Group Freeness and Certain Amalgamated Freeness

Free Entropy Dimension in Amalgamated Free Products

We calculate the microstates free entropy dimension of natural generators in an amalgamated free product of certain von Neumann algebras, with amalgamation over a hyperfinite subalgebra. In particular, some `exotic' Popa algebra generators of free group factors are shown to have the expected free entropy dimension. We also show that microstates and non--microstates free entropy dimension agree for generating sets of many groups. In the appendix by Wolfgang Lueck, the first L^2-Betti number for certain amalgamated free products of groups is calculated.

迹超乘积中的渐近自由性

Asymptotic freeness in tracial ultraproducts

本文证明了迹超乘积冯·诺伊曼代数中关于相对换位子的渐近自由性结果，并基于此推导出一般吸收结论与新的提升定理，构造了首个不具有性质Gamma且不与任何弥散迹冯·诺伊曼代数自由乘积初等等价的II_1因子实例。

Topological entropy of some automorphisms of reduced amalgamated free product C*-algebras

Certain classes of automorphisms of recued amalgamated free products of C*-algebras are shown to have Brown-Voiculescu topological entropy zero. Also, for automorphisms of exact C*-algebras, the Connes-Narnhofer-Thirring entropy is shown to be bounded above by the Brown-Voiculescu entropy. These facts are applied to generalize Stormer's result about entropy of automorphisms of the II_1-factor of a free group.

Cuntz-Krieger-Pimsner Algebras Associated with Amalgamated Free Product Groups

We give a construction of a nuclear $C^\ast$-algebra associated with an amalgamated free product of groups, generalizing Spielberg's construction of a certain Cuntz-Krieger algebra associated with a finitely generated free product of cyclic groups. Our nuclear $C^\ast$-algebras can be identified with certain Cuntz-Krieger-Pimsner algebras. We will also show that our algebras can be obtained by the crossed product construction of the canonical actions on the hyperbolic boundaries, which proves a special case of Adams' result about amenability of the boundary action for hyperbolic groups. We wil…

某些群的融合自由积的有限维逼近

Finite dimensional approximations of certain amalgamated free products of groups

本文证明具有共同正规子群的两个顺从群的融合自由积是MF群，从而提供了一类新的MF群。

Amalgamated R-diagonal Pairs

In this paper, we will consider the properties of amalgamated R-diagonal pairs. We characterize the amalgamated R-diagonality of pairs of amalgamated random variables by certain cumulant-relation.

Symmetric random walks on certain amalgamated free product groups

We consider nearest-neighbor random walks on free products of finitely many copies of the integers with amalgamation over nontrivial subgroups. When all the subgroups have index two, we find the Green function of the random walks in terms of complete elliptic integrals. Our technique is to apply Voiculescu's operator--valued R--transform.

Spectra of infinite graphs via freeness with amalgamation

We use tools from free probability to study the spectra of Hermitian operators on infinite graphs. Special attention is devoted to universal covering trees of finite graphs. For operators on these graphs we derive a new variational formula for the spectral radius and provide new proofs of results due to Sunada and Aomoto using free probability. With the goal of extending the applicability of free probability techniques beyond universal covering trees, we introduce a new combinatorial product operation on graphs and show that, in the non-commutative probability context, it corresponds to the no…

A relationship between HNN extensions and amalgamated free products in operator algebras

With a minor change made in the previous construction we observe that any reduced HNN extension is precisely a compressed algebra of a certain reduced amalgamated free product in both the von Neumann algebra and the $C^*$-algebra settings. It is also pointed out that the same fact holds true even for universal HNN extensions of C$^*$-algebras. We apply the observation to the questions of factoriality and type classification of HNN extensions of von Neumann algebras and also those of simplicity and $K$-theory of (reduced or universal) HNN extensions of $C^*$-algebras.

Tensor product decompositions of II$_1$ factors arising from extensions of amalgamated free product groups

In this paper we introduce a new family of icc groups $Γ$ which satisfy the following product rigidity phenomenon, discovered in [DHI16] (see also [dSP17]): all tensor product decompositions of the II$_1$ factor $L(Γ)$ arise only from the canonical direct product decompositions of the underlying group $Γ$. Our groups are assembled from certain HNN-extensions and amalgamated free products and include many remarkable groups studied throughout mathematics such as graph product groups, poly-amalgam groups, Burger-Mozes groups, Higman group, various integral two-dimensional Cremona groups, etc. As …

Embeddings of reduced free products of operator algebras

Given reduced amalgamated free products of C$^*$-algebras, $(A,phi)=*_i(A_i,phi_i)$ and $(D,psi)=*_i(D_i,psi_i)$, an embedding $A\to D$ is shown to exist assuming there are conditional expectation preserving embeddings $A_i\to D_i$. This result is extended to show the existance of the reduced amalgamated free product of certain classes of unital completely positive maps. Analogues of the above mentioned results are proved for von Neumann algebras.

Colimits in the correspondence bicategory

We interpret several constructions with C*-algebras as colimits in the bicategory of correspondences. This includes crossed products for actions of groups and crossed modules, Cuntz-Pimsner algebras of proper product systems, direct sums and inductive limits, and certain amalgamated free products.

Popa algebras in free group factors

For each 1<s<\infty, a Popa algebra A_s is constructed that embeds as a weakly dense C*-subalgebra of the interpolated free group factor L(F_s). Certain approximation properties for A_s are shown. It follows that L(F_s) has the weak expectation property of Lance with respect to A_s. In the course of the demonstration, it is proved that under certain conditions, full amalgamated free products of matrix algebras are residually finite dimensional.

Central amalgamation of groups and the RFD property

It is an old and challenging topic to investigate for which discrete groups G the full group C*-algebra C*(G) is residually finite-dimensional (RFD). In particular not much is known about how the RFD property behaves under fundamental constructions, such as amalgamated free products and HNN-extensions. In [CS19] it was proved that central amalgamated free products of virtually abelian groups are RFD. In this paper we prove that this holds much beyond this case. Our method is based on showing a certain approximation property for characters induced from central subgroups. In particular it allows…

Multiplicity freeness of unitary representations in sections of holomorphic Hilbert bundles

We prove several results asserting that the action of a Banach-Lie group on Hilbert spaces of holomorphic sections of a holomorphic Hilbert space bundle over a complex Banach manifold is multiplicity free. These results require the existence of compatible antiholomorphic bundle maps and certain multiplicity freeness assumptions for stabilizer groups. For the group action on the base, the notion of an $(S,σ)$-weakly visible action (generalizing T.Koboyashi's visible actions) provides an effective way to express the assumptions in an economical fashion. In particular, we derive a version for gro…

CAT(0) 空间的边界作用：拓扑自由性及其在 $C^*$-代数中的应用

Boundary actions of CAT(0) spaces: topological freeness and applications to $C^*$-algebras

本文通过验证 Myrberg 点的自由性，证明 CAT(0) 空间的多类边界作用（视觉、horofunction、Roller）是拓扑自由的强边界作用，推广了已知结果，并应用于构造新的 $C^*$-无自群及精确纯无限单的约化交叉积 $C^*$-代数。

On certain free product factors via an extended matrix model

Voiculescu's random matrix model for freeness is extended to the non-Gaussian case and also the case of constant block diagonal matrices. Thus we are able to investigate free products of free group factors with matrix algebras and with the hyperfinite II$_1$ factor, showing that $$ L(F_n) * R = L(F_(n+1)) $$ for $n \ge 1$, (where $L(F_1)=L(Z)$).

Free actions of groups on separated graph C*-algebras

In this paper we study free actions of groups on separated graphs and their \cstar{}algebras, generalizing previous results involving ordinary (directed) graphs. We prove a version of the Gross-Tucker Theorem for separated graphs yielding a characterization of free actions on separated graphs via a skew product of the (orbit) separated graph by a group labeling function. Moreover, we describe the C*-algebras associated to these skew products as crossed products by certain coactions coming from the labeling function on the graph. Our results deal with both the full and the reduced C*-algebras o…

Group measure space decomposition of II_1 factors and W*-superrigidity

We prove a "unique crossed product decomposition" result for group measure space II_1 factors arising from arbitrary free ergodic probability measure preserving (p.m.p.) actions of groups Γin a fairly large family G, which contains all free products of a Kazhdan group and a non-trivial group, as well as certain amalgamated free products over an amenable subgroup. We deduce that if T_n denotes the group of upper triangular matrices in PSL(n,Z), then any free, mixing p.m.p. action of the amalgamated free product of PSL(n,Z) with itself over T_n, is W*-superrigid, i.e. any isomorphism between L^\…

Connes-可嵌入性与扭曲群 von Neumann 代数

Connes-embeddability and twisted group von Neumann algebras

本文定义余循环-超线性群，证明其包含特定扩张与中心扩张，并在多种群运算下封闭，从而推广 Connes-可嵌入性。

A class of groups for which every action is W*-superrigid

We prove the uniqueness of the group measure space Cartan subalgebra in crossed products A \rtimes Γcovering certain cases where Γis an amalgamated free product over a non-amenable subgroup. In combination with Kida's work we deduce that if Σ< SL(3,\Z) denotes the subgroup of matrices g with g_31 = g_32 = 0, then any free ergodic probability measure preserving action of Γ= SL(3,\Z) *_ΣSL(3,\Z) is stably W*-superrigid. In the second part we settle a technical issue about the unitary conjugacy of group measure space Cartan subalgebras.

中间交叉积 $C^*$-代数

Intermediate crossed product $C^*$-algebras

本文研究可分 $C^*$-代数受离散可数群作用时的中间交叉积，给出自由性条件以刻画中间代数的交叉积结构，并推广 Suzuki 的结果。

CAT(0) cube complexes and inner amenability

We here consider inner amenability from a geometric and group theoretical perspective. We prove that for every non-elementary action of a group $G$ on a finite dimensional irreducible CAT(0) cube complex, there is a nonempty $G$-invariant closed convex subset such that every conjugation invariant mean on $G$ gives full measure to the stabilizer of each point of this subset. Specializing our result to trees leads to a complete characterization of inner amenability for HNN-extensions and amalgamated free products. One novelty of the proof is that it makes use of the existence of certain idempote…

On the Exel crossed product of topological covering maps

For dynamical systems defined by a covering map of a compact Hausdorff space and the corresponding transfer operator, the associated crossed product $C^*$-algebras $\cros$ introduced by Exel and Vershik are considered. An important property for homeomorphism dynamical systems is topological freeness. It can be extended in a natural way to in general non-invertible dynamical systems generated by covering maps. In this article, it is shown that the following four properties are equivalent: the dynamical system generated by a covering map is topologically free; the canonical imbedding of $C(X)$ i…