In this paper we set up a representation theorem for tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property in terms of Ky Fan norms. Examples of tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property include unitarily invariant norms on finite factors (type ${\rm II}\sb 1$ factors and $M_n(\cc)$) and symmetric gauge norms on $L^\infty[0,1]$ and $\cc^n$. As the first application, we obtain that the class of unitarily invariant norms on a type ${\rm II}\sb 1$ factor coincides with the class of symmetric gauge norms on $L^\infty[0,1]$ and von Neumann's classical result \cite{vN} on unitarily invariant norms on $M_n(\cc)$. As the second application, Ky Fan's dominance theorem \cite{Fan} is obtained for finite von Neumann algebras satisfying the weak Dixmier property. As the third application, some classical results in non-commutative $L^p$-theory (e.g., non-commutative H$\ddot{\text{o}}$lder's inequality, duality and reflexivity of non-commutative $L^p$-spaces) are obtained for general unitarily invariant norms on finite factors. We also investigate the extreme points of $\NN(\M)$, the convex compact set (in the pointwise weak topology) of normalized unitarily invariant norms (the norm of the identity operator is 1) on a finite factor $\M$. We obtain all extreme points of $\NN(M_2(\cc))$ and many extreme points of $\NN(M_n(\cc))$ ($n\geq 3$). For a type ${\rm II}\sb 1$ factor $\M$, we prove that if $t$ ($0\leq t\leq 1$) is a rational number then the Ky Fan $t$-th norm is an extreme point of $\NN(\M)$.
It is shown that a unital C*-algebra A has the Dixmier property if and only if it is weakly central and satisfies certain tracial conditions. This generalises the Haagerup-Zsido theorem for simple C*-algebras. We also study a uniform version of the Dixmier property, as satisfied for example by von Neumann algebras and the reduced C*-algebras of Powers groups, but not by all C*-algebras with the Dixmier property, and we obtain necessary and sufficient conditions for a simple unital C*-algebra with unique tracial state to have this uniform property. We give further examples of C*-algebras with t…
We study some general properties of tracial C*-algebras. In the first part, we consider Dixmier type approximation theorem and characterize symmetric amenability for C*-algebras. In the second part, we consider continuous bundles of tracial von Neumann algebras and classify some of them.
We define the notion of self-tracial stability for tracial von Neumann algebras and show that a tracial von Neumann algebra satisfying the Connes Embedding Problem is self-tracially stable if and only if it is amenable. We then generalize a result of Jung by showing that a separable tracial von Neumann algebra that satisfies the Connes Embedding Problem is amenable if and only if any two embeddings into $R^\mathcal{U}$ are ucp-conjugate. Moreover we show that for a II$_1$ factor $N$ satisfying CEP, the space $\mathbb{H}$om$(N, \prod_{k\to \mathcal{U}}M_k)$ of unitary equivalence classes of emb…
In his study of the relative Dixmier property for inclusions of von Neumann algebras and of $C^*$-algebras, Popa considered a certain property of automorphisms on $C^*$-algebras, that we here call the strong averaging property. In this note we characterize when an automorphism on a $C^*$-algebra has the strong averaging property. In particular, automorphisms on commutative $C^*$-algebras possess this property precisely when they are free. An automorphism on a unital separable simple $C^*$-algebra with at least one tracial state has the strong averaging property precisely when its extension to …
Several techniques together with some partial answers are given to the questions of factoriality, type classification and fullness for amalgamated free product von Neumann algebras.
Let $M=M_1*M_2$ be a nontrivial tracial free product of finite von Neumann algebras. We prove that any amenable subalgebra of $M$ that has a diffuse intersection with $M_1$ is in fact contained in $M_1$. This has been proved by C. Houdayer under more general circumstances.
We adapt a continuous logic axiomatization of tracial von Neumann algebras due to Farah, Hart and Sherman in order to prove a metatheorem for this class of structures in the style of proof mining, a research program that aims to obtain the hidden computational content of ordinary mathematical proofs using tools from proof theory.
In this paper, we study closed convex hulls of unitary orbits in various C$^*$-algebras. For unital C$^*$-algebras with real rank zero and a faithful tracial state determining equivalence of projections, a notion of majorization describes the closed convex hulls of unitary orbits for self-adjoint operators. Other notions of majorization are examined in these C$^*$-algebras. Combining these ideas with the Dixmier property, we demonstrate unital, infinite dimensional C$^*$-algebras of real rank zero and strict comparison of projections with respect to a faithful tracial state must be simple and …
We study variants of the Dixmier property that apply to elements of a unital C*-algebra, rather than to the C*-algebra itself. By a Dixmier element in a C*-algebra we understand one that can be averaged into a central element by means of a sequence of unitary mixing operators. Examples include all self-commutators and all quasinilpotent elements. We do a parallel study of an element-wise version of weak centrality, where the averaging to the centre is done using unital completely positive elementary operators (as in Magajna's characterization of weak centrality). We also obtain complete descri…
We show that every inclusion of von Neumann algebras with a faithful normal conditional expectation has the weak relative Dixmier property. This answers a question of Popa \cite{Po99}. The proof uses an improvement of Ellis' lemma for compact convex semigroup.
In this note, we show that the theory of tracial von Neumann algebras does not have a model companion. This will follow from the fact that the theory of any locally universal, McDuff II_1 factor does not have quantifier elimination. We also show how a positive solution to the Connes Embedding Problem implies that there can be no model-complete theory of II_1 factors.
We survey the developments in the model theory of tracial von Neumann algebras that have taken place in the last fifteen years. We discuss the appropriate first-order language for axiomatizing this class as well as the subclass of II$_1$ factors. We discuss how model-theoretic ideas were used to settle a variety of questions around isomorphism of ultrapowers of tracial von Neumann algebras with respect to different ultrafilters before moving on to more model-theoretic concerns, such as theories of II$_1$ factors and existentially closed II$_1$ factors. We conclude with two recent applications …
We characterize when the reduced C*-algebra of a group has unique tracial state, respectively, is simple, in terms of Dixmier-type properties of the group C*-algebra. We also give a simple proof of the recent result by Breuillard, Kalantar, Kennedy and Ozawa that the reduced C*-algebra of a group has unique tracial state if and only if the amenable radical of the group is trivial.
We extend the noncommutative L1-maximal ergodic inequality for semifinite von Neumann algebras established by Yeadon in 1977 to the framework of noncommutative L1-spaces associated with sigma-finite von Neumann algebras. Since the semifnite case of this result is one of the two essential parts in the proof of noncommutative maximal ergodic inequality for tracial Lp-spaces (1<p<infinity) by Junge-Xu in 2007, we hope our result will be helpful to establish a complete noncommutative maximal ergodic inequality for non-tracial Lp-spaces in the future.
We define the notion of von Neumann spectral triple and consider the associated index problem. We compute the analytic Chern-Connes character of such a von Neumann spectral triple and prove the corresponding local formula for its Hochschild class. This formula involves the Dixmier trace for $II_{\infty}$ von Neumann algebras. In the case of foliations, we identify this Dixmier trace with the corresponding measured Wodzicki residue.
We prove that every positive trace on a countably generated *-algebra can be approximated by positive traces on algebras of generic matrices. This implies that every countably generated tracial *-algebra can be embedded into a metric ultraproduct of generic matrix algebras. As a particular consequence, every finite von Neumann algebra with separable pre-dual can be embedded into an ultraproduct of tracial *-algebras, which as *-algebras embed into a matrix-ring over a commutative algebra.
We study the class of pseudocompact C*-algebras, which are the logical limits of finite-dimensional C*-algebras. The pseudocompact C*-algebras are unital, stably finite, real rank zero, stable rank one, and tracial. We show that the pseudocompact C*-algebras have trivial K_1 groups and the Dixmier property. The class is stable under direct sums, tensoring by finite-dimensional C*-algebras, taking corners, and taking centers. We give an explicit axiomatization of the commutative pseudocompact C*-algebras. We also study the subclass of pseudomatricial C*-algebras, which have unique tracial state…
We consider the tracial crossed product algebra $M=A\rtimesΛ$ arising from a trace preserving action $σ:Λ\curvearrowright A$ of a discrete group $Λ$ on a tracial von Neumann algebra $A$. For a unitary subgroup $\mathcal{G}\subset \mathcal{U}(M)$, we study when this $\mathcal{G}$ can be conjugated into $\mathcal{U}(A)\cdotΛ$ in $M$. We provide a general sufficient condition for this to happen. Our result generalizes a result of Ioana, Popa and Vaes, which treats the case when $M$ is the group von Neumann algebra $L(Λ)$.
In this paper, we provide a generalized version of the Voiculescu theorem for normal operators by showing that, in a von Neumann algebra with separable pre-dual and a faithful normal semifinite tracial weight $τ$, a normal operator is an arbitrarily small $(\max\{\|\cdot\|, \Vert\cdot\Vert_{2}\})$-norm perturbation of a diagonal operator. Furthermore, in a countably decomposable, properly infinite von Neumann algebra with a faithful normal semifinite tracial weight, we prove that each self-adjoint operator can be diagonalized modulo norm ideals satisfying a natural condition.
We consider the reduction of problems on general noncommutative $L_p$-spaces to the corresponding ones on those associated with finite von Neumann algebras. The main tool is a unpublished result of the first named author which approximates any noncommutative $L_p$-space by tracial ones. We show that under some natural conditions a map between two von Neumann algebras extends to their crossed products by a locally compact abelian group or to their noncommutative $L_p$-spaces. We present applications of these results to the theory of noncommutative martingale inequalities by reducing most recent…
Texas Christian University（德克萨斯基督教大学）；Universidad Nacional Autonoma de México（墨西哥国立自治大学）；University of Münster（明斯特大学）；University of Southern Denmark（南丹麦大学）
The notion of the Haagerup approximation property, originally introduced for von Neumann algebras equipped with a faithful normal tracial state, is generalized to arbitrary von Neumann algebras. We discuss two equivalent characterisations, one in terms of the standard form and the other in terms of the approximating maps with respect to a fixed faithful normal semifinite weight. Several stability properties, in particular regarding the crossed product construction are established and certain examples are introduced.
Results of Haagerup and Schultz (2009) about existence of invariant subspaces that decompose the Brown measure are extended to a large class of unbounded operators affiliated to a tracial von Neumann algebra. These subspaces are used to decompose an arbitrary operator in this class into the sum of a normal operator and a spectrally negligible operator. This latter result is used to prove that, on a bimodule over a tracial von Neumann algebra that is closed with respect to logarithmic submajorization, every trace is spectral, in the sense that the trace value on an operator depends only on the …
The von Neumann algebra free product of arbitary finite dimensional von Neumann algebras with respect to arbitrary faithful states, at least one of which is not a trace, is found to be a type~III factor possibly direct sum a finite dimensional algebra. The free product state on the type~III factor is what we call an extremal almost periodic state, and has centralizer isomorphic to $L(\freeF_\infty)$. This allows further classification the type~III factor and provides another construction of full type~III$_1$ factors having arbitrary $\Sd$~invariant of Connes. The free products considered in th…
Given a von Neumann algebra $M$ with a faithful normal finite trace, we introduce the so called finite tracial algebra $M_f$ as the intersection of $L_p$-spaces $L_p(M, μ)$ over all $p \geq 1$ and over all faithful normal finite traces $μ$ on $M.$ Basic algebraic and topological properties of finite tracial algebras are studied. We prove that all derivations on these algebras are inner.
Given a finite, directed, connected graph $Γ$ equipped with a weighting $μ$ on its edges, we provide a construction of a von Neumann algebra equipped with a faithful, normal, positive linear functional $(\mathcal{M}(Γ,μ),φ)$. When the weighting $μ$ is instead on the vertices of $Γ$, the first author showed the isomorphism class of $(\mathcal{M}(Γ,μ),φ)$ depends only on the data $(Γ,μ)$ and is an interpolated free group factor equipped with a scaling of its unique trace (possibly direct sum copies of $\mathbb{C}$). Moreover, the free dimension of the interpolated free group factor is easily com…
In 2015, Yanni Chen, Don Hadwin and Junhao Shen proved a noncommutative version of Beurling's theorems for a continuous unitarily invariant norm $% α$ on a tracial von Neumann algebra $\left( \mathcal{M},τ\right) $ where $α$ is $\left\Vert \cdot \right\Vert _{1}$-dominating with respect to $τ$. In the paper, we first define a class of norms $% N_{Δ}\left( \mathcal{M},τ\right) $ on $\mathcal{M}$, called determinant, normalized, unitarily invariant continuous norms on $\mathcal{M}$. If $α\in N_{Δ}\left( \mathcal{M},τ\right) $, then there exists a faithful normal tracial state $ρ$ on $\mathcal{M}…
Let $\Fth$ be a single vertex \textsf{k}-graph, and $π_ω(Ø_θ)"$ be the von Neumann algebra induced from the GNS representation of a distinguished state $ω$ of its $\textsf{k}$-graph C*-algebra $Ø_θ$. In this paper, we prove the factoriality of $π_ω(Ø_θ)"$ and further determine its type, when either $\Fth$ has the little pull-back property, or the intrinsic group of $\Fth$ has rank $0$. The key step to achieve this is to show that the fixed point algebra of the modular action corresponding to $ω$ has a unique tracial state.
Trace inequalities are general techniques with many applications in quantum information theory, often replacing classical functional calculus in noncommutative settings. The physics of quantum field theory and holography, however, motivate entropy inequalities in type III von Neumann algebras that lack a semifinite trace. The Haagerup and Kosaki $L_p$ spaces enable re-expressing trace inequalities in non-tracial von Neumann algebras. In particular, we show this for the generalized Araki-Lieb-Thirring and Golden-Thompson inequalities from (Sutter, Berta \& Tomamichel 2017). Then, using the Haag…
Suppose M is a hyperfinite von Neumann algebra with a tracial state $ϕ$ and $\{a_1,...,a_n\}$ is a set of selfadjoint generators for M. We calculate $δ_0(a_1,...,a_n)$, the modified free entropy dimension of $\{a_1,...,a_n\}$. Moreover we show that $δ_0(a_1,...,a_n)$ depends only on M and $ϕ$. Consequently $δ_0(a_1,...,a_n)$ is independent of the choice of generators for M. In the course of the argument we show that if $\{b_1,...,b_n\}$ is a set of selfadjoint generators for a von Neumann algebra R with a tracial state and $\{b_1,...,b_n\}$ has finite dimensional approximants, then for any $b\…
We characterize semicircular distribution by the freeness of linear and quadratic forms in noncommutative random variables from a tracial $W^*$-probability space with relaxed moment conditions.
We define a notion of tracial $\mathcal{Z}$-absorption for simple not necessarily unital C*-algebras, study it systematically, and prove its permanence properties. This extends the notion defined by Hirshberg and Orovitz for unital C*-algebras. The Razak-Jacelon algebra, simple C*-algebras with tracial rank zero, and simple purely infinite C*-algebras are tracially $\mathcal{Z}$-absorbing. We obtain the first purely infinite examples of tracially $\mathcal{Z}$-absorbing C*-algebras which are not $\mathcal{Z}$-absorbing. We use techniques from reduced free products of von~Neumann algebras to co…
We prove that every surjective unital linear mapping which preserves invertible elements from a Banach algebra onto a C*-algebra carrying a faithful tracial state is a Jordan homomorphism thus generalising Aupetit's 1998 result for finite von Neumann algebras.
Let A be a unital simple separable C*-algebra. If $A$ is nuclear and infinite-dimensional, it is known that strict comparison is equivalent to Z-stability if the extreme boundary of its tracial state space is non-empty, compact and of finite covering dimension. Here we will provide the first proof of this result on the case of certain non-compact extreme boundaries. Besides, if A has strict comparison of positive elements, it is known that the Cuntz semigroup of this C*-algebra is recovered functorially from the Murray-von Neumann semigroup and the tracial state state space whenever the extrem…
Assume $\N$ is a von Neumann algebra of type II$_1$ with a tracial state $τ_{\N}$, and $\M$ is the von Neumann algebra of the $n\times n$ matrices over $\N$ with the canonical tracial state $τ_{\M}$. Let $\mathcal D_n$ be the subalgebra of $\M$ consisting of scalar diagonal matrices in $\M$. In this article, we study the properties of semicircular elements in $\M$ that are free from $\mathcal D_n$ with respect to $τ_{\M}$. Then we define a new concept "matricial distance" of two elements in $\M$ and compute the matricial distance between two free semicircular elements in $\M$.
Let $A$ be a unital separable simple $C^*$-algebra with tracial rank zero and let $x, \, y\in A$ be two normal elements. We show that $x$ is in the closure of the convex full of the unitary obit of $y$ if and only if there exists a sequence of unital completely positive linear maps $ϕ_n$ from $A$ to $A$ such that the sequence $ϕ_n(y)$ convergent to $x$ in norm and also approximately preserves the trace values. A purely measure theoretical description for normal elements in the closure of convex hull of unitary orbit of $y$ is also given. In the case that $A$ has a unique tracial state some cla…
Let $\mathcal{M}$ be a type ${\rm II_1}$ factor and let $τ$ be the faithful normal tracial state on $\mathcal{M}$. In this paper, we prove that given an $X \in \mathcal{M}$, $X=X^*$, then there is a decomposition of the identity into $N \in \mathbb{N}$ mutually orthogonal nonzero projections $E_j\in\mathcal{M}$, $I=\sum_{j=1}^NE_j$, such that $E_jXE_j=τ(X) E_j$ for all $j=1,\cdots,N$. Equivalently, there is a unitary operator $U \in \mathcal{M}$ with $U^N=I$ and $\frac{1}{N}\sum_{j=0}^{N-1}{U^*}^jXU^j=τ(X)I.$ As the first application, we prove that a positive operator $A\in \mathcal{M}$ can be…
We show that any free product of finite-dimensional von Neumann algebras equipped with non-tracial states is isomorphic to a free Araki-Woods factor with its free quasi-free state possibly direct sum a finite-dimensional von Neumann algebra. This gives a complete answer to questions posed by Dykema and Shlyakhtenko, which had been partially answered by work of Houdayer and work of Ueda. We also extend this to suitable infinite-dimensional von Neumann algebras with almost periodic states.
We study correspondences of tracial von Neumann algebras from the model-theoretic point of view. We introduce and study an ultraproduct of correspondences and use this ultraproduct to prove, for a fixed pair of tracial von Neumann algebras M and N, that the class of M-N correspondences forms an elementary class. We prove that the corresponding theory is classifiable, all of its completions are stable, that these completions have quantifier elimination in an appropriate language, and that one of these completions is in fact the model companion. We also show that the class of triples (M, H, N), …
We extend Gour et al's characterization of quantum majorization via conditional min-entropy to the context of semifinite von Neumann algebras. Our method relies on a connection between conditional min-entropy and operator space projective tensor norm for injective von Neumann algebras. This approach also connects the tracial Hahn-Banach theorem of Helton, Klep and McCullough to noncommutative vector-valued $L_1$-space.
