We introduce and study the notions of boundary actions and of the Furstenberg boundary of a discrete quantum group. As for classical groups, properties of boundary actions turn out to encode significant properties of the operator algebras associated with the discrete quantum group in question; for example we prove that if the action on the Furstenberg boundary is faithful, the quantum group C*-algebra admits at most one KMS-state for the scaling automorphism group. To obtain these results we develop a version of Hamana's theory of injective envelopes for quantum group actions, and establish several facts on relative amenability for quantum subgroups. We then show that the Gromov boundary actions of free orthogonal quantum groups, as studied by Vaes and Vergnioux, are also boundary actions in our sense; we obtain this by proving that these actions admit unique stationary states. Moreover, we prove these actions are faithful, hence conclude a new unique KMS-state property in the general case, and a new proof of unique trace property when restricted to the unimodular case. We prove equivalence of simplicity of the crossed products of all boundary actions of a given discrete quantum group, and use it to obtain a new simplicity result for the crossed product of the Gromov boundary actions of free orthogonal quantum groups.
We introduce a general class of automorphisms of rotation algebras, the noncommutative Furstenberg transformations. We prove that fully irrational noncommutative Furstenberg transformations have the tracial Rokhlin property, which is a strong form of outerness. We conclude that crossed products by these automorphisms have stable rank one, real rank zero, and order on projections determined by traces (Blackadar's Second Fundamental Comparability Question). We also prove that several classes of simple quotients of the C*-algebras of discrete subgroups of five dimensional nilpotent Lie groups, co…
We define the Furstenberg boundary of a locally compact Hausdorff étale groupoid, generalising the Furstenberg boundary for discrete groups, by providing a construction of a groupoid-equivariant injective envelope. Using this injective envelope, we establish the absence of recurrent amenable subgroups in the isotropy as a sufficient criterion for the intersection property of a locally compact Hausdorff étale groupoid with compact unit space and no fixed points. This yields a criterion for C*-simplicity of minimal groupoids.
For a countable discrete group Γ and a minimal Γ-space X, we study the notion of (Γ, X)-boundary, which is a natural generalization of the notion of topological Γ-boundary in the sense of Furstenberg. We give characterizations of the (Γ, X)-boundary in terms of essential or proximal extensions. The characterization is used to answer a problem of Hadwin and Paulsen in negative. As an application, we find necessary and sufficient condition for the corresponding reduced crossed product to be exact.
We introduce and study a generalization of the notion of the Furstenberg boundary of a discrete group $Γ$ to the setting of a general unitary representation $π: Γ\to B(\mathcal H_π)$. This space, which we call the "Furstenberg-Hamana boundary" of the pair $(Γ, π)$, is a $Γ$-invariant subspace of $B(\mathcal H_π)$ that carries a canonical $C^*$-algebra structure. In many natural cases, including when $π$ is a quasi-regular representation, the Furstenberg-Hamana boundary of $π$ is commutative, but can be non-commutative in general. We study various properties of this boundary, and give some appl…
A C*-dynamical system is said to have the ideal separation property if every ideal in the corresponding crossed product arises from an invariant ideal in the C*-algebra. In this paper we characterize this property for unital C*-dynamical systems over discrete groups. To every C*-dynamical system we associate a "twisted" partial C*-dynamical system that encodes much of the structure of the action. This system can often be "untwisted," for example when the algebra is commutative, or when the algebra is prime and a certain specific subgroup has vanishing Mackey obstruction. In this case, we obtai…
The noncommutative Choquet boundary and the C*-envelope of operator systems of the form Span{1,T,T*}, where T is a Hilbert space operator with normal-like features, are studied. Such operators include normal operators, k-normal operators, subnormal operators, and Toeplitz operators. Our main result is the determination of the noncommutative Choquet boundary for an operator system generated by an irreducible periodic weighted unilateral shift operator.
For a discrete group G, we consider the minimal C*-subalgebra of $\ell^\infty(G)$ that arises as the image of a unital positive G-equivariant projection. This algebra always exists and is unique up to isomorphism. It is trivial if and only if G is amenable. We prove that, more generally, it can be identified with the algebra $C(\partial_F G)$ of continuous functions on Furstenberg's universal G-boundary $\partial_F G$. This operator-algebraic construction of the Furstenberg boundary has a number of interesting consequences. We prove that G is exact precisely when the G-action on $\partial_F G$…
We study an example of an index problem for a Dirac-like operator subject to Atiyah-Patodi-Singer boundary conditions on a noncommutative manifold with boundary, namely the quantum unit disk.
We define a noncommutative space we call the quantum solid torus. It is an example of a noncommutative manifold with a noncommutative boundary. We study quantum Dirac type operators subject to Atiyah-Patodi-Singer like boundary conditions on the quantum solid torus. We show that such operators have compact inverse, which means that the corresponding boundary value problem is elliptic.
A discrete group $Γ$ is C*-simple if the C*-algebra $C_λ^*(Γ)$ generated by the range of the left regular representation $λ$ on $\ell^2(Γ)$ is simple. In this case, $Γ$ acts faithfully on the Furstenberg boundary $\partial_FΓ$ and there is a unique trace on $C_λ^*(Γ)$. In this paper we study the unique trace property for the C*-algebra $C_π^*(Γ)$ generated by the range of an arbitrary unitary representation $π: Γ\to B(H_π) $ and relate it to the faithfulness of the action of $Γ$ on the Furstenberg-Hamana boundary $\mathcal B_π$. Similar relation is obtained between simplicity of $C_π^*(Γ)$ and…
We extend applications of Furstenberg boundary theory to the study of $C^*$-algebras associated to minimal actions $Γ\!\curvearrowright\! X$ of discrete groups $Γ$ on locally compact spaces $X$. We introduce boundary maps on $(Γ,X)$-$C^*$-algebras and investigate their applications in this context. Among other results, we completely determine when $C^*$-algebras generated by covariant representations arising from stabilizer subgroups are simple. We also characterize the intersection property of locally compact $Γ$-spaces and simplicity of their associated crossed products.
For a large class of word hyperbolic groups G the cross product C^*-algebra arising from the action of G on its Gromov boundary is shown to satisfy Poincare duality in K-theory. This class strictly contains fundamental groups of compact, negatively curved manifolds. The Baum-Connes Conjecture for amenable groupoids is used in a crucial way.
Let S be an operator system -- a self-adjoint linear subspace of a unital C*-algebra A such that contains 1 and A=C*(S) is generated by S. A boundary representation for S is an irreducible representation πof C*(S) on a Hilbert space with the property that $π\restriction_S$ has a unique completely positive extension to C*(S). The set $\partial_S$ of all (unitary equivalence classes of) boundary representations is the noncommutative counterpart of the Choquet boundary of a function system $S\subseteq C(X)$ that separates points of X. It is known that the closure of the Choquet boundary of a func…
We introduce the notion of a pre-spectral triple, which is a generalisation of a spectral triple $(\mathcal{A}, H, D)$ where $D$ is no longer required to be self-adjoint, but closed and symmetric. Despite having weaker assumptions, pre-spectral triples allow us to introduce noncompact noncommutative geometry with boundary. In particular, we derive the Hochschild character theorem in this setting. We give a detailed study of Dirac operators with Dirichlet boundary conditions on open subsets of $\mathbb{R}^d$, $d \geq 2$.
We classify operator systems $S\subseteq \mathcal B(H)$ that act on finite dimensional Hilbert spaces by making use of the noncommutative Choquet boundary. S is said to be {\em reduced} when its boundary ideal is 0. In the category of operator systems, that property functions as semisimplicity does in the category of complex Banach algebras. We construct explicit examples of reduced operator systems using sequences of "parameterizing maps" $Γ_k: \mathbb C^r\to \mathcal B(H_k)$, $k=1,..., N$. We show that every reduced operator system is isomorphic to one of these, and that two sequences give r…
A (finite or countably infinite) set G of generators of an abstract C*-algebra A is called hyperrigid if for every faithful representation of A on a Hilbert space $A\subseteq \mathcal B(H)$ and every sequence of unital completely positive linear maps $ϕ_1, ϕ_2,...$ from $\mathcal B(H)$ to itself, $$ \lim_{n\to\infty}\|ϕ_n(g)-g\|=0, \forall g\in G \implies \lim_{n\to\infty}\|ϕ_n(a)-a\|=0, \forall a\in A. $$ We show that one can determine whether a given set G of generators is hyperrigid by examining the noncommutative Choquet boundary of the operator space spanned by $G\cup G^*$. We present a v…
The classical Julia-Wolff-Carath{é}odory Theorem characterizes the behaviour of the derivative of an analytic self-map of a unit disc or of a half-plane of the complex plane at certain boundary points. We prove a version of this result that applies to noncommutative self-maps of noncommutative half-planes in von Neumann algebras at points of the distinguished boundary of the domain. Our result, somehow surprisingly, relies almost entirely on simple geometric properties of noncommutative half-planes, which are quite similar to certain geometric properties of classical hyperbolic spaces.
We investigate an extension of ideas of Atiyah-Patodi-Singer (APS) to a noncommutative geometry setting framed in terms of Kasparov modules. We use a mapping cone construction to relate odd index pairings to even index pairings with APS boundary conditions in the setting of KK-theory, generalising the commutative theory. We find that Cuntz-Kreiger systems provide a natural class of examples for our construction and the index pairings coming from APS boundary conditions yield complete K-theoretic information about certain graph C*-algebras.
Let $(G,{\mathfrak I},N,S)$ be an affine topological Tits system, and let $Γ$ be a torsion free cocompact lattice in $G$. This article studies the coinvariants $H_0(Γ; C(Ω,{\mathbb Z}))$, where $Ω$ is the Furstenberg boundary of $G$. It is shown that the class $[1]$ of the identity function in $H_0(Γ; C(Ω,{\mathbb Z}))$ has finite order, with explicit bounds for the order. A similar statement applies to the $K_0$ group of the boundary crossed product $C^*$-algebra $C(Ω)\rtimesΓ$. If the Tits system has type $\widetilde A_2$, exact computations are given, both for the crossed product algebra an…
Let $Γ$ be a torsion free lattice in $G = \PGL(n+1,\FF)$, where $n\ge 1$ and $\FF$ is a non-archimedean local field. Then $Γ$ acts on the Furstenberg boundary $G/P$, where $P$ is a minimal parabolic subgroup of $G$. The identity element $\id$ in the crossed product $C^*$-algebra $C(G/P)\rtimes Γ$ generates a class $[\id]$ in the $K_0$ group of $C(G/P)\rtimes Γ$. It is shown that $[\id]$ is a torsion element of $K_0$ and there is an explicit bound for the order of $[\id]$. The result is proved more generally for groups acting on affine buildings of type $\tA_n$. For $n=1, 2$ the Euler-Poincaré …
Inspired by the intimate relationship between Voiculescu's noncommutative probability theory (of type A) and large-N matrix models in physics, we look for physical models related to noncommutative probability theory of type B. These turn out to be fermionic matrix-vector models at the double large-N limit. In the context of string theory, they describe different orbifolded string worldsheets with boundaries. Their critical exponents coincide with that of ordinary string worldsheets, but their renormalised tree-level one-boundary amplitudes differ.
We study operator spaces, operator algebras, and operator modules, from the point of view of the `noncommutative Shilov boundary'. In this attempt to utilize some `noncommutative Choquet theory', we find that Hilbert C$^*-$modules and their properties, which we studied earlier in the operator space framework, replace certain topological tools. We introduce certain multiplier operator algebras and C$^*-$algebras of an operator space, which generalize the algebras of adjointable operators on a C$^*-$module, and the `imprimitivity C$^*-$algebra'. It also generalizes a classical Banach space notio…
