arXiv subjects
Publications and source records attributed to Sergey Neshveyev.
We give a self-contained account of a version of Renault's disintegration theorem for (twisted) C$^*$-algebras of not necessarily Hausdorff \'etale groupoids that can be covered by countably many open bisections. As an application we discuss the $I$-norm, crossed product decompositions by inverse semigroup actions, and KMS-states and weights for general \'etale groupoid C$^*$-algebras.
Given a pseudo-free self-similar action of a countable group $G$ on a countable directed graph $E$ with amenable stabilizers of the vertices, we identify the exact conditions under which these stabilizers do not contribute to the ideal structure of the corresponding Exel-Pardo algebra $\mathcal{O}_{G,E}$. Under these conditions, we give a complete description of the primitive ideal space of $\mathcal{O}_{G,E}$ in graph-theoretic terms. Our results apply in particular to certain crossed products $\mathcal{O}_E\rtimes G$, where $G$ acts on $E$ by graph automorphisms. When $G$ is trivial, this recovers Hong-Szymanski's description of the ideal structure of the Cuntz-Krieger algebras $\mathcal{O}_E$. Similar results are then obtained for self-similar actions of groups on row-finite higher rank graphs without sources. In order to obtain these results we formalize the notion of a graded groupoid with essentially central isotropy, which generalizes essentially principal groupoids and groupoids injectively graded by abelian groups. Under the amenability and second countability assumptions, we describe the primitive ideal spaces of the corresponding C$^*$-algebras as topological spaces.
We show how to construct unitary dual $2$-cocycles for a class of semidirect products that exhibit many similarities with the affine group ${\rm Aff}(V)=\GL(V)\ltimes V$ of a finite dimensional vector space over a local skew field. The primary source of examples comes from Lie groups whose Lie algebras are Frobenius seaweeds. The construction builds on our earlier results and relies heavily on representation theory and an associated quantization procedure of Kohn--Nirenberg type. On the technical side, the key point is the observation that any semidirect product $G=H\ltimes V$ in our class can be presented as a double crossed product $G=P\bowtie N$ with respect to which the unique square-integrable irreducible representation of $G$ takes a particularly nice form. The Kohn--Nirenberg quantization that we construct is intimately related to a scalar Fourier transform $\CF\colon L^2(N)\to L^2(P)$ intertwining the left regular representations of $P$ and $N$ with representations defined by the dressing transformations.
Given a semisimple compact Lie group $G$ and a nonzero dominant integral weight $\lambda$, the highest weight $G_q$-modules $V_{n\lambda}$ form a subproduct system of finite dimensional Hilbert spaces. Using a conjectural asymptotic behavior of Clebsch-Gordan coefficients we identify the corresponding Cuntz-Pimsner algebras with algebras of quantized functions on homogeneous spaces of $G$. We also show that the gauge-invariant part of the Toeplitz algebra provides a model for convergence of full matrix algebras to quantum flag manifolds, complementing and generalizing results of Landsman and Rieffel for $q=1$ and results of Vaes-Vergnioux in the rank one case for $q\ne1$. We verify our conjecture on Clebsch-Gordan coefficients for $G=SU(n)$ and all weights that are either regular or multiples of the fundamental weight $\omega_1$. For $\lambda=\omega_1$, we also provide a detailed description of the Toeplitz and Cuntz-Pimsner algebras, generalizing results of Arveson on symmetric subproduct systems.
Given an amenable second countable Hausdorff locally compact \'etale groupoid $\mathcal G$ such that each isotropy group $\mathcal G^x_x$ has local polynomial growth, we give a description of $\operatorname{Prim} C^*(\mathcal G)$ as a topological space in terms of the topology on $\mathcal G$ and representation theory of the isotropy groups and their subgroups. The description simplifies when either the isotropy groups are FC-hypercentral or $\mathcal G$ is the transformation groupoid $\Gamma\ltimes X$ defined by an action $\Gamma\curvearrowright X$ with locally finite stabilizers. To illustrate the class of C$^*$-algebras for which our results can provide a complete description of the ideal structure, we compute the primitive spectrum of $\mathrm{SL}_3(\mathbb Z)\ltimes C_0(\mathrm{SL}_3(\mathbb R)/U_3(\mathbb R))$, where $U_3(\mathbb R)$ is the group of unipotent upper triangular matrices.
Given a Hausdorff locally compact \'etale groupoid $\mathcal G$, we describe as a topological space the part of the primitive spectrum of $C^*(\mathcal G)$ obtained by inducing one-dimensional representations of amenable isotropy groups of $\mathcal G$. When $\mathcal G$ is amenable, second countable, with abelian isotropy groups, our result gives the description of $\operatorname{Prim} C^*(\mathcal G)$ conjectured by van Wyk and Williams. This, in principle, completely determines the ideal structure of a large class of separable C$^*$-algebras, including the transformation group C$^*$-algebras defined by amenable actions of discrete groups with abelian stabilizers and the C$^*$-algebras of higher rank graphs. As an illustration we describe the primitive spectrum of the C$^*$-algebra of any row-finite higher rank graph without sources.
We prove that the Cuntz-Pimsner algebra of every Temperley-Lieb subproduct system is KK-self-dual. We show also that every such Cuntz-Pimsner algebra has a canonical KMS-state, which we use to construct a Fredholm module representative for the fundamental class of the duality. This allows us to describe the K-homology of the Cuntz-Pimsner algebras by explicit Fredholm modules. Both the construction of the dual class and the proof of duality rely in a crucial way on quantum symmetries of Temperley-Lieb subproduct systems. In the simplest case of Arveson's $2$-shift our work establishes $U(2)$-equivariant KK-self-duality of $S^3$.
Given a C$^*$-algebra $A$ with an almost periodic time evolution $\sigma$, we define a new C$^*$-algebra $A_c$, which we call the crystal of $(A,\sigma)$, that represents the zero temperature limit of $(A, \sigma)$. We prove that there is a one-to-one correspondence between the ground states of $(A,\sigma)$ and the states on $A_c$, justifying the name. In order to investigate further the relation between low temperature equilibrium states on $A$ and traces on $A_c$, we define a Fock module $\mathcal F$ over the crystal and construct a vacuum representation of $A$ on $\mathcal F$. This allows us to show, under relatively mild assumptions, that for sufficiently large inverse temperatures $\beta$ the $\sigma$-KMS$_\beta$-states on $A$ are induced from traces on $A_c$ by means of the Fock module. In the second part, we compare the K-theoretic structures of $A$ and $A_c$. Previous work by various authors suggests that they have (rationally) isomorphic K-groups. We analyze this phenomenon in detail, confirming it under favorable conditions, but showing that, in general, there is apparently no easy way to relate these groups. As examples, we discuss in particular Exel's results on semi-saturated circle actions, and recent results of Miller on the K-theory of inverse semigroup C$^*$-algebras. In relation to the latter, we introduce the notion of a scale $N$ on an inverse semigroup $I$ and define a new inverse semigroup $I_c$, which we call the crystal of $(I,N)$.
Given an extension $0\to V\to G\to Q\to1$ of locally compact groups, with $V$ abelian, and a compatible essentially bijective $1$-cocycle $\eta\colon Q\to\hat V$, we define a dual unitary $2$-cocycle on $G$ and show that the associated deformation of $\hat G$ is a cocycle bicrossed product defined by a matched pair of subgroups of $Q\ltimes\hat V$. We also discuss an interpretation of our construction from the point of view of Kac cohomology for matched pairs. Our setup generalizes that of Etingof and Gelaki for finite groups and its extension due to Ben David and Ginosar, as well as our earlier work on locally compact groups satisfying the dual orbit condition. In particular, we get a locally compact quantum group from every involutive nondegenerate set-theoretical solution of the Yang--Baxter equation, or more generally, from every brace structure. On the technical side, the key new points are constructions of an irreducible projective representation of $G$ on $L^2(Q)$ and a unitary quantization map $L^2(G)\to{\rm HS}(L^2(Q))$ of Kohn--Nirenberg type.
Given a locally compact \'etale groupoid and an ideal $I$ in its groupoid C$^*$-algebra, we show that $I$ defines a family of ideals in group C$^*$-algebras of the isotropy groups and then study to which extent $I$ is determined by this family. As an application we obtain the following results: (a) prove that every proper ideal is contained in an induced primitive ideal; (b) describe the maximal ideals; (c) classify the primitive ideals for a class of graded groupoids with essentially central isotropy.
We apply Majid's transmutation procedure to Hopf algebra maps $H \to \mathbb C[T]$, where $T$ is a compact abelian group, and explain how this construction gives rise to braided Hopf algebras over quotients of $T$ by subgroups that are cocentral in $H$. This allows us to unify and generalize a number of recent constructions of braided compact quantum groups, starting from the braided $SU_q(2)$ quantum group, and describe their bosonizations.
We complete our analysis of the Temperley-Lieb subproduct systems, which define quantum analogues of Arveson's $2$-shift, by extending the main results of the previous paper to the general parameter case. Specifically, we show that the associated Toeplitz algebras are nuclear, find complete sets of relations for them, prove that they are equivariantly $KK$-equivalent to $\mathbb C$ and compute the $K$-theory of the associated Cuntz-Pimsner algebras. A key role is played by quantum symmetry groups, first studied by Mrozinski, preserving Temperley-Lieb polynomials up to rescaling, and their monoidal equivalence to $U_q(2)$.
We study the question whether the representations defined by a dense subset of the unit space of a locally compact \'etale groupoid are enough to determine the reduced norm on the groupoid C$^*$-algebra. We present sufficient conditions for either conclusion, giving a complete answer when the isotropy groups are torsion-free. As an application we consider the groupoid $G(S)$ associated to a left cancellative monoid $S$ by Spielberg and formulate a sufficient condition, which we call C$^*$-regularity, for the canonical map $C^*_r(G(S))\to C^*_r(S)$ to be an isomorphism, in which case $S$ has a well-defined full semigroup C$^*$-algebra $C^*(S)=C^*(G(S))$. We give two related examples of left cancellative monoids $S$ and $T$ such that both are not finitely aligned and have non-Hausdorff associated \'etale groupoids, but $S$ is C$^*$-regular, while $T$ is not.
We introduce a class of subproduct systems of finite dimensional Hilbert spaces whose fibers are defined by the Jones-Wenzl projections in Temperley-Lieb algebras. The quantum symmetries of a subclass of these systems are the free orthogonal quantum groups. For this subclass, we show that the corresponding Toeplitz algebras are nuclear C$^*$-algebras that are $KK$-equivalent to $\mathbb C$ and obtain a complete list of generators and relations for them. We also show that their gauge-invariant subalgebras coincide with the algebras of functions on the end compactifications of the duals of the free orthogonal quantum groups. Along the way we prove a few general results on equivariant subproduct systems, in particular, on the behavior of the Toeplitz and Cuntz-Pimsner algebras under monoidal equivalence of quantum symmetry groups.
Given a free unitary quantum group $G=A_u(F)$, with $F$ not a unitary $2$-by-$2$ matrix, we show that the Martin boundary of the dual of $G$ with respect to any $G$-$\hat G$-invariant, irreducible, finite range quantum random walk coincides with the topological boundary defined by Vaes and Vander Vennet. This can be thought of as a quantum analogue of the fact that the Martin boundary of a free group coincides with the space of ends of its Cayley tree.
We show that two approaches to equivariant strict deformation quantization of C*-algebras by actions of negatively curved Kahlerian Lie groups, one based on oscillatory integrals and the other on quantizations maps defined by dual 2-cocycles, are equivalent.
Given a discrete quantum group $H$ with a finite normal quantum subgroup $G$, we show that any positive, possibly unbounded, harmonic function on $H$ with respect to an irreducible invariant random walk is $G$-invariant. This implies that, under suitable assumptions, the Poisson and Martin boundaries of $H$ coincide with those of $H/G$. A similar result is also proved in the setting of exact sequences of C$^*$-tensor categories. As an immediate application, we conclude that the boundaries of the duals of the group-theoretical easy quantum groups are classical.
We give a dynamical characterization of categorical Morita equivalence between compact quantum groups. More precisely, by a Tannaka-Krein type duality, a unital C*-algebra endowed with commuting actions of two compact quantum groups corresponds to a bimodule category over their representation categories. We show that this bimodule category is invertible if and only if the actions are free, with finite dimensional fixed point algebras, which are in duality as Frobenius algebras in an appropriate sense. This extends the well-known characterization of monoidal equivalence in terms of bi-Hopf-Galois objects.