We will introduce the notion of inductive limits of compact quantum groups as $W^*$-bialgebras equipped with some additional structures. We also formulate their unitary representation theories. Those give a more explicit representation-theoretic meaning to our previous study of quantized characters associated with a given inductive system of compact quantum groups. As a byproduct, we will give an explicit representation-theoretic interpretation to some transformations that play an important role in the analysis of $q$-central probability measures on the paths in the Gelfand-Tsetlin graph.
In this paper, we study Markov dynamics on unitary duals of compact quantum groups. We construct such dynamics from characters of quantum groups. Then we show that the dynamics have generators, and we give an explicit formula of the generators using the representation theory. Moreover, we construct Markov dynamics on the unitary dual of an inductive limit of compact quantum groups.
We study the character theory of inductive limits of $q$-deformed classical compact groups. In particular, we clarify the relationship between the representation theory of Drinfeld-Jimbo quantized universal enveloping algebras and our previous work on the quantized characters. We also apply the character theory to construct Markov semigroups on unitary duals of $SO_q(2N+1)$, $Sp_q(N)$, and their inductive limits.
We propose a natural quantized character theory for inductive systems of compact quantum groups based on KMS states on AF-algebras following Stratila-Voiculescu's work (Stratila-Voiculescu, 1975) (or (Enomoto-Izumi, 2016)), and give its serious investigation when the system consists of quantum unitary groups $U_q(N)$ with $q$ in $(0,1)$. The key features of this work are: The "quantized trace" of a unitary representation of a compact quantum group can be understood as a quantized character associated with the unitary representation and its normalized one is captured as a KMS state with respect…
Given a unital inductive limit of C*-algebras for which each C*-algebra of the inductive sequence comes equipped with a compact quantum metric of Rieffel, we produce sufficient conditions to build a compact quantum metric on the inductive limit from the quantum metrics on the inductive sequence by utilizing the completeness of the dual Gromov-Hausdorff propinquity of Latremoliere on compact quantum metric spaces. This allows us to place new quantum metrics on all unital AF algebras that extend our previous work with Latremoliere on unital AF algebras with faithful tracial state. As a consequen…
Let $A=\underrightarrow{\lim}{A_n}$ be an AF algebra, $G$ be a compact group. We consider inductive limit actions of the form $α=\underrightarrow{\lim}{α_n}$, where $α_n\colon G\curvearrowright A_n$ is an action on the finite dimensional C*-algebra $A_n$ which fixes each matrix summand. If each $α_n$ is inner, such actions are classified by equivariant K-theory by Handelman and Rossmann. However, if the actions $α_n$ are not inner, we show that such actions are not classifiable by equivariant K-theory. We give a complete classification of such actions using twisted equivariant K-theory.
General semifinite factor representations of the diffeomorphism group of euclidean space are constructed by means of a canonical correspondence with the finite factor representations of the inductive limit unitary group. This construction includes the quasi-free representations of the canonical commutation and anti-commutation relations. To establish this correspondence requires a non-linear form of complete positivity as developed by Arveson. We also compare the asymptotic character formula for the unitary group with the thermodynamic ($N/V$) limit construction for diffeomorphism group repres…
Generalizing the notion of continuous Hilbert space representations of compact topological groups we define unitary continuous correpresentations of $C^*$-completions of compact quantum group Hopf algebras on arbitrary Hilbert spaces. It is proved that the unitary continuous correpresentations decompose in finite dimensional irreducible correpresentations.
We determine a substantial part of the unitary representation theory of the Drinfeld double of a $q$-deformation of a compact Lie group in terms of the complexification of the compact Lie group. Using this, we show that the dual of every $q$-deformation of a higher rank compact Lie group has central property ${\rm (T)}$. We also determine the unitary dual of $SL_q(n,\mathbb{C})$.
We study unitary representations of semidirect products of a compact quantum group with a finite group. We give a classification of all irreducible unitary representations, a description of the conjugate representation of irreducible unitary representations in terms of this classification, and the fusion rules for the semidirect product.
We study some examples of minimal length curves in homogeneous spaces of B(H) under a left action of a unitary group. Recent results relate these curves with the existence of minimal (with respect to a quotient norm) anti-Hermitian operators Z in the tangent space of the starting point. We show minimal curves that are not of this type but nevertheless can be approximated uniformly by those.
Let v be the right regular representation of a compact quantum group G. Then (S.L.Woronowicz, "Compact quantum groups") v contains all irreducible representations of G and each irreducible representation enters v with the multiplicity equal to its dimension. The result is certainly known for classical compact groups. We give a short survey on the subject and provide a different proof of Woronowicz's result above. The proof is an adaptation of the corresponding result for classical compact groups and provides a concrete decomposition of the right regular representation in irreducible components…
We show that for every $N\ge 3$ the free unitary group $U^+_N$ is topologically generated by its classical counterpart $U_N$ and the lower-rank $U^+_{N-1}$. This allows for a uniform inductive proof that a number of finiteness properties, known to hold for all $N\ne 3$, also hold at $N=3$. Specifically, all discrete quantum duals $\widehat{U^+_N}$ and $\widehat{O^+_N}$ are residually finite, and hence also have the Kirchberg factorization property and are hyperlinear. As another consequence, $U^+_N$ are topologically generated by $U_N$ and their maximal tori $\widehat{\mathbb{Z}^{*N}}$ (dual t…
In this short note, further to Ng's study, we extend Bekka amenability and weak Bekka amenability to general locally compact quantum groups. We generalize some Ng's results to the general case. In particular, we show that, a locally compact quantum group $\mathbb{G}$ is co-amenable if and only if the contra-corepresentation of its fundamental multiplicative unitary $W_{\mathbb{G}}$ is Bekka amenable, and $\mathbb{G}$ is amenable if and only if its dual quantum group's fundamental multiplicative unitary $W_{\widehat{\mathbb{G}}}$ is weakly Bekka amenable.
We prove that amenability of a unitary co-representation $U$ of a locally compact quantum group passes to unitary co-representations that weakly contain $U$. This generalizes a result of Bekka, and answers affirmatively a question of Bédos, Conti and Tuset. As a corollary, we extend to locally compact quantum groups a result of the first-named author, which characterizes amenability of a locally compact group $G$ by nuclearity of the reduced group $C^{*}$-algebra $C_{r}^{*}(G)$ and an additional condition.
We show that the unitary conjugacy relation for unitary representations of a second countable locally compact group on a separable Hilbert space is a Borel equivalence relation.
Suppose that $G$ is a locally compact group and $π$ is a (not necessarily irreducible) unitary representation of a closed normal subgroup $N$ of $G$ on a Hilbert space $H$. We extend results of Clifford and Mackey to determine when $π$ extends to a unitary representation of $G$ on the same space $H$ in terms of a cohomological obstruction.
Let H be a separable Hilbert space, and D(B(H))^ah the anti-Hermitian bounded diagonals in some fixed orthonormal basis and K(H) the compact operators. We study the group of unitary operators U_kd = {u in U(H): such that u-e^D in K(H) for D in D(B(H))^ah} in order to obtain a concrete description of short curves in unitary Fredholm orbits Ob={ e^K b e^{-K} : K in K(H)^ah } of a compact self-adjoint operator b with spectral multiplicity one. We consider the rectifiable distance on Ob defined as the infimum of curve lengths measured with the Finsler metric defined by means of the quotient space …
In this paper we study actions of locally compact quantum groups on von Neumann algebras and prove that every action has a canonical unitary implementation, paralleling Haagerup's classical result on the unitary implementation of a locally compact group action. This result is an important tool in the study of quantum groups in action. We will use it in this paper to study subfactors and inclusions of von Neumann algebras. When alpha is an action of a locally compact quantum group on the von Neumann algebra N we can give necessary and sufficient conditions under which the inclusion of the fixed…
Wieler has shown that every irreducible Smale space with totally disconnected stable sets is a solenoid (i.e., obtained via a stationary inverse limit construction). Using her construction, we show that the associated stable C*-algebra is the stationary inductive limit of a C*-stable Fell algebra that has compact spectrum and trivial Dixmier-Douady invariant. This result applies in particular to Williams solenoids along with other examples. Beyond the structural implications of this inductive limit, one can use this result to in principle compute the K-theory of the stable C*-algebra. A specif…
We study the unitary orbit of a compact Hermitian diagonal operator with spectral multiplicity one under the action of the unitary group U_(K+C) of the unitization of the compact operators K(H)+C, or equivalently, the quotient U_(K+C)/ U_Diag(K+C). We relate this and the action of different unitary subgroups to describe metric geodesics (using a natural distance) which join end points. As a consequence we obtain a local Hopf-Rinow theorem. We also explore cases about the uniqueness of short curves and prove that there exist some of these that cannot be parameterized using minimal anti-Hermitia…
We construct equivariant $KK$-theory with coefficients in $\mathbb{R}$ and $\mathbb{R}/\mathbb{Z}$ as suitable inductive limits over ${\rm II}_1$-factors. We show that the Kasparov product, together with its usual functorial properties, extends to $KK$-theory with real coefficients. Let $Γ$ be a group. We define a $Γ$-algebra $A$ to be $K$-theoretically free and proper (KFP) if the group trace ${\bf tr}$ of $Γ$ acts as the unit element in $KK^Γ_{\mathbb{R}}(A,A)$. We show that free and proper $Γ$-algebras (in the sense of Kasparov) have the (KFP) property. Moreover, if $Γ$ is torsion free and …
Weyl-von Neumann Theorem asserts that two bounded self-adjoint operators $A,B$ on a Hilbert space $H$ are unitarily equivalent modulo compacts, i.e., $uAu^*+K=B$ for some unitary $u\in \mathcal{U}(H)$ and compact self-adjoint operator $K$, if and only if $A$ and $B$ have the same essential spectra: $σ_{\rm{ess}}(A)=σ_{\rm{ess}}(B)$. In this paper we consider to what extent the above Weyl-von Neumann's result can(not) be extended to unbounded operators using descriptive set theory. We show that if $H$ is separable infinite-dimensional, this equivalence relation for bounded self-adjoin operators…
We propose a definition of compact quantum groupoids in the setting of C*-algebras, associate to such a quantum groupoid a regular C*-pseudo-multiplicative unitary, and use this unitary to construct a dual Hopf C*-bimodule and to pass to a measurable quantum groupoid in the sense of Enock and Lesieur. Moreover, we discuss examples related to compact and to étale groupoids and study principal compact C*-quantum groupoids.
The generalization of multiplicative unitary notion from compact quantum groups to compact quantum semigroups is considered. We show why the same construction doesn't work in this case by giving examples of C*-algebras with non-trivial comultiplication which do not admit multiplicative unitaries. By the use of the pentagon equation we suggest a notion of an operator which gives comultiplication on any C*-algebra. The multiplicative unitary turns out to be its special case. We prove for some compact quantum semigroups that the comultiplication is given by such operator.
K-Theory for hermitian symmetric spaces of non-compact type, as developed recently by the authors, allows to put Cartan's classification into a homological perspective. We apply this method to the case of inductive limits of finite dimensional hermitian symmetric spaces. This might be seen as an indication of how much more powerful the homological theory is in comparison to the more classical approach. When seen from high above, we follow the path laid out by a similar result in the theory of C*-algebras. Important is a clear picture of the behavior of morphisms between bounded symmetric domai…
Let $\G$ be a unimodular type I second countable locally compact group and $\wG$ its unitary dual. Motivated by a recent pseudo-differential calculus, we develop a positive Berezin-type quantization with operator-valued symbols defined on $\wG\times\G$.
Mimicking the von Neumann version of Kustermans and Vaes' locally compact quantum groups, Franck Lesieur had introduced a notion of measured quantum groupoid, in the setting of von Neumann algebras. In a former article, the author had introduced the notion of actions, crossed-product, dual actions of a measured quantum groupoid: a biduality theorem for actions had been proved. This article continues that program : we prove the existence of a standard implementation for an action, and a bidulaity theorem for weights. We generalize this way results which were proved, for locally compact quantum …
We study irreducible spherical unitary representations of the Drinfeld double of a $q$-deformation of a connected simply connected compact Lie group, which can be considered as a quantum analogue of the complexification of the Lie group. In the case of $SU_q(3)$, we give a complete classification of such representations. As an application, we show the Drinfeld double of the quantum group $SU_q(2n+1)$ has property (T), which also implies central property (T) of the dual of $SU_q(2n+1)$.
Let $\mathbf{G}$ be an algebraic group over a local field $\mathbf k$ of characteristic zero. We show that the locally compact group $\mathbf G(\mathbf k)$ consisting of the $\mathbf k$-rational points of $\mathbf G$ is of type I. Moreover, we complete Lipsman's characterization of the groups $\mathbf G$ for which every irreducible unitary representation of $\mathbf G(\mathbf k)$ is a CCR representation and show at the same time that such groups $\mathbf G(\mathbf k)$ are trace class as studied recently by Deitmar and van Dijk.
Let G be a locally compact group and rho a non-unitary finite dimensional representation of G. We consider tensor products of rho by some unitary representations of G in order to define two Banach algebras analogous to the group C*-algebras, C*(G) and C*_r(G). We calculate the K-theory of such algebras for a large class of groups satisfying the Baum-Connes conjecture.
Let $H$ be a complex Hilbert space of dimension not less than $3$ and let ${\mathcal C}$ be a conjugacy class of compact self-adjoint operators on $H$. Suppose that the dimension of the kernels of operators from ${\mathcal C}$ not less than the dimension of their ranges. In the case when ${\mathcal C}$ is formed by operators of finite rank $k$ and $\dim H=2k$, we assume that $k\ge 4$. We show that every bijective transformation of C preserving the commutativity in both directions is induced by a unitary or anti-unitary operator up to a permutation of eigenspaces of the same dimension.
Let $U_c(H)={u: u is unitary and u-1 is compact}$ stand for the unitary Fredholm group. We prove the following convexity result. Denote by $d_\infty$ the rectifiable distance induced by the Finsler metric given by the operator norm in $U_c(H)$. If $u_0,u_1,u\in U_c(H)$ and the geodesic $β$ joining $u_0$ and $u_1$ in $U_c(H)$ verifies $d_\infty(u,β)<π/2$, then the map $f(s)=d_\infty(u,β(s))$ is convex for $s\in[0,1]$. In particular the convexity radius of the geodesic balls in $U_c(H)$ is $π/4$. The same convexity property holds in the $p$-Schatten unitary groups $U_p(H)={u: u is unitary and u-…
We introduce the construction of induced corepresentations in the setting of locally compact quantum groups and prove that the resulting induced corepresentations are unitary under some mild integrability condition. We also establish a quantum analogue of the classical bijective correspondence between quasi-invariant measures and certain measures on the larger locally compact group.
Admissible vectors for unitary representations of locally compact groups are the basis for group-frame and covariant coherent state expansions. Main tools in the study of admissible vectors have been Plancherel and central integral decomposition, of applicability only under certain separability and semifiniteness restrictions. In this work we present a study of admissible vectors in terms of convolution Hilbert algebras valid for arbitrary unitary representations of general locally compact groups.
We find a combinatorial formula for the Haar functional of the orthogonal and unitary quantum groups. As an application, we consider diagonal coefficients of the fundamental representation, and we investigate their spectral measures.
We prove that any II$_1$ factor $L^{\infty}(X)\rtimesΓ$ arising from a compact, free, ergodic, probability measure preserving action of a countable group $Γ$ with positive first $\ell^2$-Betti number, has a unique group measure space Cartan subalgebra, up to unitary conjugacy.
