Faithful Tracial State Research Articles

On Unital Full Amalgamated Free Products of Quasidiagonal C*-Algebras

In the paper, we consider the question as to whether a unital full amalgamated free product of quasidiagonal C*-algebras is itself quasidiagonal. We give a sufficient condition for a unital full amalgamated free product of quasidiagonal C*-algebras with amalgamation over a finite-dimensional C*-algebra to be quasidiagonal. By applying this result, we conclude that the unital full free product of two AF algebras with amalgamation over a finite-dimensional C*-algebra is AF if there exists a faithful tracial state on each of the two AF algebras such that the restrictions of these states to the common subalgebra coincide.

Closed convex hulls of unitary orbits in C⁎-algebras of real rank zero

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 have a unique tracial state. Also, closed convex hulls of unitary orbits of self-adjoint operators are fully described in unital, simple, purely infinite C$^*$-algebras.

GROUP ACTION PRESERVING THE HAAGERUP PROPERTY OF -ALGEBRAS

From the viewpoint of $C^{\ast }$-dynamical systems, we define a weak version of the Haagerup property for the group action on a $C^{\ast }$-algebra. We prove that this group action preserves the Haagerup property of $C^{\ast }$-algebras in the sense of Dong [‘Haagerup property for $C^{\ast }$-algebras’, J. Math. Anal. Appl.377 (2011), 631–644], that is, the reduced crossed product $C^{\ast }$-algebra $A\rtimes _{{\it\alpha},\text{r}}{\rm\Gamma}$ has the Haagerup property with respect to the induced faithful tracial state $\widetilde{{\it\tau}}$ if $A$ has the Haagerup property with respect to ${\it\tau}$.

On the properties of some sets of von Neumann algebras under perturbation

Let ℒ be a type II1 factor with separable predual and τ be a normal faithful tracial state of ℒ. We first show that the set of subfactors of ℒ with property Γ, the set of type II1 subfactors of ℒ with similarity property and the set of all McDuff subfactors of ℒ are open and closed in the Hausdorff metric d 2 induced by the trace norm; then we show that the set of all hyperfinite von Neumann subalgebras of ℒ is closed in d 2. We also consider the connection of perturbation of operator algebras under d 2 with the fundamental group and the generator problem of type II1 factors. When Open image in new window is a finite von Neumann algebra with a normal faithful trace, the set of all von Neumann subalgebras Open image in new window of Open image in new window such that Open image in new window is rigid is closed in the Hausdorff metric d 2.

Generalisations of the Haagerup approximation property to arbitrary von Neumann algebras

The notion of the Haagerup approximation property, originally introduced for von Neumann algebras equipped with a faithful normal tracial state, is generalised to arbitrary von Neumann algebras. We discuss two equivalent characterisations, one in term of the standard form and the other in term 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.

A non-commutative Wiener–Wintner theorem

For a von Neumann algebra $\mathcal{M}$ with a faithful normal tracial state $\tau$ and a positive ergodic homomorpsism $\alpha :\mathcal{L}^{1}(\mathcal{M},\tau)\to\mathcal{L}^{1}(\mathcal{M},\tau)$ such that $\tau\circ\alpha =\tau$ and $\alpha $ does not increase the norm in $\mathcal{M}$, we establish a non-commutative counterpart of the classical Wiener–Wintner ergodic theorem.

Haagerup property for [formula omitted]-algebras and rigidity of [formula omitted]-algebras with property (T)

We study the Haagerup property for C⁎-algebras. We first give new examples of C⁎-algebras with the Haagerup property. A nuclear C⁎-algebra with a faithful tracial state always has the Haagerup property, and the permanence of the Haagerup property for C⁎-algebras is established. As a consequence, the class of all C⁎-algebras with the Haagerup property turns out to be quite large. We then apply Popaʼs results and show the C⁎-algebras with property (T) have a certain rigidity property. Unlike the case of von Neumann algebras, for the reduced group C⁎-algebras of groups with relative property (T), the rigidity property strongly fails in general. Nevertheless, for some groups without nontrivial property (T) subgroups, we show a rigidity property in some cases. As examples, we prove the reduced group C⁎-algebras of the (non-amenable) affine groups of the affine planes have a rigidity property.

Purely infinite simple reduced [formula omitted]-algebras of one-relator separated graphs

Given a separated graph (E,C), there are two different C∗-algebras associated to it: the full graph C∗-algebra C∗(E,C) and the reduced one Cred∗(E,C). For a large class of separated graphs (E,C), we prove that Cred∗(E,C) either is purely infinite simple or admits a faithful tracial state. The main tool we use to show pure infiniteness of reduced graph C∗-algebras is a generalization to the amalgamated case of a result on purely infinite simple free products due to Dykema.

Characterization of subdiagonal algebras

Let M be a finite von Neumann algebra with a faithful normal tracial state T , and let A be a tracial subalgebra of M. We show that A has L P -factorization (1 ≤ p < oo) if and only if A is a subdiagonal algebra. Also, we obtain some characterizations of subdiagonal algebras.

Burkholder’s inequalities in noncommutative Lorentz spaces

We prove Burkholder’s inequalities in noncommutative Lorentz spaces L p , q ( M ) , 1 &gt; p &gt; ∞ , L^{p,q}(\mathcal {M}), 1&gt;p&gt;\infty , 1 ≤ q &gt; ∞ 1\leq q&gt;\infty , associated with a von Neumann algebra M \mathcal {M} equipped with a faithful normal tracial state. These estimates generalize the classical inequalities in the commutative case.

The noncommutative Wiener lemma, linear independence, and spectral properties of the algebra of time-frequency shift operators

In this paper we analyze the Banach *-algebra of time-frequency shifts with absolutely summable coefficients. We prove a noncommutative version of the Wiener lemma. We also construct a faithful tracial state on this algebra which proves the algebra contains no compact operators. As a corollary we obtain a special case of the Heil-Ramanathan-Topiwala conjecture regarding linear independence of finitely many time-frequency shifts of one $L^2$ function. We also estimate the coefficient decay of the inverse of finite linear combinations of time-frequency shifts.

AF-embedding of crossed products of AH-algebras by $\mathbb{Z}$ and asymptotic AF-embedding

Let A be a unital AH-algebra and let α ∈ Aut(A) be an automorphism. A necessary condition for A X α ∥ being embedded into a unital simple AF-algebra is the existence of a faithful tracial state. If in addition, there is an automorphism K with κ *1 = -id K1 (A) such that α o K and K ° α are asymptotically unitarily equivalent, then A X α Z can be embedded into a unital simple AF-algebra. Consequently, in the case that A is a unital AH-algebra (not necessarily simple) with torsion K 1 (A), A X α Z can be embedded into a unital simple AF-algebra if and only if A admits a faithful α-invariant tracial state. We also show that if A is a unital AT-algebra then A X α Z can be embedded into a unital simple AF-algebra if and only if A admits a faithful α-invariant tracial state. Consequently, for any unital simple AT-algebra A, A X α Z can always be embedded into a unital simple AF-algebra. If X is a compact metric space and A: Z 2 → Aut(C(X)) is a homomorphism, then C(X) X Λ Z 2 can be asymptotically embedded into a unital simple AF-algebra provided that X admits a strictly positive A-invariant probability measure. Consequently C(X) X Λ Z 2 is quasidiagonal if X admits a strictly positive A-invariant Borel probability measure.

Operator-algebraic superridigity for SL n (ℤ), n≥3

For n≥3, let Γ=SL n (ℤ). We prove the following superridigity result for Γ in the context of operator algebras. Let L(Γ) be the von Neumann algebra generated by the left regular representation of Γ. Let M be a finite factor and let U(M) be its unitary group. Let π:Γ→U(M) be a group homomorphism such that π(Γ)”=M. Then either (i) M is finite dimensional, or (ii) there exists a subgroup of finite index Λ of Γ such that π|Λ extends to a homomorphism U(L(Λ))→U(M). This answers, in the special case of SL n (ℤ), a question of A. Connes discussed in [Jone00, p. 86]. The result is deduced from a complete description of the tracial states on the full C *–algebra of Γ. As another application, we show that the full C *–algebra of Γ has no faithful tracial state, thus answering a question of E. Kirchberg.

On the operator space $UMD$ property for noncommutative $L_p$-spaces

We study the operator space UMD property, introduced by Pisier in the context of noncommutative vector-valued L p -spaces. It is unknown whether the property is independent of p in this setting. We prove that for 1 < p, q < oo, the Schatten q-classes Sq are OUMDp. The proof relies on properties of the Haagerup tensor product and complex interpolation. Using ultra-product techniques, we extend this result to a large class of noncommutative Lq-spaces. Namely, we show that if M is a QWEP von Neumann algebra (i.e., a quotient of a C*-algebra with Lance's weak expectation property) equipped with a normal, faithful tracial state T, then L q (M, τ) is OUMDp for 1 < p, q < oo.

THE SPECTRAL SCALE AND THE NUMERICAL RANGE

Suppose that c is an operator on a Hilbert Space H such that the von Neumann algebra N generated by c is finite. Let τ be a faithful normal tracial state on N and set b1= (c + c*)/2 and b2= (c - c*)/2i. Also write B for the spectral scale of {b1, b2} relative to τ. In previous work by the present authors, some joint with Nik Weaver, B has been shown to contain considerable spectral information about the operator c. In this paper we expand that information base by showing that the numerical range of c is encoded in B also.We begin by proving that the k-numerical range of an arbitrary operator d in B(H) coincides with the numerical range of d when the von Neumann algebra generated by d contains no finite rank operators. Thus, the k-numerical range is not useful for most operators considered here.We next show that the boundary of the numerical range of c is exactly the set of radial complex slopes on B at the origin. Further, we show that points on this boundary that lie in the numerical range are visible as line segments in the boundary of B. Also, line segments on the boundary which lie in the numerical range show up as faces of dimension two in the boundary of B. Finally, when N is abelian, we prove that the point spectrum of c appears as complex slopes of 1-dimensional faces of B.

Brown's Spectral Distribution Measure for R-Diagonal Elements in Finite von Neumann Algebras

In 1983 L. G. Brown introduced a spectral distribution measure for non-normal elements in a finite von Neumann algebra M with respect to a fixed normal faithful tracial state τ. In this paper we compute Brown's spectral distribution measure in case T has a polar decomposition T=UH where U is a Haar unitary and U and H are *-free. (When KerT={0} this is equivalent to that (T, T*) is an R-diagonal pair in the sense of Nica and Speicher.) The measure μT is expressed explicitly in terms of the S-transform of the distribution μT*T of the positive operator T*T. In case T is a circular element, i.e., T=(X1+iX2)/2 where (X1, X2) is a free semicircular system, then spT=D, the closed unit disk, and μT has constant density 1/π on D.

Uniqueness of the trace and simplicity

It is proved that for certain large classes of unital C*-algebras, the existence of a unique, faithful tracial state implies simplicity. An example is given to show that this implication does not hold for all unital C*-algebras.

Toeplitz operators with noncommuting symbols

LetM be a von Neumann algebra with a faithful normal tracial state τ and letH∞ be a finite maximal subdiagonal subalgebra ofM. LetH2 be the closure ofH∞ in the noncommutative Lebesgue spaceL2(M). We consider Toeplitz operators onH2 whose symbol belong toM, and find that they possess several of the properties of Toeplitz operators onH2(\(\mathbb{T}\)) with symbol fromL∞(\(\mathbb{T}\)), including norm estimates, a Hartman-Wintner spectral inclusion theorem, and a characterisation of the weak* continuous linear functionals on the space of Toeplitz operators.

Automorphisms and nonselfadjoint crossed products

We are interested in the invariant subspace structure of the nonselfadjoint crossed product determined by a finite von Neumann algebra M and a trace preserving automorphism α. In this paper we investigate the form of two-sided invariant subspaces for the case that a is ergodic on the center of M. 1. Introduction* In this paper, we consider the typical finite maximal subdiagonal algebras which are called nonselfadjoint crossed products. These algebras are constructed as certain subalgebras of crossed products of finite von Neumann algebras by trace preserving automorphisms. Recently, McAsey, Muhly and the author studied the invariant subspace structure and the maximality of these algebras (cf. [4], [5], [6], [7]). Let M be a von Neumann algebra with a faithful normal tracial state τ and let a be a *-automorphism of M such that τ°α = r. We regard M as acting on the noncommutative Lebesgue space L2(M, τ) (cf. [10]) and consider the Hubert space Π - {/: Z >L\M, τ)\Σ\\f(n)\\l } which may be identified with l\Z) (x) L\M, τ). Let 2 (resp. 31) be the left (resp. right) crossed product of M and a, and let £+ (resp. 9t+) be the left (resp. right) nonselfadjoint crossed product of S (resp. 3t) (cf. §2). In [6], we showed that the following three conditions are equivalent; (i) M is a factor; (ii) a conditioned form of the Beurling-Lax-Halmos theorem is valid; and (iii) 2+ is a maximal (M). We now suppose that a is ergodic on S(Λf). Then every two-sided invariant subspace of U which is not left-reducing is left-pure, left-full, right-pure and right-full (Theorems 3.2 and 4.5). Further, if S is a factor, then every proper two-sided invariant subspace of U is of