arXiv subjects
Publications and source records attributed to Changying Ding.
We show that the \(q\)-Gaussian von Neumann algebras associated with an infinite-dimensional real Hilbert space are pairwise nonisomorphic as \(q\) ranges over \((-1,1)\).
It is proved that the $q$-Araki-Woods factor $\Gamma_q(\sH_\R, U)''$ associated with a strongly continuous orthogonal representation $U:\R\to \cO(\sH_\R)$ is strongly solid for all $q\in (-1,1)$ if the representation $U$ is almost periodic. We also show that the $q$-Araki-Woods factor $\Gamma_q(\sH_\R, U)''$ is not isomorphic to any free Araki-Woods factor for any $q\in (-1,1)\setminus\{0\}$ if the representation $U$ has nontrivial weakly mixing part or infinite dimensional almost periodic part with bounded spectrum.
We demonstrate a relative solidity property for the product of a nonamenable biexact group with an arbitrary infinite group in the measure equivalence setting. Among other applications, we obtain the following unique product decomposition for products of nonamenable biexact groups, strengthening \cite{Sa09}: for any nonamenable biexact groups $\Gamma_1,\cdots, \Gamma_n$, if a product group $\Lambda_1\times \Lambda_2$ is measure equivalent to $\times_{k=1}^n\Gamma_k$, then there exists a partition $T_1\sqcup T_2=\{1,\dots, n\}$ such that $\Lambda_i$ is measure equivalent to $\times_{k\in T_i}\Gamma_k$ for $i=1,2$.
We show that if $\Gamma\curvearrowright (X^\Gamma,\mu^\Gamma)$ is a Bernoulli action of an i.c.c. nonamenable group $\Gamma$ which is weakly amenable with Cowling-Haagerup constant $1$, and $\Lambda\curvearrowright(Y,\nu)$ is a free ergodic p.m.p. algebraic action of a group $\Lambda$, then the isomorphism $L^\infty(X^\Gamma)\rtimes\Gamma\cong L^\infty(Y)\rtimes\Lambda$ implies that $L^\infty(X^\Gamma)$ and $L^\infty(Y)$ are unitarily conjugate. This is obtained by showing a new rigidity result of non properly proximal groups and combining it with a rigidity result of properly proximal groups from \cite{BIP21}.
We introduce the notion of biexactness for general von Neumann algebras, naturally extending the notion from group theory. We show that biexactness implies solidity for von Neumann algebras, and that many of the examples of solid von Neumann algebras contained in the literature are, in fact, biexact. We also give examples of certain crossed products arising from Gaussian actions that are solid but not biexact, and we give examples of certain $q$-Gaussian von Neumann algebras that are strongly solid but not biexact. The techniques developed involve studying a certain weak form of nuclear embeddings, and we use this setting to give a new description of weak exactness for von Neumann algebras, which allows us to answer several open problems in the literature about weakly exact von Neumann algebras.
Using computations in the bidual of $\mathbb{B}(L^2M)$ we develop a new technique at the von Neumann algebra level to upgrade relative proper proximality to full proper proximality. This is used to structurally classify subalgebras of $LΓ$ where $Γ$ is an infinite group that is biexact relative to a finite family of subgroups $\{Λ_i\}_{i\in I}$ such that each $Λ_i$ is almost malnormal in $Γ$. This generalizes the result of \cite{DKEP21} which classifies subalgebras of von Neumann algebras of biexact groups. By developing a combination with techniques from Popa's deformation-rigidity theory we obtain a new structural absorption theorem for free products and a generalized Kurosh type theorem in the setting of properly proximal von Neumann algebras.
We introduce the notion of proper proximality for finite von Neumann algebras, which naturally extends the notion of proper proximality for groups. Apart from the group von Neumann algebras of properly proximal groups, we provide a number of additional examples, including examples in the settings of free products, crossed products, and compact quantum groups. Using this notion, we answer a question of Popa by showing that the group von Neumann algebra of a nonamenable inner amenable group cannot embed into a free group factor. We also introduce a notion of proper proximality for probability measure preserving actions, which gives an invariant for the orbit equivalence relation. This gives a new approach for establishing strong ergodicity type properties, and we use this in the setting of Gaussian actions to expand on solid ergodicity results first established by Chifan and Ioana, and later generalized by Boutonnet. The techniques developed also allow us to answer a problem left open by Anantharaman-Delaroche in 1995, by showing the equivalence between the Haagerup property and the compact approximation property for II$_1$ factors.
We show that for a countable exact group, having positive first $\ell^2$-Betti number implies proper proximality in this sense of \cite{BoIoPe21}. This is achieved by showing a cocycle superrigidty result for Bernoulli shifts of non-properly proximal groups. We also obtain that Bernoulli shifts of countable, nonamenable, i.c.c., exact, non-properly proximal groups are OE-superrigid.
In this paper, the notion of proper proximality (introduced in [BIP18]) is studied and classified in various families of groups. We show that if a group acts non-elementarily by isometries on a tree such that for any two edges, the intersection of their edge stabilizers is finite, then G is properly proximal. We show that the wreath product G\wr H is properly proximal if and only if H is non-amenable. We then completely classify proper proximality among graph products of non-trivial groups. Our results generalize recent work of Duchesne, Tucker-Drob and Wesolek classifying inner amenability for these families of groups. Our results also recover some rigidity results associated to the group von Neumann algebras, by virtue of being properly proximal. A key idea in the proofs of our theorems is a technique to upgrade from relative proper proximality using computations in the double dual of the small at infinity boundary.