Quanhua Xu
We consider Arveson's problem on the maximality of subdiagonal algebras. We prove that a subdiagonal algebra is maximal if it is invariant under the modular group of a faithful normal state which is preserved by the conditional expectation associated with the subdiagonal algebra.
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Let $\mathfrak A$ be a type 1 subdiagonal algebra in a $σ$-finite von Neumann algebra $\mathcal M$ with respect to a faithful normal conditional expectation $Φ$. We give necessary and sufficient conditions for which $\mathfrak A$ is maximal among the $σ$-weakly closed subalgebras of $\mathcal M$. In addition, we show that a type 1 subdiagonal algebra in a finite von Neumann algebra is automatically finite which gives a positive answer of Arveson's finiteness problem in 1967 in type 1 case.
Interpolation problems in subdiagonal algebras
本文研究了子对角代数中具有普遍因子分解性质的插值问题，通过构造周期流证明类型1子对角代数与von Neumann代数中的分析算子代数一致，并推导出回旋定理的非交换变体。
Let $\mathfrak A$ be a maximal subdiagonal algebra in a $σ$-finite von Neumann algebra $\mathcal M$. If every right invariant subspace of $\mathfrak A$ in the non-commutative Hardy space $H^2$ is of Beurling type, then we say $\mathfrak A$ to be type 1. We determine generators of these algebras and consider a Riesz type factorization theorem for the non-commutative $H^1$ space. We show that the right analytic Toeplitz algebra on the non-commutative Hardy space $H^p$ associated with a type 1 subdiagonal algebra with multiplicity 1 is hereditary reflexive.
On a class of subdiagonal algebras
本文提出 semi-$σ$-finite subdiagonal 与 Riesz approximable 算子代数类，构建了基于 Arveson's subdiagonal algebras 的最一般非交换 Hardy 空间理论，并发展该理论与研究新代数类的性质。
This article shows how the work of Henry Helson, especially the two papers of Helson and Lowdenslager, came to influence the development of the theory of non self adjoint operator algebras acting on Hilbert space.
Let $\mathfrak A$ be a type 1 subdiagonal algebra in a $σ$-finite von Neumann algebra $\mathcal M$ with respect to a faithful normal conditional expectation $Φ$. We consider a Riesz type factorization theorem in noncommutative $H^p$ spaces associated with $\mathfrak A$. It is shown that if $1\leq r,p,q<\infty$ such that $\frac1r=\frac1p+\frac1q$, then for any $h\in H^r$, there exist $h_p\in H^p$ and $h_q\in H^q$ such that $h=h_ph_q$. Beurling type invariant subspace theorem for noncommutative $L^p(1< p<\infty)$ space is obtained. Furthermore, we show that a $σ$-weakly closed subalgebra contain…
Let $\M$ be a von Neumann algebra with a faithful normal trace $\T$, and let $H^\infty$ be a finite, maximal, subdiagonal algebra of $\M$. Fundamental theorems on conjugate functions for weak$^*$\!-Dirichlet algebras are shown to be valid for non-commutative $H^\infty$. In particular the conjugation operator is shown to be a bounded linear map from $L^p(\M, \T)$ into $L^p(\M, \T)$ for $1 < p < \infty$, and to be a continuous map from $L^1(\M,\T)$ into $L^{1, \infty}(\M,\T)$. We also obtain that if an operator $a$ is such that $|a|\log^+|a| \in L^1(\M,\T)$ then its conjugate belongs to $L^1(\M,…
We extend Beurling's invariant subspace theorem, by characterizing subspaces $K$ of the noncommutative $L^p$ spaces which are invariant with respect to Arveson's maximal subdiagonal algebras, sometimes known as noncommutative $H^\infty$. It is significant that a certain subspace, and a certain quotient, of $K$ are $L^p({\mathcal D})$-modules in the recent sense of Junge and Sherman, and therefore have a nice decomposition into cyclic submodules. We also give general inner-outer factorization formulae for elements in the noncommutative $L^p$. These facts generalize the classical ones, and shoul…
An extension of Haagerup's reduction theorem with applications to subdiagonal subalgebras of general von Neumann algebras
本文将 Haagerup 约化定理推广到带任意忠实正规半有限权的一般 von Neumann 代数，并据此扩展 $H^p$ 空间应用、提出近似次对角子代数概念。
Jensen's inequality in finite subdiagonal algebras
本文将有限次对角代数中的Jensen不等式推广至更一般的递增连续函数情形，证明了迹不等式及其取等条件，并给出了关于点谱包含关系的应用。
In this paper we give a simple proof of the maximality of dual coactions on full cross-sectional C*-algebras of Fell bundles over locally compact groups. As applications we extend certain exotic crossed-product functors in the sense of Baum, Guentner and Willett to the category of Fell bundles and the category of partial actions and we obtain results about the K-theory of (exotic) cross-sectional algebras of Fell-bundles over K-amenable groups. As a bonus, we give a characterisation of maximal coactions of discrete groups in terms of maximal tensor products.
We generalize to the setting of Arveson's maximal subdiagonal subalgebras of finite von Neumann algebras, the Szegö $L^p$-distance estimate, and classical theorems of F. and M. Riesz, Gleason and Whitney, and Kolmogorov. In so doing, we are finally able to provide a complete noncommutative analog of the famous cycle of theorems characterizing the function theoretic generalizations of $H^\infty$. A sample of our other results: we prove a Kaplansky density result for a large class of these algebras, and give a necessary condition for when every completely contractive homomorphism on a unital sub…
Maximality and symmetry related to the \(2\)-adic ring \(C^*\)-algebra
本文证明Cuntz代数$\mathcal{O}_2$是2-adic环$C^*$-代数$\mathcal{Q}_2$的极大$C^*$-子代数，并将该极大性延拓到交叉积与不动点代数，从而解决了若干关于$\mathcal{Q}_2$的公开问题。
Choquet-Type Relations and a State Space Level Amendment of Arveson's Hyperrigidity Conjecture
本文针对Arveson超刚性猜想，建立态空间方法，引入强扩张与积分细分关系，证明其性质并给出Clouâtre-Thompson定理的另一证明，还明确了两类关系在交换情形下的一致性。
Let $\A$ be a finite subdiagonal algebra in Arveson's sense. Let $H^p(\A)$ be the associated noncommutative Hardy spaces, $0<p\le\8$. We extend to the case of all positive indices most recent results about these spaces, which include notably the Riesz, Szegö and inner-outer type factorizations. One new tool of the paper is the contractivity of the underlying conditional expectation on $H^p(\A)$ for $p<1$.
Around 1967, Arveson invented a striking noncommutative generalization of classical $H^\infty$, known as {\em subdiagonal algebras}, which include a wide array of examples of interest to operator theorists. Their theory extends that of the generalized $H^p$ spaces for function algebras from the 1960s, in an extremely remarkable, complete, and literal fashion, but for reasons that are `von Neumann algebraic'. Most of the present paper consists of a survey of our work on Arveson's algebras, and the attendant $H^p$ theory, explaining some of the main ideas in their proofs, and including some impr…
We continue our study of outer elements of the noncommutative H^p spaces associated with Arveson's subdiagonal algebras. We extend our generalized inner-outer factorization theorem, and our characterization of outer elements, to include the case of elements with zero determinant. In addition, we make several further contributions to the theory of outers. For example, we generalize the classical fact that outers in H^p actually satisfy the stronger condition that there exist a_n in A with h a_n in Ball(A) and h a_n \to 1 in p-norm.
We undertake a systematic study of the so-called $2$-adic ring $C^*$-algebra $\mathcal{Q}_2$. This is the universal $C^*$-algebra generated by a unitary $U$ and an isometry $S_2$ such that $S_2U=U^2S_2$ and $S_2S_2^*+US_2S_2^*U^*=1$. Notably, it contains a copy of the Cuntz algebra $\mathcal{O}_2=C^*(S_1, S_2)$ through the injective homomorphism mapping $S_1$ to $US_2$. Among the main results, the relative commutant $C^*(S_2)'\cap \mathcal{Q}_2$ is shown to be trivial. This in turn leads to a rigidity property enjoyed by the inclusion $\mathcal{O}_2\subset\mathcal{Q}_2$, namely the endomorphis…
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