On $\mathcal{OL}_\infty$ structure of nuclear, quasidiagonal C*-algebras

P. Jones'Interpolation theorem for noncommutative martingale Hardy spaces

Let $\mathcal{M}$ be a semifinite von Nemann algebra equipped with an increasing filtration $(\mathcal{M}_n)_{n\geq 1}$ of (semifinite) von Neumann subalgebras of $\M$. For $0<p \leq\infty$, let $\h_p^c(\mathcal{M})$ denote the noncommutative column conditioned martingale Hardy space associated with the filtration $(\mathcal{M}_n)_{n\geq 1}$ and the index $p$. We prove that for $0<p<\infty$, the compatible couple $\big(\h_p^c(\mathcal{M}), \h_\infty^c(\mathcal{M})\big)$ is $K$-closed in the couple $\big(L_p(\mathcal{N}), L_\infty(\mathcal{N}) \big)$ for an appropriate amplified semifinite von …

Interpolation between $L_0({\mathcal M},τ)$ and $L_\infty({\mathcal M},τ)$

Let ${\mathcal M}$ be a semifinite von Neumann algebra with a faithful semifinite normal trace $τ$. We show that the symmetrically $Δ$-normed operator space $E({\mathcal M},τ)$ corresponding to an arbitrary symmetrically $Δ$-normed function space $E(0,\infty)$ is an interpolation space between $L_0({\mathcal M},τ)$ and ${\mathcal M}$, which is in contrast with the classical result that there exist symmetric operator spaces $E({\mathcal M},τ)$ which are not interpolation spaces between $L_1({\mathcal M},τ)$ and ${\mathcal M}$. Besides, we show that the ${\mathcal K}$-functional of every $X\in L…

Notes on real interpolation of operator $L_p$-spaces

Let $\mathcal{M}$ be a semifinite von Neumann algebra. We equip the associated noncommutative $L_p$-spaces with their natural operator space structure introduced by Pisier via complex interpolation. On the other hand, for $1<p<\infty$ let $$L_{p,p}(\mathcal{M})=\big(L_{\infty}(\mathcal{M}),\,L_{1}(\mathcal{M})\big)_{\frac1p,\,p}$$ be equipped with the operator space structure via real interpolation as defined by the second named author ({\em J. Funct. Anal}. 139 (1996), 500--539). We show that $L_{p,p}(\mathcal{M})=L_{p}(\mathcal{M})$ completely isomorphically if and only if $\mathcal{M}$ is f…

群胚 $C^*$-代数上的迹态与本质自由性

Tracial states on groupoid $C^*$-algebras and essential freeness

本文研究群胚 $C^*$-代数上迹态的典范性与不动点迹的可延拓性，证明二者共同刻画群胚本质自由，并给出群作用与拟对角迹的应用。

Hyperreflexivity of the space of module homomorphisms between non-commutative $L^p$-spaces

Let $\mathcal{M}$ be a von Neumann algebra, and let $0<p,q\le\infty$. Then the space $\Hom_\mathcal{M}(L^p(\mathcal{M}),L^q(\mathcal{M}))$ of all right $\mathcal{M}$-module homomorphisms from $L^p(\mathcal{M})$ to $L^q(\mathcal{M})$ is a reflexive subspace of the space of all continuous linear maps from $L^p(\mathcal{M})$ to $L^q(\mathcal{M})$. Further, the space $\Hom_\mathcal{M}(L^p(\mathcal{M}),L^q(\mathcal{M}))$ is hyperreflexive in each of the following cases: (i) $1\le q<p\le\infty$; (ii) $1\le p,q\le\infty$ and $\mathcal{M}$ is injective, in which case the hyperreflexivity constant is a…

Amenability and paradoxicality in semigroups and C*-algebras

We analyze the dichotomy amenable/paradoxical in the context of (discrete, countable, unital) semigroups and corresponding semigroup rings. We consider also Følner's type characterizations of amenability and give an example of a semigroup whose semigroup ring is algebraically amenable but has no Følner sequence. In the context of inverse semigroups $S$ we give a characterization of invariant measures on $S$ (in the sense of Day) in terms of two notions: $domain$ $measurability$ and $localization$. Given a unital representation of $S$ in terms of partial bijections on some set $X$ we define a n…

Nuclear dimension of extensions of $\mathcal{O}_\infty$-stable algebras

We obtain an improved upper bound for the nuclear dimension of extensions of $\mathcal{O}_\infty$-stable $\rm{C}^*$-algebras. In particular, we prove that the nuclear dimension of a full extension of an $\mathcal{O}_\infty$-stable $\rm{C}^*$-algebra by a stable AF algebra is one.

核C*-代数嵌入II_1因子的唯一性

Uniqueness for embeddings of nuclear $C^*$-algebras into type II$_{1}$ factors

研究核C*-代数到II_1因子的嵌入唯一性，证明在迹和K-理论一致时，单位元满核映射在单位元共轭下等价，得出准diagonal近似和II_1因子中核映射的唯一性结论。

On crossed products of the Cuntz algebra ${\mathcal O}_\infty$ by quasi-free actions of abelian groups

We investigate the structures of crossed products of the Cuntz algebra ${\mathcal O}_\infty$ by quasi-free actions of abelian groups. We completely determine their ideal structures and compute the strong Connes spectra and K-groups.

Banach limits and traces on $\mathcal L_{1,\infty}$

We introduce a new approach to traces on the principal ideal $\mathcal L_{1,\infty}$ generated by any positive compact operator whose singular value sequence is the harmonic sequence. Distinct from the well-known construction of J.~Dixmier, the new approach provides the explicit construction of every trace of every operator in $\mathcal L_{1,\infty}$ in terms of translation invariant functionals applied to a sequence of restricted sums of eigenvalues. The approach is based on a remarkable bijection between the set of all traces on $\mathcal L_{1,\infty}$ and the set of all translation invarian…

A Beurling-Blecher-Labuschagne Theorem for noncommutative Hardy spaces associated with semifinite von Neumann algebras

In 2008, Blecher and Labuschagne extended Beurling's classical theorem to $H^\infty$-invariant subspaces of $L^p(\mathcal{M},τ)$ for a finite von Neumann algebra $\mathcal{M}$ with a finite, faithful, normal tracial state $τ$ when $1\le p\le \infty$. In this paper, using Arveson's non-commutative Hardy space $H^\infty$ in relation to a von Neumann algebra $\mathcal{M}$ with a semifinite, faithful, normal tracial weight $τ$, we prove a Beurling-Blecher-Labuschagne theorem for $H^\infty$-invariant spaces of $L^p(\mathcal{M},τ)$ when $0<p\leq\infty$. The proof of the main result relies on proofs …

通过 $\mathcal O_\infty$ 的包含关系分离点

Separating points through inclusions of $\mathcal O_\infty$

本文定义了 C*-代数包含关系 $\mathcal O_\infty\subset A$ 的一种基性质，给出了其成立条件与应用，并刻画了余冠代数中的开投影。

Schur-Horn theorems in II$_\infty$-factors

We describe majorization between selfadjoint operators in a $σ$-finite II$_\infty$ factor $(\mathcal{M},τ)$ in terms of simple spectral relations. For a diffuse abelian von Neumann subalgebra $\mathcal{A}\subset \mathcal{M}$ with trace-preserving conditional expectation $E_{\mathcal{A}}$, we characterize the closure in the measure topology of the image through $E_{\mathcal{A}}$ of the unitary orbit of a selfadjoint operator in $\mathcal{M}$ in terms of majorization (i.e., a Schur-Horn theorem). We also obtain similar results for the contractive orbit of positive operators in $\mathcal{M}$ and …

Surjective separating maps on noncommutative $L^p$-spaces

Let $1\leq p<\infty$ and let $T\colon L^p({\mathcal M})\to L^p({\mathcal N})$ be a bounded map between noncommutative $L^p$-spaces. If $T$ is bijective and separating (i.e., for any $x,y\in L^p({\mathcal M})$ such that $x^*y=xy^*=0$, we have $T(x)^*T(y)=T(x)T(y)^*=0$), we prove the existence of decompositions ${\mathcal M}={\mathcal M}_1\mathop{\oplus}\limits^\infty{\mathcal M}_2$, ${\mathcal N}={\mathcal N}_1 \mathop{\oplus}\limits^\infty{\mathcal N}_2$ and maps $T_1\colon L^p({\mathcal M}_1)\to L^p({\mathcal N}_1)$, $T_2\colon L^p({\mathcal M}_2)\to L^p({\mathcal N}_2)$, such that $T=T_1+T_2…

Weak type commutator and Lipschitz estimates: resolution of the Nazarov-Peller conjecture

Let $\mathcal{M}$ be a semi-finite von Neumann algebra and let $f: \mathbb{R} \rightarrow \mathbb{C}$ be a Lipschitz function. If $A,B\in\mathcal{M}$ are self-adjoint operators such that $[A,B]\in L_1(\mathcal{M}),$ then $$\|[f(A),B]\|_{1,\infty}\leq c_{abs}\|f'\|_{\infty}\|[A,B]\|_1,$$ where $c_{abs}$ is an absolute constant independent of $f$, $\mathcal{M}$ and $A,B$ and $\|\cdot\|_{1,\infty}$ denotes the weak $L_1$-norm. If $X,Y\in\mathcal{M}$ are self-adjoint operators such that $X-Y\in L_1(\mathcal{M}),$ then $$\|f(X)-f(Y)\|_{1,\infty}\leq c_{abs}\|f'\|_{\infty}\|X-Y\|_1.$$ This result re…

Classification of $\mathcal O_\infty$-stable $C^\ast$-algebras

I present a proof of Kirchberg's classification theorem: two separable, nuclear, $\mathcal O_\infty$-stable $C^\ast$-algebras are stably isomorphic if and only if they are ideal-related $KK$-equivalent. In particular, this provides a more elementary proof of the Kirchberg--Phillips theorem which is isolated in the paper to increase readability of this important special case.

A non-commutative Beurling's theorem with respect to unitarily invariant norms

In 1967, Arveson invented a non-commutative generalization of classical $H^{\infty},$ known as finite maximal subdiagonal subalgebras, for a finite von Neumann algebra $\mathcal M$ with a faithful normal tracial state $τ$. In 2008, Blecher and Labuschagne proved a version of Beurling's theorem on $H^\infty$-right invariant subspaces in a non-commutative $L^{p}(\mathcal M,τ)$ space for $1\le p\le \infty$. In the present paper, we define and study a class of norms ${\mathcal{N}}_{c}(\mathcal M, τ)$ on $\mathcal{M},$ called normalized, unitarily invariant, $\Vert \cdot \Vert_{1}$-dominating, cont…