Faithful Normal Semifinite Trace Research Articles

Notes on noncommutative ergodic theorems

Given a semifinite von Neumann algebra M \mathcal M equipped with a faithful normal semifinite trace τ \tau , we prove that the spaces L 0 ( M , τ ) L^0(\mathcal M,\tau ) and R τ \mathcal R_\tau are complete with respect to pointwise—almost uniform and bilaterally almost uniform—convergences in L 0 ( M , τ ) L^0(\mathcal M,\tau ) . Then we show that the pointwise Cauchy property for a special class of nets of linear operators in the space L 1 ( M , τ ) L^1(\mathcal M,\tau ) can be extended to pointwise convergence of such nets in any fully symmetric space E ⊂ R τ E\subset \mathcal R_\tau , in particular, in any space L p ( M , τ ) L^p(\mathcal M,\tau ) , 1 ≤ p > ∞ 1\leq p>\infty . Some applications of these results in the noncommutative ergodic theory are discussed.

A block projection operator in the algebra of measurable operators

Let τ be a faithful normal semifinite trace on a von Neumann algebra M. We investigate the block projection operator Pn (n ≥ 2) in the ∗-algebra S(M, τ ) of all τ -measurable operators. We show that A ≤ nPn(A) for any operator A ∈ S(M, τ )+. If an operator A ∈ S(M, τ )+ is invertible in S(M, τ ) then Pn(A) is invertible in S(M, τ ). Consider A = A∗ ∈ S(M, τ ). Then (i) if Pn(A) ≤ A (or if Pn(A) ≥ A) then Pn(A) = A; (ii) Pn(A) = A if and only if PkA = APk for all k = 1, . . . , n; (iii) if A, n(A) are projections then n(A) = A. We obtain 4 corollaries. We also refined and reinforced one example from the paper “A. Bikchentaev, F. Sukochev, Inequalities for the block projection operators, J. Funct. Anal. 280 (7), article 108851, 18 p. (2021)”.

The algebra of thin measurable operators is directly finite

Let $\mathcal{M}$ be a semifinite von Neumann algebra on a Hilbert space $\mathcal{H}$ equipped with a faithful normal semifinite trace $\tau$, $S(\mathcal{M},\tau)$ be the ${}^*$-algebra of all $\tau$-measurable operators. Let $S_0(\mathcal{M},\tau)$ be the ${}^*$-algebra of all $\tau$-compact operators and $T(\mathcal{M},\tau)=S_0(\mathcal{M},\tau)+\mathbb{C}I$ be the ${}^*$-algebra of all operators $X=A+\lambda I$ with $A\in S_0(\mathcal{M},\tau)$ and $\lambda \in \mathbb{C}$. It is proved that every operator of $T(\mathcal{M},\tau)$ that is left-invertible in $T(\mathcal{M},\tau)$ is in fact invertible in $T(\mathcal{M},\tau)$. It is a generalization of Sterling Berberian theorem (1982) on the subalgebra of thin operators in $\mathcal{B} (\mathcal{H})$. For the singular value function $\mu(t; Q)$ of $Q=Q^2\in S(\mathcal{M},\tau)$, the inclusion $\mu(t; Q)\in \{0\}\bigcup [1, +\infty)$ holds for all $t>0$. It gives the positive answer to the question posed by Daniyar Mushtari in 2010.

Derivations with Values in the Ideal of tau -Compact Operators Affiliated with a Semifinite von Neumann Algebra

Let {{mathcal {M}}} be a semifinite von Neumann algebra with a faithful normal semifinite trace tau and let {{mathcal {A}}} be an arbitrary von Neumann subalgebra of {{mathcal {M}}}. We characterize the class of symmetric ideals {{mathcal {E}}} in {{mathcal {M}}} such that derivations delta :{{mathcal {A}}}rightarrow {{mathcal {E}}} are necessarily inner, which is a unification and far-reaching extension of the results due to Johnson and Parrott (J Funct Anal 11:39–61, 1972), due to Kaftal and Weiss (J Funct Anal 62:202–220, 1985), and due to Popa (J Funct Anal 71:393–408, 1987). In particular, we show that every derivation from {{mathcal {A}}} into the ideal {{mathcal {C}}}_0({{mathcal {M}}},tau ) of all tau -compact operators is inner, establishing a semifinite version of the Johnson–Parrott–Popa Theorem which is different from Popa and Rădulescu (Duke Math J 57(2):485–518, 1988, Theorem 1.1) and contrasts to the example of a non-inner derivation established in Popa and Rădulescu (1988, Theorem 1.2).

Noncommutative Calderón–Lozanovskiĭ–Hardy spaces

Let $${\mathcal {M}}$$ be a diffuse von Neumann algebra with a faithful normal semi-finite trace $$\tau $$ , and let E be a symmetric quasi-Banach space. Then for any Orlicz function $$\varphi $$ , we can define the noncommutative Calderon–Lozanovskiĭ spaces $$E_\varphi ({\mathcal {M}})$$ . These spaces share many properties with their classical counterparts. In particular, new multiplication operator spaces and complex interpolation spaces of such spaces are given under a wide range of conditions. Moreover, letting $${\mathcal {A}}$$ be a maximal subdiagonal algebra of $${\mathcal {M}}$$ , we introduce the noncommutative Calderon–Lozanovskiĭ–Hardy spaces $$H^\varphi ({\mathcal {A}})$$ and transfer the recent results of the noncommutative Hardy spaces to the noncommutative Calderon–Lozanovskiĭ–Hardy spaces.

Convergence in Measure and τ-Compactness of τ-Measurable Operators, Affiliated with a Semifinite von Neumann Algebra

Let τ be a faithful normal semifinite trace on a von Neumann algebra. We establish the Leibniz criterion for sign-alternating series of τ-measurable operators and present an analogue of the criterion of series “sandwich” series for τ-measurable operators. We prove a refinement of this criterion for the τ-compact case. In terms of measure convergence topology, the criterion of τ-compactness of an arbitrary τ-measurable operator is established. We also give a sufficient condition of 1) τ-compactness of the commutator of a τ-measurable operator and a projection; 2) convergence of τ-measurable operator and projection commutator sequences to the zero operator in the measure τ.

Rearrangements of Tripotents and Differences of Isometries in Semifinite von Neumann Algebras

Let τ be a faithful normal semifinite trace on a von Neumann algebra ℳ, and ℳu be a unitary part of ℳ. We prove a new property of rearrangements of some tripotents in ℳ. If V ∈ ℳ is an isometry (or a coisometry) and U − V is τ-compact for some U ∈ ℳu then V ∈ ℳu. Let ℳ be a factor with a faithful normal trace τ on it. If V ∈ ℳ is an isometry (or a coisometry) and U − V is compact relative to ℳ for some U ∈ ℳu then V ∈ ℳu. We also obtain some corollaries.

M-embedded symmetric operator spaces and the derivation problem

AbstractLet ℳ be a semifinite von Neumann algebra with a faithful semifinite normal trace τ. Assume that E(0, ∞) is an M-embedded fully symmetric function space having order continuous norm and is not a superset of the set of all bounded vanishing functions on (0, ∞). In this paper, we prove that the corresponding operator space E(ℳ, τ) is also M-embedded. It extends earlier results by Werner [48, Proposition 4∙1] from the particular case of symmetric ideals of bounded operators on a separable Hilbert space to the case of symmetric spaces (consisting of possibly unbounded operators) on an arbitrary semifinite von Neumann algebra. Several applications are given, e.g., the derivation problem for noncommutative Lorentz spaces ℒp,1(ℳ, τ), 1 < p < ∞, has a positive answer.

On τ-essentially Invertibility of τ-measurable Operators

Let $$ \mathcal{M} $$ be a von Neumann algebra of operators on a Hilbert space and τ be a faithful normal semifinite trace on $$ \mathcal{M} $$ . Let I be the unit of the algebra $$ \mathcal{M} $$ . A τ-measurable operator A is said to be τ-essentially right (or left) invertible if there exists a τ-measurable operator B such that the operator I − AB (or I − BA) is τ-compact. A necessary and sufficient condition for an operator A to be τ-essentially left invertible is that A∗A (or, equivalently, $$ \sqrt{A^{\ast }A} $$ ) is τ-essentially invertible. We present a sufficient condition that a τ-measurable operator A not be τ-essentially left invertible. For τ-measurable operators A and P = P2 the following conditions are equivalent: 1. A is τ-essential right inverse for P; 2. A is τ-essential left inverse for P; 3. I − A,I − P are τ-compact; 4. PA is τ-essential left inverse for P. For τ-measurable operators A = A3, B = B3 the following conditions are equivalent: 1. B is τ-essential right inverse for A; 2. B is τ-essential left inverse for A. Pairs of faithful normal semifinite traces on $$ \mathcal{M} $$ are considered.

Interpolation between $$L_0({\mathcal M},\tau )$$ L 0 ( M , τ ) and $$L_\infty ({\mathcal M},\tau )$$ L ∞ ( M , τ )

Let ${\mathcal M}$ be a semifinite von Neumann algebra with a faithful semifinite normal trace $\tau$. We show that the symmetrically $\Delta$-normed operator space $E({\mathcal M},\tau)$ corresponding to an arbitrary symmetrically $\Delta$-normed function space $E(0,\infty)$ is an interpolation space between $L_0({\mathcal M},\tau)$ and ${\mathcal M}$, which is in contrast with the classical result that there exist symmetric operator spaces $E({\mathcal M},\tau)$ which are not interpolation spaces between $L_1({\mathcal M},\tau)$ and ${\mathcal M}$. Besides, we show that the ${\mathcal K}$-functional of every $X\in L_0({\mathcal M},\tau)+ {\mathcal M} $ coincides with the ${\mathcal K}$-functional of its generalized singular value function $\mu(X)$. Several applications are given, e.g., it is shown that the pair $(L_0({\mathcal M},\tau),{\mathcal M})$ is ${\mathcal K}$-monotone when ${\mathcal M}$ is a non-atomic finite factor.

When weak and local measure convergence implies norm convergence

Let M be a von Neumann algebra of operators on a Hilbert space H and τ be a faithful normal semifinite trace on M . Let t τ be the measure topology on the ⁎-algebra S ( M , τ ) of all τ -measurable operators. We prove that for B ∈ S ( M , τ ) + the sets I B = { A ∈ S ( M , τ ) h : − B ≤ A ≤ B } and K B = { A ∈ S ( M , τ ) : A ⁎ A ≤ B } are convex and t τ -closed in S ( M , τ ) . In this case, we have I B = { B T B : T ∈ M h and ‖ T ‖ ≤ 1 } and, for invertible B , we describe the set of extreme points of the set I B . Let M be an atomic von Neumann algebra. We prove that an operator B ∈ S ( M , τ ) + is τ -compact if and only if the set I B is t τ -compact. The t τ -compactness of I B for all τ -compact operators B characterizes these algebras.

On Sets of Measurable Operators Convex and Closed in Topology of Convergence in Measure

For a von Neumann algebra with a faithful normal semifinite trace, the properties of operator “intervals” of three types for operators measurable with respect to the trace are investigated. The first two operator intervals are convex and closed in the topology of convergence in measure, while the third operator interval is convex for all nonnegative operators if and only if the von Neumann algebra is Abelian. A sufficient condition for the operator intervals of the second and third types not to be compact in the topology of convergence in measure is found. For the algebra of all linear bounded operators in a Hilbert space, the operator intervals of the second and third types cannot be compact in the norm topology. A nonnegative operator is compact if and only if its operator interval of the first type is compact in the norm topology. New operator inequalities are proved. Applications to Schatten–von Neumann ideals are obtained. Two examples are considered.

2-Local Derivations on Some $$ C^* $$ C ∗ -Algebras

In this paper, we introduce the concept of trace-open projections in the second dual $$ \mathcal {A}^{**} $$ , of a $$ C^* $$ -algebra $$ \mathcal {A} $$ . This new concept is applied to show that if there is a faithful normal semi-finite trace $$ \tau $$ on $$ \mathcal {A}^{**} $$ such that $$ 1_{ \mathcal {A}^{**} } $$ is a $$ \tau $$ -open projection, then every 2-local derivation $$ \Delta $$ from $$ \mathcal {A} $$ to $$ \mathcal {A}^{**} $$ is an inner derivation. We also prove that the same conclusion holds for approximately 2-local derivations when $$ \mathcal {A}^{**} $$ is a finite von Neumann algebra without extra assumptions on its unit element.

Paranormal Measurable Operators Affiliated with a Semifinite von Neumann Algebra

Let M be a von Neumann algebra of operators on a Hilbert space H, τ be a faithful normal semifinite trace on M. We define two (closed in the topology of convergence in measure τ) classes P1 and P2 of τ-measurable operators and investigate their properties. The class P2 contains P1. If a τ-measurable operator T is hyponormal, then T lies in P1; if an operator T lies in Pk, then UTU* belongs to Pk for all isometries U from Mand k = 1, 2; if an operator T from P1 admits the bounded inverse T−1 then T−1 lies in P1. If a bounded operator T lies in P1 then T is normaloid, Tn belongs to P1 and a rearrangement μt(Tn) ≥ μt(T )n for all t > 0 and natural n. If a τ-measurable operator T is hyponormal and Tn is τ-compact operator for some natural number n then T is both normal and τ-compact. If an operator T lies in P1 then T 2 belongs to P1. If M= B(H) and τ = tr, then the class P1 coincides with the set of all paranormal operators onH. If a τ-measurable operator A is q-hyponormal (1 ≥ q > 0) and |A*| ≥ μ∞(A)I then Ais normal. In particular, every τ-compact q-hyponormal (or q-cohyponormal) operator is normal. Consider a τ-measurable nilpotent operator Z ≠ 0 and numbers a, b ∈ R. Then an operator Z*Z − ZZ* + aRZ + bSZ cannot be nonpositive or nonnegative. Hence a τ-measurable hyponormal operator Z ≠ 0 cannot be nilpotent.

Ideal Spaces of Measurable Operators Affiliated to A Semifinite Von Neumann Algebra

Suppose that M is a von Neumann algebra of operators on a Hilbert space H and τ is a faithful normal semifinite trace on M. Let E, F and G be ideal spaces on (M, τ). We find when a τ-measurable operator X belongs to E in terms of the idempotent P of M. The sets E+F and E·F are also ideal spaces on (M, τ); moreover, E·F = F·E and (E+F)·G = E·G+F·G. The structure of ideal spaces is modular. We establish some new properties of the L1(M, τ) space of integrable operators affiliated to the algebra M. The results are new even for the *-algebra M = B(H) of all bounded linear operators on H which is endowed with the canonical trace τ = tr.

On dynamical systems preserving weights

The canonical unitary representation of a locally compact separable group arising from an ergodic action of the group on a von Neumann algebra with separable predual preserving a faithful normal semifinite (infinite) weight is weak mixing. On the contrary, there exists a non-ergodic automorphism of a von Neumann algebra preserving a faithful normal semifinite trace such that the spectral measure and the spectral multiplicity of the induced action are respectively the Haar measure (on the unit circle) and $\infty$. Despite not even being ergodic, this automorphism has the same spectral data as that of a Bernoulli shift.

Derivations with values in ideals of semifinite von Neumann algebras

Let M be a semifinite von Neumann algebra with a faithful semifinite normal trace τ and let A be an arbitrary C⁎-subalgebra of M. Assume that E is a fully symmetric function space on (0,∞) having Fatou property and order continuous norm and E(M,τ) is the corresponding symmetric operator space. We prove that every derivation δ:A→E(M,τ):=E(M,τ)∩M is inner, strengthening earlier results by Kaftal and Weiss [28]. In the case when M is a semifinite non-finite factor, we show that our assumptions on E(0,∞) are sharp.

On τ-Compactness of Products of τ-Measurable Operators

Let \(\mathcal {M}\) be a von Neumann algebra of operators on a Hilbert space \(\mathcal {H}\), τ be a faithful normal semifinite trace on \(\mathcal {M}\). We obtain some new inequalities for rearrangements of τ-measurable operators products. We also establish some sufficient τ-compactness conditions for products of selfadjoint τ-measurable operators. Next we obtain a τ-compactness criterion for product of a nonnegative τ-measurable operator with an arbitrary τ-measurable operator. We construct an example that shows importance of nonnegativity for one of the factors. The similar results are obtained also for elementary operators from \(\mathcal {M}\). We apply our results to symmetric spaces on \((\mathcal {M}, \tau )\). The results are new even for the *-algebra \(\mathcal {B}(\mathcal {H})\) of all linear bounded operators on \(\mathcal {H}\) endowed with the canonical trace τ = tr.

Two classes of τ-measurable operators affiliated with a von Neumann algebra

Let M be a von Neumann algebra of operators on a Hilbert space H, τ be a faithful normal semifinite trace on M. We define two (closed in the topology of convergence in measure τ) classes P1 and P2 of τ-measurable operators and investigate their properties. The class P2 contains P1. If a τ-measurable operator T is hyponormal, then T lies in P1; if an operator T lies in Pk, then UTU* belongs to Pk for all isometries U from M and k = 1, 2; if an operator T from P1 admits the bounded inverse T−1, then T−1 lies in P1. We establish some new inequalities for rearrangements of operators from P1. If a τ-measurable operator T is hyponormal and Tn is τ-compact for some natural number n, then T is both normal and τ-compact. If M = B(H) and τ = tr, then the class P1 coincides with the set of all paranormal operators on H.

The fidelity of density operators in an operator-algebraic framework

Josza’s definition of fidelity [R. Jozsa, J. Mod. Opt. 41(12), 2315–2323 (1994)] for a pair of (mixed) quantum states is studied in the context of two types of operator algebras. The first setting is mainly algebraic in that it involves unital C∗-algebras A that possess a faithful trace functional τ. In this context, the role of quantum states (that is, density operators) in the classical quantum-mechanical framework is assumed by positive elements ρ ∈ A for which τ(ρ) = 1. The second setting is more operator theoretic: by fixing a faithful normal semifinite trace τ on a semifinite von Neumann algebra M, we define and consider the fidelity of pairs of positive operators in M of unit trace. The main results of this paper address monotonicity and preservation of fidelity under the action of certain trace-preserving positive linear maps of A or of the predual M∗. Our results in the von Neumann algebra setting are novel in that we focus on the Schrödinger picture rather than the Heisenberg picture, and they also yield a new proof of a theorem of Molnár [Rep. Math. Phys. 48(3), 299–303 (2001)] on the structure of fidelity-preserving quantum channels on the trace-class operators.