arXiv subjects
Publications and source records attributed to Narutaka Ozawa.
We construct an MF C*-algebra A such that the spatial tensor product of A and the hyperfinite II_1 factor R contains a proper isometry. Consequently, stable finiteness of C*-algebras is not stable under tensor product and the hyperfinite II_1 factor is not a quasidiagonal C*-algebra. The C*-algebra A is non-simple and has only non-faithful tracial states. This result was obtained using OpenAI's Chat GPT Pro 6.0.
We introduce the notion of uniform exactness, or uniform amenability at infinity, for discrete groups and prove it for a wide class of groups containing free groups and their limit groups. This shows a novel strong convergence phenomenon that any convergent sequence of such groups in the space of marked groups converges strongly in the operator algebraic sense. In particular, convergence of the spectral radius formula is uniform over probability measures on such groups whose supports have a fixed cardinality.
We prove that the reduced group C*-algebras of infinite countable discrete groups having topologically-free extreme boundaries, or more generally groups that satisfy certain combinatorial property including all acylindrically hyperbolic groups with no nontrivial finite normal subgroups and all Zariski-dense subgroups of PSL(n,R), are selfless in the sense of L. Robert. This generalizes the recent results of Amrutam, Gao, Kunnawalkam Elayavalli, and Patchell, and of Vigdorovich. We also prove that selflessness is stable under tensor product among exact C*-algebras and that a C*-probability space is selfless provided that it is either simple and purely infinite or simple, exact, Z-stable, and uniquely tracial.
We prove that every total sequence in a Hilbert space satisfies the lower frame inequality after scaling. This solves A. Kulikov's "exercise" at mathoverflow.
We prove that the approximately inner automorphism group of a separable strongly stable von Neumann algebra is contractible in the u-topology. Thus the automorphism group of the hyperfinite type III_1 factor is contractible.
The Paszkiewicz conjecture about a product of positive contractions asserts that given a decreasing sequence $T_1\ge T_2\ge \dots$ of positive contractions on a separable infinite-dimensional Hilbert space, the product $S_n=T_n\dots T_1$ converges strongly. Recently, the first named author verified the conjecture for certain classes of sequences. In this paper, we prove the Paszkiewicz conjecture in full generality. Moreover, we show that in some cases, a generalized version of the Paszkiewicz conjecture also holds.
We answer the recent problem posed by Baudier, Braga, Farah, Vignati, and Willett that asks whether the $\ell_\infty$-direct sum of the matrix algebras embeds into the uniform Roe algebra or the quasi-local algebra of a uniformly locally finite metric space. The answers are no and yes, respectively. Hence the inclusion of the uniform Roe algebra into the quasi-local algebra can be proper.
We prove the following two results. First, the isometry semigroup of a unital properly infinite nuclear C*-algebra is right amenable. Second, the unitary group of a unital simple monotracial C*-algebra whose tracial GNS representation is hyperfinite is skew-amenable in the weak topology. This answers in part a conjecture of Alekseev, Schmidt, and Thom and a question of Pestov.
The well-known theorem of Shalom--Vaserstein and Ershov--Jaikin-Zapirain states that the group $\mathrm{EL}_n(\mathcal{R})$, generated by elementary matrices over a finitely generated commutative ring $\mathcal{R}$, has Kazhdan's property (T) as soon as $n\geq3$. This is no longer true if the ring $\mathcal{R}$ is replaced by a commutative rng (a ring but without the identity) due to nilpotent quotients $\mathrm{EL}_n(\mathcal{R}/\mathcal{R}^k)$. In this paper, we prove that even in such a case the group $\mathrm{EL}_n(\mathcal{R})$ satisfies a certain property that can substitute property (T), provided that $n$ is large enough.
We show the equivalence of several amenability type conditions for C*-dynamical systems. As an application, based on the Pimsner--Meyer construction, we give a new method to produce amenable actions on simple C*-algebras.
It is recently proved by Lubetzky and Peres that the simple random walk on a Ramanujan graph exhibits a cutoff phenomenon, that is to say, the total variation distance of the random walk distribution from the uniform distribution drops abruptly from near $1$ to near $0$. There are already a few alternative proofs of this fact. In this note, we give yet another proof based on functional analysis and entropic consideration.
We show that if $M$ is a full factor and $N \subset M$ is a co-amenable subfactor with expectation, then $N$ is also full. This answers a question of Popa from 1986. We also generalize a theorem of Tomatsu by showing that if $M$ is a full factor and $σ\colon G \curvearrowright M$ is an outer action of a compact group $G$, then $σ$ is automatically minimal and $M^G$ is a full factor which has w-spectral gap in $M$. Finally, in the appendix, we give a proof of the fact that several natural notions of co-amenability for an inclusion $N\subset M$ of von Neumann algebras are equivalent, thus closing the cycle of implications given in Anantharaman-Delaroche's paper in 1995.
We give a constructive, computer-assisted proof that $\operatorname{Aut}(\mathbb{F}_5)$, the automorphism group of the free group on $5$ generators, has Kazhdan's property $(T)$.
We prove that every unital C*-algebra $A$ has the Mazur--Ulam property. Namely, every surjective isometry from the unit sphere $S_A$ of $A$ onto the unit sphere $S_Y$ of another normed space $Y$ extends to a real linear map. This extends the result of A. M. Peralta and F. J. Fernandez-Polo who have proved the same under the additional assumption that both $A$ and $Y$ are von Neumann algebras. In the course of the proof, we strengthen Mankiewicz's theorem and prove that every surjective isometry from a closed unit ball with enough extreme points onto an arbitrary convex subset of a normed space is necessarily affine.
Given a $1$-cocycle $b$ with coefficients in an orthogonal representation, we show that any finite dimensional summand of $b$ is cohomologically trivial if and only if $\| b(X_n) \|^2/n$ tends to a constant in probability, where $X_n$ is the trajectory of the random walk $(G,μ)$. As a corollary, we obtain sufficient conditions for $G$ to satisfy Shalom's property $H_{\mathrm{FD}}$. Another application is a convergence to a constant in probability of $μ^{*n}(e) -μ^{*n}(g)$, $n\gg m$, normalized by its average with respect to $μ^{*m}$, for any finitely generated amenable group without infinite virtually Abelian quotients. Finally, we show that the harmonic equivariant mapping of $G$ to a Hilbert space obtained as an $U$-ultralimit of normalized $μ^{*n}- g μ^{*n}$ can depend on the ultrafilter $U$ for some groups.
We prove an inverse theorem for the Gowers $U^2$-norm for maps $G\to\mathcal M$ from an countable, discrete, amenable group $G$ into a von Neumann algebra $\mathcal M$ equipped with an ultraweakly lower semi-continuous, unitarily invariant (semi-)norm $\Vert\cdot\Vert$. We use this result to prove a stability result for unitary-valued $\varepsilon$-representations $G\to\mathcal U(\mathcal M)$ with respect to $\Vert\cdot \Vert$.
The celebrated theorem of Gromov asserts that any finitely generated group with polynomial growth contains a nilpotent subgroup of finite index. Alternative proofs have been given by Kleiner and others. In this note, we give yet another proof of Gromov's theorem, along the lines of Shalom and Chifan--Sinclair, which is based on the analysis of reduced cohomology and Shalom's property H_FD.
Recently C. Houdayer and Y. Isono have proved among other things that every biexact group $Γ$ has the property that for any non-singular strongly ergodic action $Γ\curvearrowright (X,μ)$ on a standard measure space the group measure space von Neumann algebra $Γ\ltimes L^\infty(X)$ is full. In this note, we prove the same property for a wider class of groups, notably including $\mathrm{SL}(3,{\mathbb Z})$. We also prove that for any connected simple Lie group $G$ with finite center, any lattice $Γ\le G$, and any closed non-amenable subgroup $H\le G$, the non-singular action $Γ\curvearrowright G/H$ is strongly ergodic and the von Neumann factor $Γ\ltimes L^\infty(G/H)$ is full.