Faithful Tracial State Research Articles

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

Let \mathcal{G} be a locally compact Hausdorff étale groupoid. We call a tracial state \tau on a general groupoid C^{*} -algebra C_{\nu}^{*}(\mathcal{G}) canonical if \tau=\tau|_{C_0(\mathcal{G}^{(0)})}\circ E , where E:C^{*}_{\nu}(\mathcal{G}) \to C_{0}(\mathcal{G}^{(0)}) is the canonical conditional expectation. In this paper, we consider so-called fixed point traces on C_{c}(\mathcal{G}) , and prove that \mathcal{G} is essentially free if and only if any tracial state on C_{\nu}^{*}(\mathcal{G}) is canonical and any fixed point trace is extendable to C_{\nu}^{*}(\mathcal{G}) .As applications, we obtain the following: (1) a group action is essentially free if every tracial state on the reduced crossed product is canonical and every isotropy group is amenable; (2) if the groupoid \mathcal{G} is second-countable, amenable and essentially free then every (not necessarily faithful) tracial state on the reduced groupoid C^{*} -algebra is quasidiagonal.

On finite sums of projections and Dixmier's averaging theorem for type II1 factors

Let M be a type II1 factor and let τ be the faithful normal tracial state on M. In this paper, we prove that given an X∈M, X=X⁎, then there is a decomposition of the identity into N∈N mutually orthogonal nonzero projections Ej∈M, I=∑j=1NEj, such that EjXEj=τ(X)Ej for all j=1,…,N. Equivalently, there is a unitary operator U∈M with UN=I and 1N∑j=0N−1U⁎jXUj=τ(X)I. As the first application, we prove that a positive operator A∈M can be written as a finite sum of projections in M if and only if τ(A)≥τ(RA), where RA is the range projection of A. This result answers affirmatively Question 6.7 of [9]. As the second application, we show that if X∈M, X=X⁎ and τ(X)=0, then there exists a nilpotent element Z∈M such that X is the real part of Z. This result answers affirmatively Question 1.1 of [4]. As the third application, we show that for X1,…,Xn∈M, there exist unitary operators U1,…,Uk∈M such that 1k∑i=1kUi⁎XjUi=τ(Xj)I,∀1≤j≤n. This result is a stronger version of Dixmier's averaging theorem for type II1 factors.

A stronger version of Dixmier's averaging theorem and some applications

Let M be a type II1 factor and let τ be the faithful normal tracial state on M. In this paper, we prove that given finite elements X1,⋯,Xn∈M, there is a finite decomposition of the identity into integer N∈N mutually orthogonal nonzero projections Ej∈M, I=∑j=1NEj, such that EjXiEj=τ(Xi)Ej for all j=1,⋯,N and i=1,⋯,n. Equivalently, there is a unitary operator U∈M such that 1N∑j=0N−1U⁎jXiUj=τ(Xi)I for i=1,⋯,n. This result is a stronger version of Dixmier's averaging theorem for type II1 factors. As the first application, we show that all elements of trace zero in a type II1 factor are single commutators and any self-adjoint elements of trace zero are single self-commutators. This result answers affirmatively Question 1.1 in [6]. As the second application, we prove that any self-adjoint element in a type II1 factor can be written a linear combination of 4 projections. This result answers affirmatively Question 6(2) in [12]. As the third application, we show that if (M,τ) is a finite factor, X∈M, then there exists a normal operator N∈M and a nilpotent operator K such that X=N+K.

On isotropic subspaces of operators in II1 factors

Let M be a type II1 factor and let τ be the faithful normal tracial state on M. In this paper, we firstly prove that given an X∈M, then there is a decomposition of the identity into N∈N⋃{+∞} mutually orthogonal nonzero projections {En}n=1N⊆M, I=∑n=1NEn, such that EnXEn=τ(X)En for all n=1,⋯,N. Secondly, we show that if R is the hyperfinite II1 factor and (Rω,τω) is the ultrapower of R, then for any X∈Rω, there is a finite family of mutually orthogonal nonzero projections {Ei}i=1N in Rω such that ∑i=1NEi=I and EnXEn=τ(X)En, for all n=1,2,⋯,N. Equivalently, there is a unitary operator U∈Rω such that 1N∑i=1N(U⁎)iXUi=τω(X)I.

The Fell topology and the modular Gromov-Hausdorff propinquity

We show that any (norm-closed two-sided) ideal of a unital C*-algebra that comes equipped with certain natural quantum metric structure is a metrized quantum vector bundle, when canonically viewed as a module over A A . Next, given a unital AF-algebra A A equipped with a faithful tracial state, we equip each ideal of A A with a metrized quantum vector bundle structure using previous work of the first author and Latrémolière. Moreover, we show that convergence of ideals in the Fell topology implies convergence of the associated metrized quantum vector bundles in the modular Gromov-Hausdorff propinquity of Latrémolière. In a similar vein but requiring a different approach, given a compact metric space ( X , d ) (X,d) , we equip each ideal of C ( X ) C(X) with a metrized quantum vector bundle structure, and show that convergence in the Fell topology implies convergence in the modular Gromov-Hausdorff propinquity.

A tracial characterization of Furstenberg’s conjecture

AbstractWe investigate almost minimal actions of abelian groups and their crossed products. As an application, given multiplicatively independent integers p and q, we show that Furstenberg’s $\times p,\times q$ conjecture holds if and only if the canonical trace is the only faithful extreme tracial state on the $C^*$ -algebra of the group $\mathbb {Z}[\frac {1}{pq}]\rtimes \mathbb {Z}^2$ . We also compute the primitive ideal space and K-theory of $C^*(\mathbb {Z}[\frac {1}{pq}]\rtimes \mathbb {Z}^2)$ .

Invertibility preserving mappings onto finite $C^*$-algebras

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

Complete Gradient Estimates of Quantum Markov Semigroups

In this article we introduce a complete gradient estimate for symmetric quantum Markov semigroups on von Neumann algebras equipped with a normal faithful tracial state, which implies semi-convexity of the entropy with respect to the recently introduced noncommutative 2-Wasserstein distance. We show that this complete gradient estimate is stable under tensor products and free products and establish its validity for a number of examples. As an application we prove a complete modified logarithmic Sobolev inequality with optimal constant for Poisson-type semigroups on free group factors.

On Murray-von Neumann algebras—I: topological, order-theoretic and analytical aspects

For a countably decomposable finite von Neumann algebra \({\mathscr {R}}\), we show that any choice of a faithful normal tracial state on \({\mathscr {R}}\) engenders the same measure topology on \({\mathscr {R}}\) in the sense of Nelson (J Funct Anal 15:103–116, 1974). Consequently it is justified to speak of ‘the’ measure topology of \({\mathscr {R}}\). Having made this observation, we extend the notion of measure topology to general finite von Neumann algebras and denominate it the \({\mathfrak {m}}\)-topology. We note that the procedure of \({\mathfrak {m}}\)-completion yields Murray-von Neumann algebras in a functorial manner and provides them with an intrinsic description as unital ordered complex topological \(*\)-algebras. This enables the study of abstract Murray-von Neumann algebras avoiding reference to a Hilbert space. Furthermore, it makes apparent the appropriate notion of Murray-von Neumann subalgebras, and the intrinsic nature of the spectrum and point spectrum of elements, independent of their ambient Murray-von Neumann algebra. In this context, we show the well-definedness of the Borel function calculus for normal elements and use it along with approximation techniques in the \({\mathfrak {m}}\)-topology to transfer many standard operator inequalities involving bounded self-adjoint operators to the setting of (unbounded) self-adjoint operators in Murray-von Neumann algebras. On the algebraic side, Murray-von Neumann algebras have been described as the Ore localization of finite von Neumann algebras with respect to their corresponding multiplicative subset of non-zero-divisors. Our discussion reveals that, in addition, there are fundamental topological, order-theoretic and analytical facets to their description.

Property T C⁎-algebras with amenable tracial states

As a non-unital analogue of the main result in [6], we show in this article that if A is a separable quasi-central C⁎-algebra with property T and is nuclear, then there is a sequence of positive integers {nk}k∈N such that the c0-direct sum of the family {Mnk}k∈N of matrix algebras is an ideal JA of A and that A/JA has no tracial state. In particular, a separable C⁎-algebra A is a c0-direct sum of matrix algebras if and only if A is quasi-central, is nuclear, has property T and every unital simple quotient of A has a tracial state. We also show that a separable property TC⁎-algebra A is residually finite dimensional if and only if A admits an amenable faithful tracial state and every ideal of A has non-zero center.

INDUCTIVE LIMITS OF C*-ALGEBRAS AND COMPACT QUANTUM METRIC SPACES

AbstractGiven a unital inductive limit of C*-algebras for which each C*-algebra of the inductive sequence comes equipped with a Rieffel compact quantum metric, we produce sufficient conditions to build a compact quantum metric on the inductive limit from the quantum metrics on the inductive sequence by utilizing the completeness of the dual Gromov–Hausdorff propinquity of Latrémolière on compact quantum metric spaces. This allows us to place new quantum metrics on all unital approximately finite-dimensional (AF) algebras that extend our previous work with Latrémolière on unital AF algebras with faithful tracial state. As a consequence, we produce a continuous image of the entire Fell topology on the ideal space of any unital AF algebra in the dual Gromov–Hausdorff propinquity topology.

Matrix algebras over algebras of unbounded operators

Let $${{\mathscr {M}}}$$ be a $$II_1$$ factor acting on the Hilbert space $${{\mathscr {H}}}$$ , and $${\mathscr {M}} _{\text {aff}}$$ be the Murray–von Neumann algebra of closed densely-defined operators affiliated with $${{\mathscr {M}}}$$ . Let $$\tau $$ denote the unique faithful normal tracial state on $$\mathscr {M}$$ . By virtue of Nelson’s theory of non-commutative integration, $${\mathscr {M}} _{\text {aff}}$$ may be identified with the completion of $${{\mathscr {M}}}$$ in the measure topology. In this article, we show that $$M_n({\mathscr {M}} _{\text {aff}}) \cong M_n({{\mathscr {M}}})_{\text {aff}}$$ as unital ordered complex topological $$*$$ -algebras with the isomorphism extending the identity mapping of $$M_n({{\mathscr {M}}}) \rightarrow M_n({{\mathscr {M}}})$$ . Consequently, the algebraic machinery of rank identities and determinant identities are applicable in this setting. As a step further in the Heisenberg–von Neumann puzzle discussed by Kadison–Liu (SIGMA Symmetry Integrability Geom. Methods Appl., 10:Paper 009, 40, 2014), it follows that if there exist operators P, Q in $${\mathscr {M}} _{\text {aff}}$$ satisfying the commutation relation $$Q \; {{\hat{\cdot }}} \;P \; {\hat{-}} \;P \; {{\hat{\cdot }}} \;Q = {i\,}I$$ , then at least one of them does not belong to $$L^p({{\mathscr {M}}}, \tau )$$ for any $$0 < p \le \infty $$ . Furthermore, the respective point spectrums of P and Q must be empty. Hence the puzzle may be recasted in the following equivalent manner - Are there invertible operators P, A in $${{\mathscr {M}}}_{\text {aff}}$$ such that $$P^{-1} \; {{\hat{\cdot }}} \;A \; {{\hat{\cdot }}} \;P = I \; {\hat{+}} \;A$$ ? This suggests that any strategy towards its resolution must involve the study of conjugacy invariants of operators in $${{\mathscr {M}}}_{\text {aff}}$$ in an essential way.

An extension of the Beurling-Chen-Hadwin-Shen theorem for noncommutative Hardy spaces associated with finite von Neumann algebras

In 2015, Yanni Chen, Don Hadwin and Junhao Shen proved a noncommutative version of Beurling’s theorem for a continuous unitarily invariant norm α on a tracial von Neumann algebra (M, τ) such that α is one dominating with respect to τ . The role of H∞ is played by a maximal subdiagonal algebra A. In the talk, we first will show that if α is a continuous normalized unitarily invariant norm on (M, τ), then there exists a faithful normal tracial state ρ on M and a constant c > 0 such that α is a c times one norm-dominating norm on (M,ρ). Moreover, ρ(x) = τ(xg), where x ∈M , g is positive in L1(Z, τ), where Z is the center of M . Here c and ρ are not unique. However, if there is a c and ρ so that the Fuglede-Kadison determinant of g is positive, then Beurling-Chen-Hadwin-Shen theorem holds for L(α)(M, τ). The key ingredients in the proof of our result include a factorization theorem and a density theorem for for L(α)(M,ρ). Talk time: 2016-07-19 03:00 PM— 2016-07-19 03:20 PM Talk location: Cupples I Room 207

Manifolds of classical probability distributions and quantum density operators in infinite dimensions

The manifold structure of subsets of classical probability distributions and quantum density operators in infinite dimensions is investigated in the context of C^{*}-algebras and actions of Banach-Lie groups. Specificaly, classical probability distributions and quantum density operators may be both described as states (in the functional analytic sense) on a given C^{*}-algebra mathscr {A} which is Abelian for Classical states, and non-Abelian for Quantum states. In this contribution, the space of states mathscr {S} of a possibly infinite-dimensional, unital C^{*}-algebra mathscr {A} is partitioned into the disjoint union of the orbits of an action of the group mathscr {G} of invertible elements of mathscr {A}. Then, we prove that the orbits through density operators on an infinite-dimensional, separable Hilbert space mathcal {H} are smooth, homogeneous Banach manifolds of mathscr {G}=mathcal {GL}(mathcal {H}), and, when mathscr {A} admits a faithful tracial state tau like it happens in the Classical case when we consider probability distributions with full support, we prove that the orbit through tau is a smooth, homogeneous Banach manifold for mathscr {G}.

Simultaneous Upper Triangular Forms for Commuting Operators in a Finite von Neumann Algebra

AbstractThe joint Brown measure and joint Haagerup–Schultz projections for tuples of commuting operators in a von Neumann algebra equipped with a faithful tracial state are investigated, and several natural properties are proved for these. It is shown that the support of the joint Brown measure is contained in the Taylor joint spectrum of the tuple, and also in the ostensibly smaller left Harte spectrum. A simultaneous upper triangularization result for finite commuting tuples is proved, and the joint Brown measure and joint Haagerup–Schultz projections are shown to behave well under the Arens multivariate holomorphic functional calculus of such a commuting tuple.

Jensen's inequality in finite subdiagonal algebras

Let $\mathscr{M}$ be a finite von Neumann algebra with a faithful normal tracial state $\tau$ and $\mathfrak{A}$ be a finite subdiagonal subalgebra of $\mathscr{M}$ with respect to a $\tau$-preserving faithful normal conditional expectation $\Phi$ on $\mathscr{M}$. Let $\Delta$ denote the Fuglede-Kadison determinant corresponding to $\tau$. For $X \in \mathscr{M}$, define $|X| := (X^*X)^{\frac{1}{2}}$. In 2005, Labuschagne proved the so-called Jensen's inequality for finite subdiagonal algebras i.e. $\Delta(\Phi(A)) \le \Delta(A)$ for an operator $A \in \mathfrak{A}$, thus resolving a long-standing open problem posed by Arveson in 1967. In this article, we prove the following more general result: $\tau(f( |\Phi(A)|)) \le \tau(f( |A|))$ for $A \in \mathfrak{A}$ and any increasing continuous function $f : [0, \infty) \to \mathbb{R}$ such that $f \circ \exp$ is convex on $\mathbb{R}$. Under the additional hypotheses that $A$ is invertible in $\mathscr{M}$ and $f \circ \exp$ is strictly convex, we have $\tau(f( |\Phi(A)|)) = \tau(f( |A|))$ $\Longleftrightarrow \Phi(A) = A$. As an application, we show that for $A \in \mathfrak{A}$ the point spectrum of $A$ is contained in the point spectrum of $\Phi(A)$, though such a conclusion does not hold in general for their spectra.

On a class of determinant preserving maps for finite von Neumann algebras

Let R be a finite von Neumann algebra with a faithful tracial state τ and let Δ denote the associated Fuglede–Kadison determinant. In this paper, we characterize all unital bijective maps ϕ on the set of invertible positive elements in R which satisfyΔ(ϕ(A)+ϕ(B))=Δ(A+B). We show that any such map originates from a τ-preserving Jordan ⁎-automorphism of R (either ⁎-automorphism or ⁎-anti-automorphism in the more restrictive case of finite factors). In establishing the aforementioned result, we make crucial use of the solutions to the equation Δ(A+B)=Δ(A)+Δ(B) in the set of invertible positive operators in R. To this end, we give a new proof of the inequalityΔ(A+B)≥Δ(A)+Δ(B), using a generalized version of the Hadamard determinant inequality and conclude that equality holds for invertible B if and only if A is a nonnegative scalar multiple of B.

On outer elements of noncommutative Orlicz–Hardy spaces

Let M be a von Neumann algebra equipped with a faithful normal tracial state τ, A be a subdiagonal subalgebra of M, and let Φ be a growth function. We transfer the results of Studia Math. 217, No. 3, 265–287 (2013) to the noncommutative HΦ(A) space case.

The (**)-Haagerup property for C*-algebras

Extending the notion of Haagerup property for finite von Neumann algebras to the general von Neumann algebras, the authors define and study the (**)-Haagerup property for C*-algebras in this paper. They first give an answer to Suzuki’s question (2013), and then obtain several results of (**)-Haagerup property parallel to those of Haagerup property for C*-algebras. It is proved that a nuclear unital C*-algebra with a faithful tracial state always has the (**)-Haagerup property. Some heredity results concerning the (**)-Haagerup property are also proved.

Quantum ultrametrics on AF algebras and the Gromov–Hausdorff propinquity

We construct quantum metric structures on unital AF algebras with a faithful tracial state, and prove that for such metrics, AF algebras are limits of their defining inductive sequences of finite dimensional C*-algebras for the quantum propinquity. We then study the geometry, for the quantum propinquity, of three natural classes of AF algebras equipped with our quantum metrics: the UHF algebras, the Effros-Shen AF algebras associated with continued fraction expansions of irrationals, and the Cantor space, on which our construction recovers traditional ultrametrics. We also exhibit several compact classes of AF algebras for the quantum propinquity and show continuity of our family of Lip-norms on a fixed AF algebra. Our work thus brings AF algebras into the realm of noncommutative metric geometry.