and b) a new generator β commuting with a and anticommuting with b. β (or more precisely β ⊗ β) is used to select a suitable selfadjoint extension of a ⊗ b + b ⊗ I. Furthermore we have to assume that [Formula: see text], where k = 0,1,2, ·. In this case, q is a root of 1. To construct the group, we write an explicit formula for the Kac–Takesaki operator W. It is shown that W is a manageable multiplicative unitary in the sense of [3,19]. Then using the general theory we construct a C*-algebra A and a comultiplication Δ ∈ Mor (A,A ⊗ A). A should be interpreted as the algebra of all continuous functions vanishing at infinity on quantum 'ax + b'-group. The group structure is encoded by Δ. The existence of coinverse also follows from the general theory [19].

An example of amenable Kac systems

By giving an interesting characterisation of amenable multiplicative unitaries, we show, in a very simple way, that bicrossproducts of amenable locally compact groups are both amenable and coamenable.

AN ALGEBRAIC DUALITY THEORY FOR MULTIPLICATIVE UNITARIES

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 characterizations of minimal representation categories of a multiplicative unitary. Endomorphisms of the Cuntz algebra related algebraically to the grading are discussed as is the notion of braided symmetry in a tensor C*-category.