Free Entropy Dimension in Amalgamated Free Products
Nathanial P. Brown, Kenneth J. Dykema, Kenley Jung
备注：
The second revised version significantly generalized the main result of the original one, (see the abstract) and contains a new appendix by Wolfgang Lueck. The third and fourth revisions correct some minor mistakes. The fifth version adds a result about embeddability of amalgmated 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.
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.
Some completely positive maps on reduced amalgamated free products of C*-algebras are constructed; these allow a proof that the class of exact unital C*-algebras is closed under taking reduced amalgamated free products. Consequently, the class of exact discrete groups is closed under taking amalgamated free products.
We investigate the use of Popa's asymptotic orthogonality to establish maximal amenability for amalgamated free product von Neumann algebras. Although new techniques have recently been developed to consider amalgamated free products, we find that the technique developed by Popa in the 1980s can be used to demonstrate maximal amenability for a certain family of amalgamated free product von Neumann algebras.
We prove that any reduced amalgamated free product C*-algebra is KK-equivalent to the corresponding full amalgamated free product C*-algebra. The main ingredient of its proof is Julg--Valette's geometric construction of Fredholm modules with Connes's view for representation theory of operator algebras.
Given two unital continuous C*-bundles A and B over the same Hausdorff compact base space X, we study the continuity properties of their different amalgamated free products over C(X).
In this paper we describe the amalgamated free product of two hyperfinite von Neumann algebras over a finite dimensional subalgebra. In general the free product is a finite direct sum of interpolated free group factors and a hyperfinite von Neumann algebra. We then show that the class of von Neumann algebras of this form is closed under taking amalgamated free products over finite dimensional subalgebras.
Several techniques together with some partial answers are given to the questions of factoriality, type classification and fullness for amalgamated free product von Neumann algebras.
In this paper, we observevd the amalgamated free probability of direct product of noncommutative probability spaces. We defined the amalgamated R-transforms, amalgamated moment series and the amalgamated boxed convolution. They maks us to do the amalgamated R-transform calculus, like the scalar-valued case.
We construct flows on an amalgamated free product factor and characterize the Rohlin property for them by freeness of $\mathbf{R}\times \mathbf{Z}$ actions on a measured space. As a corolally, it turns out that there exist Rohlin flows on an amalgamated free product factor. This shows that the classification theorem of Masuda and Tomatsu for the Rohlin flows is also applicable to flows on non-McDuff factors.
The basic notions and results of equivariant KK-theory concerning crossed products can be extended to the case of locally compact quantum groups. We recall these constructions and prove some usefull properties of subgroups and amalgamated free products of discrete quantum groups. Using these properties and a quantum analogue of the Bass-Serre tree, we establish the K-amenability of amalgamated free products of amenable discrete quantum groups.
We prove a relative analogue of equivalence between nuclearity and CPAP. In its proof, the notion of weak containment for C$^*$-correspondences plays an important role. As an application we prove $KK$-equivalence between full and reduced amalgamated free products of C$^*$-algebras under a strengthened variant of `relative nuclearity'.
In this paper we describe the amalgamated free product of finite and semifinite hyperfinite von Neumann algebras over atomic type I subalgebras. To do this we extend the notions of free dimension and standard embeddings used in the related results for finite von Neumann algebras to ones which work better for the semifinite case. We also define classes R3 (of finite von Neumann algebras) and R4 (of semifinite von Neumann algebras) which are closed under such amalgamated free products.
In this paper, we concentrate on the MF property of reduced free products of unital C*-algebras with amalgamation over finite dimensional C*-algebras. More specifically, we give a necessary and sufficient condition for a reduced free product of two UHF algebras amalgamated over a finite-dimensional C*-algebra with respect to trace preserving conditional expectations to be MF.
For any reduced amalgamated free product $\mathrm{C}^*$-algebra $(A,E)=(A_1, E_1) \ast_D (A_2,E_2)$, we introduce and study a canonical ambient $\mathrm{C}^*$-algebra $Δ\mathbf{T}(A,E)$ of $A$ which generalizes the crossed product arising from the canonical action of an amalgamated free product group on the compactification of the associated Bass--Serre tree. Using an explicit identification of $Δ\mathbf{T}(A,E)$ with a Cuntz--Pimsner algebra we prove two kinds of "amenability" results for $Δ\mathbf{T}(A,E)$; nuclearity and universality. As applications of our framework, we provide new concept…
In the paper, we consider the question whether a unital full amalgamated free product of quasidiagonal C*-algebras is quasidiagonal again. We give a sufficient condition such that a unital full amalgamated free product of quasidiagonal C*-algebras with amalgamation over a finite dimensional C*- algebra is quasidiagonal. Applying this result, we conclude that a unital full free product of two AF algebras with amalgamation over a finite-dimensional C*-algebra is AF if there are faithful tracial states on each of these two AF algebras such that the restrictions on the common subalgebra agree.
Here we generalize the concept of spatial tensor product, introduced by Skeide, of two product systems via a pair of normalized units. This new notion is called amalgamated tensor product of product systems, and now the amalgamation can be done using a contractive morphism. Index of amalgamation product (when done through units) adds up for normalized units but for non-normalized units, the index is one more than the sum. We define inclusion systems and use it as a tool for index computations. It is expected that this notion will have other uses.
Supplementary results obtained after the completion of our previous paper are given together with discussing some examples. A quick review of the previous paper is also included.
We investigate the position of amenable subalgebras in arbitrary amalgamated free product von Neumann algebras $M = M_1 \ast_B M_2$. Our main result states that under natural analytic assumptions, any amenable subalgebra of $M$ that has a large intersection with $M_1$ is actually contained in $M_1$. The proof does not rely on Popa's asymptotic orthogonality property but on the study of non normal conditional expectations.
We examine the question of when the *-homomorphism of full amalgamated free product C*-algebras λ: A *_D B --> A' *_{D'} B', arising from compatible inclusions of C*-algebras A in A', B in B' and D in D', is an embedding. Results giving sufficient conditions for λto be injective, as well of classes of examples where λfails to be injective, are obtained. As an application, we give necessary and sufficient condition for the full amalgamated free product of finite dimensional C*-algebras to be residually finite dimensional.
In this paper we study conditions under which, for an inclusion of finite von Neumann algebras $N \subseteq M$, we have the reduced amalgamated free product $\ast_N M$ is embeddable into $(R \bar{\otimes} N_1)^ω$ for some other finite von Neumann algebra $N_1$, where $R$ is the hyperfinite $\textrm{II}_1$ factor.
We provide a fairly large family of amalgamated free product groups $Γ=Γ_1\ast_ΣΓ_2$ whose amalgam structure can be completely recognized from their von Neumann algebras. Specifically, assume that $Γ_i$ is a product of two icc non-amenable bi-exact (e.g., hyperbolic) groups, and $Σ$ is icc amenable and has trivial one-sided commensurator in $Γ_i$, for every $i\in\{1,2\}$. Then $Γ$ satisfies the following rigidity property: any group $\La$ such that $L(\La)$ is isomorphic to $L(\G)$ admits an amalgamated free product decomposition $\La=\La_1\ast_Δ\La_2$ such that the inclusions $L(Δ)\subseteq L…
The $K$-groups of the crossed product of the rotation C*-algebra $A_θ$ by free and amalgamated products of the cyclic groups $\mathbb Z_n$, for $n=2,3,4,6$, are calculated. The actions here arise from the canonical actions of these groups on the rotation algebra under the flip, cubic, Fourier, and hexic automorphisms, respectively. An interesting feature in this study is that although the inclusion $A_θ\to A_θ\rtimes \mathbb Z_n$ induces injective maps on their $K_0$-groups, the same is not the case for the inclusions $A_θ\rtimes \mathbb Z_d \to A_θ\rtimes \mathbb Z_n$ for $2\le d < n \le 6$ a…
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…
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.
Let $\mathcal{M}_i$ be a family of $\mathrm{II}_1$-factors, containing a common $\mathrm{II}_1$-subfactor $\mathcal{N}$, such that $[\mathcal{M}_i:\mathcal{N}] \in \mathbb{N}_0$ for all $i$. Furthermore, let $ϕ\colon \mathbb{N}_0 \to \mathbb{C}$. We show that if a Hankel matrix related to $ϕ$ is trace-class, then there exists a unique completely bounded map $M_ϕ$ on the amalgamated free product of the $\mathcal{M}_i$ with amalgamation over $\mathcal{N}$, which acts as an radial multiplier. Hereby we extend a result of U. Haagerup and the author for radial multipliers on reduced free products o…
We consider amalgamated unital full free products of the form $A_1*_DA_2$, where $A_1, A_2$ and $D$ are finite dimensional C*-algebras and there are faithful traces on $A_1$ and $A_2$ whose restrictions to $D$ agree. We provide several conditions on the matrices of partial multiplicities of the inclusions $D\hookrightarrow A_1$ and $D\hookrightarrow A_2$ that guarantee that the C*-algebra $A_1*_DA_2$ is primitive. If the ranks of the matrices of partial multiplicities are one, we prove that the algebra $A_1*_DA_2$ is primitive if and only if it has a trivial center.
We investigate factoriality, Connes' type ${\rm III}$ invariants and fullness of arbitrary amalgamated free product von Neumann algebras using Popa's deformation/rigidity theory. Among other things, we generalize many previous structural results on amalgamated free product von Neumann algebras and we obtain new examples of full amalgamated free product factors for which we can explicitely compute Connes' type ${\rm III}$ invariants.
We prove a long exact sequence in KK-theory for both full and reduced amalgamated free products in the presence of conditional expectations. In the course of the proof, we established the KK-equivalence between the full amalgamated free product of two unital C*-algebras and a newly defined reduced amalgamated free product that is valid even for non GNS-faithful conditional expectations. Our results unify, simplify and generalize all the previous results obtained before by Cuntz, Germain and Thomsen.
In this paper, we consider the question whether a unital full free product of MF algebras with amalgamation over a finite dimensional C*-algebra is an MF algebra. First, we show that, under a natural condition, a unital full free product of two separable residually finite dimensional (RFD) C*-algebras with amalgamation over a finite dimensional C*-algebra is again a separable RFD C*-algebra. Applying this result on MF C*-algebras, we show that, under a natual condition, a unital full free product of two MF algebras is again an MF algebra. As an application, we show that a unital full free prod…
We show that a free product of a II_1-factor and a finite von Neumann algebra with amalgamation over a finite dimensional subalgebra is always a II_1-factor, and provide an algorithm for describing it in terms of free products (with amalgamation over the scalars) and compression/dilation. As an application, we show that the class of direct sums of finitely many von Neumann algebras that are interpolated free group factors, hyperfinite II_1-factors, type I_n algebras for n finite, and finite dimensional algebras, is closed under taking free products with amalgamation over finite dimensional sub…
Different (fibrewise) amalgamated products of continuous C*-bundles have been studied over the last years, one of the main question being to know when these amalgamated products are continuous C*-bundles. In order to gather these approaches in a joint framework, we first recall a few definitions from the theory of deformations of C*-algebras and we fix several notations that will be used in the sequel. Then we characterise the continuity properties of different amalgamated products of (continuous) C(X)-algebras.
We establish the MF property of the reduced group $ C^* $-algebra of an amalgamated free product of countable Abelian discrete groups. This result is then used to give a characterization of the amalgamated free products of Abelian groups for which the BDF semigroup of the reduced group $ C^* $-algebra is a group. Along the way we get a tensor product factorization of the corresponding group von Neumann algebra. We end the exposition by applying the ideas from the first part to give a few more examples of groups with a reduced group $ C^* $-algebra which is MF.
Let A and B be C*-algebras whose quotients are all RFD, and let C be a central C*-subalgebra in both A and B. We prove that the full amalgamated free product of A and B over C is then RFD. This generalizes Korchagin's result that amalgamated free products of commutative C*-algebras are RFD. When applied to the case of a trivial amalgam, our methods recover the result of Exel-Loring for separable C*-algebras. As corollaries to our theorem, we give sufficient conditions for amalgamated free products of maximally almost periodic (MAP) groups to have RFD C*-algebras and hence to be MAP.
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 …
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.
We study some reduced free products of C*-algebras with amalgamations. We give sufficient conditions for the positive cone of the K_0 group to be the largest possible. We also give sufficient conditions for simplicity and uniqueness of trace. We use the later result to give a necessary and sufficient condition for simplicity and uniqueness of trace of the reduced C*-algebras of the Baumslag-Solitar groups BS(m,n).
We construct a generalized version for the free product of unital C*-algebras over a family of unital C*-subalgebras, starting from the group-analogue. When all the subalgebras are the same, we recover the free product with amalgamation over a common subalgebra. We reduce the problem to the study of minimal amalgams. We specialize to triangles of algebras and subalgebras, study freeness in this context, and give some examples of constructions of minimal amalgams derived from triangles of operator algebras.
Recently, Adrian Ioana proved that all crossed products by free ergodic probability measure preserving actions of a nontrivial free product group Γ_1 * Γ_2 have a unique Cartan subalgebra up to unitary conjugacy. Ioana deduced this result from a more general dichotomy theorem on the normalizer N_M(A)'' of an amenable subalgebra A of an amalgamated free product von Neumann algebra M = M_1 *_B M_2. We improve this dichotomy theorem by removing the spectral gap assumptions and obtain in particular a simpler proof for the uniqueness of the Cartan subalgebra in crossed products by Γ_1 * Γ_2.
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.
In this paper, we observe the amalgamated free product structure of a Graph W*-probability space. In [16] and [17], we already observed the operator-valued freeness conditions on a graph W*-algebra. By using the conditions, we will consider the amalgamated free product structure of a graph W*-algebra. In particular, we can make the algebra be blocked, by so-called the loop free blocks (D_G-semicircular blocks) and non-loop blocks (R-diagonal blocks). Finally, we can see the the graph W*-algebra is the free product of edge blocks of the algebra.
Let $(M_i)_{i}$ be a (finite or infinite) family of finite von Neumann algebras with a common subalgebra $P$. When $φ:\IN\rightarrow\IC$ is a function, we define the radial multiplier $M_φ$ on the amalgamated free product $M=M_1\free_P M_2\free_P\ldots$ setting $M_φ(x)=φ(n)x$ for every reduced expression $x$ of length $n$. In this paper we give a sufficient condition on $φ$ to ensure that the corresponding radial multiplier $M_φ$ is a completely bounded map, and moreover we give an upper bound on its completely bounded norm. Our condition on $φ$ does not depend on the choice of von Neumann alg…
We establish a platform to transfer $L_p$-completely bounded maps on tensor products of von Neumann algebras to $L_p$-completely bounded maps on the corresponding amalgamated free products. As a consequence, we obtain a Hörmander-Mikhlin multiplier theory for free products of groups. Let $\mathbb{F}_\infty$ be a free group on infinite generators $\{g_1, g_2,\cdots\}$. Given $d\ge1$ and a bounded symbol $m$ on $\mathbb{Z}^d$ satisfying the classical Hörmander-Mikhlin condition, the linear map $M_m:\mathbb{C}[\mathbb{F}_\infty]\to \mathbb{C}[\mathbb{F}_\infty]$ defined by $λ(g)\mapsto m(k_1,\cdo…
We establish sufficient conditions for the C$^*$-simplicity of two classes of groups. The first class is that of groups acting on trees, such as amalgamated free products, HNN-extensions, and their normal subgroups; for example normal subgroups of Baumslag-Solitar groups. The second class is that of fundamental groups of compact 3-manifolds, related to the first class by their Kneser-Milnor and JSJ-decompositions. Much of our analysis deals with conditions on an action of a group $Γ$ on a tree $T$ which imply the following three properties: abundance of hyperbolic elements, better called stron…
We investigate Cartan subalgebras in nontracial amalgamated free product von Neumann algebras $M_1 \ast_B M_2$ over an amenable von Neumann subalgebra $B$. First, we settle the problem of the absence of Cartan subalgebra in arbitrary free product von Neumann algebras. Namely, we show that any nonamenable free product von Neumann algebra $(M_1, φ_1) \ast (M_2, φ_2)$ with respect to faithful normal states has no Cartan subalgebra. This generalizes the tracial case that was established in \cite{Io12a}. Next, we prove that any countable nonsingular ergodic equivalence relation $\mathcal R$ defined…
We study Cartan subalgebras in the context of amalgamated free product II$_1$ factors and obtain several uniqueness and non-existence results. We prove that if $Γ$ belongs to a large class of amalgamated free product groups (which contains the free product of any two infinite groups) then any II$_1$ factor $L^{\infty}(X)\rtimesΓ$ arising from a free ergodic probability measure preserving action of $Γ$ has a unique Cartan subalgebra, up to unitary conjugacy. We also prove that if $\mathcal R=\mathcal R_1*\mathcal R_2$ is the free product of any two non-hyperfinite countable ergodic probability …
Using a non-negative curvature condition, we prove the complete version of modified log-Sobolev inequalities for central Markov semigroups on various compact quantum groups, including group von Neumann algebras, free orthogonal group and quantum automorphism groups. We also prove that the "geometric Ricci curvature lower bound" introduced by Junge-Li-LaRacuente is stable under tensor products and amalgamated free products. As an application, we obtain the geometric Ricci curvature lower bound and complete modified logarithmic Sobolev inequality for word-length semigroups on free group factors …
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…
We consider amalgamated free product II$_1$ factors $M = M_1 *_B M_2 *_B ...$ and use ``deformation/rigidity'' and ``intertwining'' techniques to prove that any relatively rigid von Neumann subalgebra $Q\subset M$ can be intertwined into one of the $M_i$'s. We apply this to the case $M_i$ are w-rigid II$_1$ factors, with $B$ equal to either $\Bbb C$, to a Cartan subalgebra $A$ in $M_i$, or to a regular hyperfinite II$_1$ subfactor $R$ in $M_i$, to obtain the following type of unique decomposition results, à la Bass-Serre: If $M = (N_1 *_C N_2 *_C ...)^t$, for some $t>0$ and some other similar …
It has recently been proved that the class of unital exact C*-algebras is closed under taking reduced amalgamated free products. In particular, this implied that the class of exact discrete groups is closed under taking amalgamated free products. Here a proof is presented of a special case: that the class of exact discrete groups is closed under taking free products (with amalgamation over the identity element). The proof of this special case is considerably simpler than in full generality.
We get three types of results on measure equivalence rigidity; direct product groups of Ozawa's class $\mathcal{S}$ groups, wreath product groups and amalgamated free products. We prove measure equivalence factorization results on direct product groups of Ozawa's class $\mathcal{S}$ groups. As consequences, Monod--Shalom type orbit equivalence rigidity theorems follow. We prove that if two wreath product groups $A \wr G$, $B \wr Γ$ of non-amenable exact direct product groups $G$, $Γ$ with amenable bases $A$, $B$ are measure equivalent, then $G$ and $Γ$ are measure equivalent. We get Bass--Serr…
