D. Dauitbek, K. S. Tulenov
In this paper we investigate the conditional expectation on the non-commutative $H^{(r,s)}_{p}(\mathcal A;\ell_{\infty})$ and $H_{p}(\mathcal A;\ell_{1})$ spaces associated with semifinite subdiagonal algebra, and prove the contractibility of the underlying conditional expectation on these spaces.
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In the current paper, we generalize the "compact operator" part of the Voiculescu's non-commutative Weyl-von Neumann theorem on approximate equivalence of unital $*$-homomorphisms of an commutative C$^*$ algebra $\mathcal{A}$ into a semifinite von Neumann algebra. A result of D. Hadwin for approximate summands of representations into a finite von Neumann factor $\mathcal{R}$ is also extended.
Let $E=E(0,\infty)$ be a symmetric function space and $E(\mathcal{M},τ)$ be a symmetric operator space associated with a semifinite von Neumann algebra with a faithful normal semifinite trace. Our main result identifies the class of spaces $E$ for which every derivation $δ:\mathcal{A}\to E(\mathcal{M},τ)$ is necessarily inner for each $C^*$-subalgebra $\mathcal{A}$ in the class of all semifinite von Neumann algebras $\mathcal{M}$ as those with the Levi property.
Utilizing the notion of uniform equicontinuity for sequences of functions with the values in the space of measurable operators, we present a non-commutative version of the Banach Principle for $L^\infty$.
The inter-relationship between isomorphisms of commutative and isomorphisms of non-commutative $\log$-algebras
本文研究了非交换 $\log$-代数的同构问题，建立了由不同精确正规半有限迹构造的非交换 $\log$-代数重合及同构的充分必要条件，并揭示了其与 von Neumann 代数中心上 $\log$-代数同构的联系。
Non-Linear Traces on Semifinite Factors and Generalized Singular Numbers
本文在半有限因子上引入Choquet型和Sugeno型非线性迹，利用加权维数函数刻画其部分可加性，并揭示其与Lorentz空间的关系及在因子上诱导度量的性质。
We prove the version of interpolation theorem for non-commutative vector-valued fully symmetric spaces associated with fully symmetric Banach function spaces and a von Neumann algebra equipped with a faithful semifinite normal trace.
Let $\mathcal{M}$ be a von Neumann algebra with a normal semifinite faithful trace $τ$. We prove that every continuous $m$-homogeneous polynomial $P$ from $L^p(\mathcal{M},τ)$, with $0<p<\infty$, into each topological linear space $X$ with the property that $P(x+y)=P(x)+P(y)$ whenever $x$ and $y$ are mutually orthogonal positive elements of $L^p(\mathcal{M},τ)$ can be represented in the form $P(x)=Φ(x^m)$ $(x\in L^p(\mathcal{M},τ))$ for some continuous linear map $Φ\colon L^{p/m}(\mathcal{M},τ)\to X$.
The UMD property of symmetric operator spaces
本文证明了对称Banach函数空间的UMD性质可传递至半有限von Neumann代数上的非交换空间，解决了四十多年的开放问题。
Quantum Doubly Stochastic Operators on Non-commutative $L_p$-Spaces
本文研究了非交换 $L_p$-空间上与半有限von Neumann代数相关的正迹保持映射，即量子双随机算子，探讨了其基本性质、严格范数不等式、Schatten理想意义下的紧性条件以及有限和无限维中的新例子，并讨论了其在量子主次性和插值稳定性中的应用。
On the non-commutative Neveu decomposition and stochastic ergodic theorems
该研究证明了半有限及σ-有限von Neumann代数上顺从半群/群作用的Neveu分解，并得到非交换L¹空间上正压缩的随机遍历定理，填补了Danford-Schwartz范畴外的相关空白。
We give a proof of the Khintchine inequalities in non-commutative $L_p$-spaces for all $0< p<1$. These new inequalities are valid for the Rademacher functions or Gaussian random variables, but also for more general sequences, e.g. for the analogues of such random variables in free probability. We also prove a factorization for operators from a Hilbert space to a non commutative $L_p$-space, which is new for $0<p<1$. We end by showing that Mazur maps are Hölder on semifinite von Neumann algebras. The main tool is a new form of Hölder inequality for non commutative Lp spaces with weights.
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 …
Kuroda's theorem for $n$-tuples in semifinite von Neumann algebras
针对半有限冯·诺依曼代数中满足特定条件的对称函数空间，证明交换自伴n元组可被交换对角算子n元组在对应非交换对称空间范数和无穷范数下逼近，且在$\boldsymbol{\mathcal{M}}=B(H)$时覆盖经典黑田定理等结果。
In this paper, we extend the classical Bohr's inequality to the setting of the non-commutative Hardy space $H^1$ associated with a semifinite von Neumann algebra. As a consequence, we obtain Bohr's inequality for operators in the von Neumann-Schatten class $\cl C_1$ and square matrices of any finite order. Interestingly, we establish that the optimal bound for $r$ in the above mentioned Bohr's inequality concerning von Neumann-Shcatten class is 1/3 whereas it is 1/2 in the case of $2\times 2$ matrices and reduces to $\sqrt{2}-1$ for the case of $3\times 3$ matrices. We also obtain a generaliza…
We give a systematic study on the Hardy spaces of functions with values in the non-commutative $L^p$-spaces associated with a semifinite von Neumann algebra ${\cal}M.$ This is motivated by the works on matrix valued Harmonic Analysis (operator weighted norm inequalities, operator Hilbert transform), and on the other hand, by the recent development on the non-commutative martingale inequalities. Our non-commutative Hardy spaces are defined by the non-commutative Lusin integral function.
The paper makes the first steps into the study of extensions ("twisted sums") of noncommutative $L^p$-spaces regarded as Banach modules over the underlying von Neumann algebra $\mathcal M$. Our approach combines Kalton's description of extensions by centralizers (these are certain maps which are, in general, neither linear nor bounded) with a general principle, due to Rochberg and Weiss saying that whenever one finds a given Banach space $Y$ as an intermediate space in a (complex) interpolation scale, one automatically gets a self-extension $ 0\longrightarrow Y\longrightarrow X\longrightarrow …
We prove a factorization of completely bounded maps from a $C^*$-algebra $A$ (or an exact operator space $E\subset A$) to $\ell_2$ equipped with the operator space structure of $(C,R)_θ$ ($0<θ<1$) obtained by complex interpolation between the column and row Hilbert spaces. More precisely, if $F$ denotes $\ell_2$ equipped with the operator space structure of $(C,R)_θ$, then $u: A \to F$ is completely bounded iff there are states $f,g$ on $A$ and $C>0$ such that \[ \forall a\in A\quad \|ua\|^2\le C f(a^*a)^{1-θ}g(aa^*)^θ.\] This extends the case $θ=1/2$ treated in a recent paper with Shlyakhtenk…
A note on the column-row property
本文完整分类了半有限冯·诺依曼代数上非交换 Lp 空间的列-行性质，并讨论了相关性质在自然算子空间例子中的存在性。
A Non-commutative Individual Ergodic Theorem Along Sparse Random Subsequences
本文证明半有限冯·诺依曼代数上沿稀疏随机子序列的约当同构平均双侧几乎一致收敛，将LaVictoire随机遍历定理推广至非交换设置。
This is a survey article of geometric properties of noncommutative symmetric spaces of measurable operators $E(\mathcal{M},τ)$, where $\mathcal{M}$ is a semifinite von Neumann algebra with a faithful, normal, semifinite trace $τ$, and $E$ is a symmetric function space. If $E\subset c_0$ is a symmetric sequence space then the analogous properties in the unitary matrix ideals $C_E$ are also presented. In the preliminaries we provide basic definitions and concepts illustrated by some examples and occasional proofs. In particular we list and discuss the properties of general singular value functio…
We consider commutative C* -algebras of Toeplitz operators in the weighted Bergman space on the unit ball in $\mathbb{C}^{\mathbf{n}}$. For the algebras of elliptic type we find a new representation, namely as the algebra of operators which are functions of certain collections of commuting unbounded self-adjoint operators in the Bergman space.
Pointwise Convergence for Random Ergodic Averages in Non-commutative $L^p$-spaces
本文研究非交换 $L^p$-空间中由伯努利变量确定的随机稀疏子序列的遍历平均，证明其几乎处处一致收敛到遍历投影，扩展了 Bourgain 的定理。
We will show that if $\sM$ is a factor, then for any pair $\f, \p\in\sMdsup$ of normal positive linear functionals on $\sM$, the inequality: $$ \lrnorm{\f}\leq \lrnorm{\p} $$ is equivalent to the fact that there exist a countable family $\lrbrace{\ffdi: i\in I}\subset \sMdsup$ in $\sMdsup$ and a family $\lrbrace{\udi: i\in I}ı\sM$ of partial isometries in \cM such that $$ \f=\sumd{i\in I} \ffdi,\quad \sumd{i\in I} \udi{\ffdi}\udius\leq \p, \quad \text{and} \quad \udius\udi=s\lr{\ffdi}, i\in I, $$ where $s(ω), ω\in\sMdsup$, means the support projection of $ω$. Furthermore, if $\lrnorm{\f}=\lrno…
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…
This paper gives a survey of the index theory of tangentially elliptic and transversally elliptic operators on foliated manifolds as well as of related notions and results in non-commutative geometry.
This paper introduces a new approach to the non-normal Dixmier and Connes-Dixmier traces (introduced by Dixmier and adapted to non-commutative geometry by Connes) on a general Marcinkiewicz space associated with an arbitrary semifinite von Neumann algebra. By unifying various constructions, and translating the situation of Dixmier traces into the theory of singular symmetric functionals on Marcinkiewicz function/operator spaces, we obtain the results (i) and (ii) below. The results are stated here, for the reader, in terms of the ideal $L^{(1,\infty)}$ of compact operators whose partial sums o…
In this paper, we consider the real rank zero $\mathrm{C}^*$-algebras which can be written as an inductive limit of the Elliott-Thomsen building blocks and prove a decomposition result for the connecting homomorphisms; this technique will be used in the classification theorem.
This article extends the classical Real Nullstellensatz to matrices of polynomials in a free $\ast$-algebra $\RR\axs$ with $x=(x_1, \ldots, x_n)$. This result is a generalization of a result of Cimpri\vc, Helton, McCullough, and the author. In the free left $\RR\axs$-module $\RR^{1 \times \ell}\axs$ we introduce notions of the (noncommutative) zero set of a left $\RR\axs$-submodule and of a real left $\RR\axs$-submodule. We prove that every element from $\RR^{1 \times \ell}\axs$ whose zero set contains the intersection of zero sets of elements from a finite subset $S \subset \RR^{1 \times \ell…
We show that the validity of the non-commutative Khintchine inequality for some $q$ with $1<q<2$ implies its validity (with another constant) for all $1\le p<q$. We prove this for the inequality involving the Rademacher functions, but also for more general "lacunary" sequences, or even non-commutative analogues of the Rademacher functions. For instance, we may apply it to the "Z(2)-sequences" previously considered by Harcharras. The result appears to be new in that case. It implies that the space $\ell^n_1$ contains (as an operator space) a large subspace uniformly isomorphic (as an operator s…
We prove stability theorems in the Cuntz semigroup of a commutative C*-algebra which are analogues of classical stability theorems for topological vector bundles over compact Hausdorff spaces. Several applications to simple unital AH algebras of slow dimension growth are then given: such algebras have strict comparison of positive elements; their Cuntz semigroups are recovered functorially from the Elliott invariant; the lower-semicontinuous dimension functions are dense in the space of all dimension functions, and the latter is a Choquet simplex.
Let $\Free_n$ denote the free group with $n$ generators $g_1, g_2, ..., g_n$. Let $λ$ stand for the left regular representation of $\Free_n$ and let $τ$ be the standard trace associated to $λ$. Given any positive integer $d$, we study the operator space structure of the subspace $\Word_p(n,d)$ of $L_p(τ)$ generated by the family of operators $λ(g_{i_1}g_{i_2} ... g_{i_d})$ with $1 \le i_k \le n$. Moreover, our description of this operator space holds up to a constant which does not depend on $n$ or $p$, so that our result remains valid for infinitely many generators. We also consider the subsp…
For any irrational theta and rational number p/q such that q|qtheta-p|<1, a projection e of trace q|qtheta-p| is constructed in the the irrational rotation algebra A_theta that is invariant under the Fourier transform. (The latter is the order four automorphism U mapped to V, V mapped to U^{-1}, where U, V are the canonical unitaries generating A_theta.) Further, the projection e is approximately central, the cut down algebra eA_theta e contains a Fourier invariant q x q matrix algebra whose unit is e, and the cut downs eUe, eVe are approximately inside the matrix algebra. (In particular, ther…
As many will agree, it feels good to complement a cup of tea by a donut or two. This sweet relationship is also a guiding principle of non-commutative geometry known as Serre Theorem. We explain the algebra behind this theorem and prove that elliptic curves are complementary to the so-called non-commutative tori.
On representations and topological aspects of positive maps on non-unital quasi *-algebras
研究非单位拟 *-代数上 C*-值正映射的表示，结合 Banach 拟 *-代数实例验证，并推导范数不等式及构造非交换 L2 空间上的拓扑传递算子。
For differential operators which are invariant under the action of an abelian group Bloch theory is the tool of choice to analyze spectral properties. By shedding some new non-commutative light on this we motivate the introduction of a non-commutative Bloch theory for elliptic operators on Hilbert C*-modules. It relates properties of C*-algebras to spectral properties of module operators such as band structure, weak genericity of cantor spectra, and absence of discrete spectrum. It applies e.g. to differential operators invariant under a projective group action, such as Schroedinger operators …
For differential operators which are invariant under the action of an abelian group Bloch theory is the preferred tool to analyze spectral properties. By shedding some new non-commutative light on this we motivate the introduction of a non-commutative Bloch theory for elliptic operators on Hilbert C*-modules. It relates properties of C*-algebras to spectral properties of module operators such as band structure, weak genericity of cantor spectra, and absence of discrete spectrum. It applies e.g. to differential operators invariant under a projective group action, such as Schroedinger, Dirac and…
We prove that cocompact (and more generally: undistorted) lattices on $\tilde{A}_2$-buildings satisfy Lafforgue's strong property (T), thus exhibiting the first examples that are not related to algebraic groups over local fields. Our methods also give two further results. First, we show that the first $\ell^p$-cohomology of an $\tilde{A}_2$-building vanishes for any finite $p$. Second, we show that the non-commutative $L^p$-space for $p$ not in $[\frac 4 3,4]$ and the reduced $C^*$-algebra associated to an $\tilde{A}_2$-lattice do not have the operator space approximation property and, consequ…
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