Von Neumann algebraM Research Articles

Operator algebras with the partial factorization property

A subalgebra A \mathfrak A in a σ \sigma -finite von Neumann algebra M \mathcal M has the left (resp. right) partial factorization property if for any invertible operator S ∈ M S\in \mathcal M , there exists an isometry (resp. a co-isometry) U ∈ M U\in \mathcal M such that U ∗ S , S − 1 U ∈ A U^*S, S^{-1}U\in \mathfrak A . We show that the relative invariant subspace lattice L a t M A {Lat}_{\mathcal M}\mathfrak A consisting of those projections in M \mathcal M whose ranges are invariant under A \mathfrak A is commutative. Moreover if M \mathcal M has a separable predual, then L a t M A {Lat}_{\mathcal M}\mathfrak A is generated by a nest together with the lattice of projections in the center of M \mathcal M . In particular, if A \mathfrak A is a von Neumann subalgebra, then A = M \mathfrak A=\mathcal M .

A noncommutative Gretsky–Ostroy theorem and its applications

Let H \mathcal {H} be a separable Hilbert space and let B ( H ) B(\mathcal {H}) be the ∗ * -algebra of all bounded linear operators on H \mathcal {H} . In the present paper, we prove that a positive/regular operator from L 1 ( 0 , 1 ) L_1(0,1) into an arbitrary separable operator ideal in B ( H ) B(\mathcal {H}) is necessarily Dunford–Pettis, extending and strengthening results due to Gretsky and Ostroy [Glasgow Math. J. 28 (1986), pp. 113–114], and Holub [Proc. Amer. Math. Soc. 104 (1988), pp. 89–95]. Consequently, for an arbitrary atomless von Neumann algebra M \mathcal {M} and an arbitrary KB-ideal C E C_E in B ( H ) B(\mathcal {H}) , the predual M ∗ \mathcal {M}_* of M \mathcal {M} is not isomorphic to any subspace of C E C_E . This observation complements several earlier results.

Derivations of Murray–von Neumann algebras

AbstractIn this paper, we answer in the affirmative the long-standing conjecture that the first cohomology group of the Murray–von Neumann algebraS⁢(ℳ){S(\mathcal{M})}of all operators affiliated with a typeII1{\mathrm{II}_{1}}von Neumann algebraℳ{\mathcal{M}}is 0. That is, we show that all derivations ofS⁢(ℳ){S(\mathcal{M})}are inner.

ℓ₂-strictly singular operators on the predual of a hyperfinite von Neumann algebra

We show that every ℓ 2 \ell _2 -strictly singular operator on the predual of any atomless hyperfinite finite von Neumann algebra M \mathcal {M} is Dunford–Pettis, which extends a Rosenthal’s theorem for the case of commutative algebra M = L ∞ ( 0 , 1 ) \mathcal {M}=L_\infty (0,1) . We also apply our result to the study of noncommutative symmetric spaces X = E ( M , τ ) X=E(\mathcal {M},\tau ) for which every ℓ 2 \ell _2 -strictly singular operator from L p ( 0 , 1 ) L_p(0,1) into X X is narrow.

Maximality and finiteness of type 1 subdiagonal algebras

Let A \mathfrak A be a type 1 subdiagonal algebra in a σ \sigma -finite von Neumann algebra M \mathcal M with respect to a faithful normal conditional expectation Φ \Phi . We give necessary and sufficient conditions for which A \mathfrak A is maximal among the σ \sigma -weakly closed subalgebras of M \mathcal M . In addition, we show that a type 1 subdiagonal algebra in a finite von Neumann algebra is automatically finite which gives a positive answer of Arveson’s finiteness problem in 1967 in type 1 case.

Spectral gap characterization of full type III factors

Abstract We give a spectral gap characterization of fullness for type III {\mathrm{III}} factors which is the analog of a theorem of Connes in the tracial case. Using this criterion, we generalize a theorem of Jones by proving that if M is a full factor and σ : G → Aut ⁢ ( M ) {\sigma:G\rightarrow\mathrm{Aut}(M)} is an outer action of a discrete group G whose image in Out ⁢ ( M ) {\mathrm{Out}(M)} is discrete, then the crossed product von Neumann algebra M ⋊ σ G {M\rtimes_{\sigma}G} is also a full factor. We apply this result to prove the following conjecture of Tomatsu–Ueda: the continuous core of a type III 1 {\mathrm{III}_{1}} factor M is full if and only if M is full and its τ invariant is the usual topology on ℝ {\mathbb{R}} .

Ultraproducts, QWEP von Neumann algebras, and the Effros–Maréchal topology

Abstract Based on the analysis on the Ocneanu/Groh–Raynaud ultraproducts and the Effros–Maréchal topology on the space vN ( H ) ${\mathrm {vN}(H)}$ of von Neumann algebras acting on a separable Hilbert space H, we show that for a von Neumann algebra M ∈ vN ( H ) ${M\in \mathrm {vN}(H)}$ , the following conditions are equivalent: (1) M has the Kirchberg's quotient weak expectation property (QWEP). (2) M is in the closure of the set ℱ inj ${\mathcal {F}_{\rm inj}}$ of injective factors on H with respect to the Effros–Maréchal topology. (3) M admits an embedding i into the Ocneanu ultrapower R ∞ ω ${R_{\infty }^{\omega }}$ of the injective III 1 ${\mathrm {III}_1}$ factor R ∞ with a normal faithful conditional expectation ε : R ∞ ω → i ( M ) ${\varepsilon \colon R_{\infty }^{\omega }\rightarrow i(M)}$ . (4) For every ε > 0 ${\varepsilon >0}$ , n ∈ ℕ and ξ 1 , ... , ξ n ∈ 𝒫 M ♮ ${\xi _1,\ldots ,\xi _n\in \mathcal {P}_M^{\natural }}$ , there are k ∈ ℕ and a 1 , ... , a n ∈ M k ( ℂ ) + ${a_1,\ldots ,a_n\in M_k(\mathbb {C})_+}$ such that | 〈 ξ i , ξ j 〉 - tr k ( a i a j ) | < ε ${|\langle \xi _i,\xi _j\rangle -\operatorname{tr}_k(a_ia_j)|<\varepsilon }$ ( 1 ≤ i , j ≤ n ${1\le i,j\le n}$ ) holds, where tr k ${\operatorname{tr}_k}$ is the tracial state on M k ( ℂ ) ${M_k(\mathbb {C})}$ , and 𝒫 M ♮ ${\mathcal {P}_M^{\natural }}$ is the natural cone in the standard form of M.

The $p$-Banach Saks property in symmetric operator spaces

Let E be a separable r.i. space which is an interpolation space between Lr(R+) and Lq(R+), 1 < r < q < ∞. We give sufficient conditions on E implying that the symmetric operator space E(M, τ) has the p-Banach-Saks property for a suitable p, for an arbitrary semifinite von Neumann algebraM.

Relative weak compactness of orbits in Banach spaces associated with locally compact groups

We study analogues of weak almost periodicity in Banach spaces on locally compact groups. i) If μ \mu is a continous measure on the locally compact abelian group G G and f ∈ L ∞ ( μ ) f\in L^\infty (\mu ) , then { γ f : γ ∈ G ^ } \{\gamma f:\gamma \in \widehat G\} is not relatively weakly compact. ii) If G G is a discrete abelian group and f ∈ ℓ ∞ ( G ) ∖ C o ( G ) f\in \ell ^\infty (G)\backslash C_o(G) , then { γ f : γ ∈ E } \{\gamma f:\gamma \in E\} is not relatively weakly compact if E ⊂ G ^ E\subset \widehat G has non-empty interior. That result will follow from an existence theorem for I o I_o -sets, as follows. iii) Every infinite subset of a discrete abelian group Γ \Gamma contains an infinite I o I_o -set such that for every neighbourhood U U of the identity of Γ ^ \widehat \Gamma the interpolation (except at a finite subset depending on U U ) can be done using at most 4 point masses. iv) A new proof that B ( G ) ⊂ W A P ( G ) B(G)\subset WAP(G) for abelian groups is given that identifies the weak limits of translates of Fourier-Stieltjes transforms. v) Analogous results for C o ( G ) C_o(G) , A p ( G ) A_p(G) , and M p ( G ) M_p(G) are given. vi) Semigroup compactifications of groups are studied, both abelian and non-abelian: the weak* closure of G ^ \widehat G in L ∞ ( μ ) L^\infty (\mu ) , for abelian G G ; and when ρ \rho is a continuous homomorphism of the locally compact group Γ \Gamma into the unitary elements of a von Neumann algebra M \mathcal {M} , the weak* closure of ρ ( Γ ) \rho (\Gamma ) is studied.

A note on faithful traces on a von Neumann algebra

In this short note we give some techniques for constructing, starting from asufficient family ℱ of semifinite or finite traces on a von Neumann algebraM, a new trace which is faithful.

Linear-topological classification of separableL p-spaces associated with von Neumann algebras of type I

We classify, up to a linear-topological isomorphism, all separableLp-spaces, 1≤p<∞, associated with von Neumann algebras of type I. In particular, anyLp-space associated with an infinite-dimensional atomic von Neumann algebra is isomorphic tolp, or toCp, or to\(Sp = (\sum {_{n = 1}^\infty C_p^n )_{l_p } } \). Further, anyLp-space,p∈[1,∞),p∈2 associated with an infinite-dimensional von Neumann algebraM of type I is isomorphic to one of the following nine Banach spaces: lp, Lp, SP, Cp, Sp ⊕ Lp, Lp(Sp), Cp ⊕ Lp, Lp(Cp), Cp ⊕ Lp(Sp). In the casep=1 all the spaces in this list are pairwise non-isomorphic.

Relative Yoneda Cohomology for Operator Spaces

We provide two alternate presentations of the completely bounded Hochschild cohomology. One as a relative Yoneda cohomology, i.e., as equivalence classes ofn-resolutions which are relatively split, and the second as a derived functor. The first presentation makes clear the importance of certain relative notions of injectivity, projectivity and amenability which we introduce and study. We prove that every von Neumann algebra is relatively injective as a bimodule over itself and consequently,Hncb(M,M)=0 for any von Neumann algebraM. A result obtained earlier by Christensen and Sinclair. We prove that the relatively amenableC*-algebras are precisely the nuclearC*-algebras, and hence exactly those which are amenable as Banach algebras. In a similar vein we prove that the only relatively projectiveC*-algebras are finite dimensional. This result implies that the onlyC*-algebras that are projective as Banach algebras are finite dimensional, a result first obtained by Selivanov and Helemskii. In a slightly different direction we prove thatB(H) viewed as a bimodule over the disk algebra with the left action given by multiplication by a coisometry and the right action given by multiplication by an isometry is an injective module. This result is in some sense a generalization of the Sz.-Nagy-Foias commutant lifting theorem or of the hypoprojectivity introduced by Douglas and the author.

Real reduction theory

On a Borel bar space (E,B,-), the following concepts are introduced in a suitable way: measurable fields of real Hilbert spaces, real measurable fields of vectors, operators and Von Neumann (VN) algebras: ξ (⋅),a (⋅),M (⋅). Then a satisfactory real reduction theory is obtained: a real VN algebraM can be represented as a direct integral\(M = \int_{\left( {E, - } \right)}^ \oplus {M(t)dv(t)} \) where each VN algebraM(t) in this field will be simpler.

Pelczynski’s property (V*) for symmetric operator spaces

We show that if a rearrangement invariant Banach function space E E on the positive semi-axis contains no subspace isomorphic to c 0 c_0 then the corresponding space E ( M ) E(\mathcal {M}) of τ \tau -measurable operators, affiliated with an arbitrary semifinite von-Neumann algebra M \mathcal {M} equipped with a distinguished faithful, normal and semifinite trace τ \tau , has Pełczyński’s property (V*).

Minimal upper bounds of commuting operators

Let ( x i ) (x_{i}) be a finite collection of commuting self-adjoint elements of a von Neumann algebra M \mathcal {M} . Then within the (abelian) C*-algebra they generate, these elements have a least upper bound x x . We show that within M \mathcal {M} , x x is a minimal upper bound in the sense that if y y is any self-adjoint element such that x i ≤ y ≤ x x_{i} \leq y \leq x for all i i , then y = x y = x . The corresponding assertion for infinite collections ( x i ) (x_{i}) is shown to be false in general, although it does hold in any finite von Neumann algebra. We use this sort of result to show that if N ⊂ M \mathcal {N} \subset \mathcal {M} are von Neumann algebras, Φ : M → N \Phi : \mathcal {M} \to \mathcal {N} is a faithful conditional expectation, and x ∈ M x \in \mathcal {M} is positive, then Φ ( x n ) 1 / n \Phi (x^{n})^{1/n} converges in the strong operator topology to the “spectral order majorant” of x x in N \mathcal {N} .

A variational expression for the relative entropy

We prove that for the relative entropy of faithful normal states ϕ and ω on the von Neumann algebraM the formula $$S(\varphi ,\omega ) = \sup \{ \omega (h) - \log \varphi ^h (I):h = h^* \in M\}$$ holds.

Sufficient subalgebras and the relative entropy of states of a von Neumann algebra

A subalgebraM 0 of a von Neumann algebraM is called weakly sufficient with respect to a pair (φ,ω) of states if the relative entropy of φ and ω coincides with the relative entropy of their restrictions toM 0. The main result says thatM 0 is weakly sufficient for (φ,ω) if and only ifM 0 contains the Radon-Nikodym cocycle [Dφ,Dω] t . Other conditions are formulated in terms of generalized conditional expectations and the relative Hamiltonian.

An inequality for Hilbert-Schmidt norm

For the absolute value |C|=(C*C)1/2 and the Hilbert-Schmidt norm ∥C∥HS=(trC*C)1/2 of an operatorC, the following inequality is proved for any bounded linear operatorsA andB on a Hilbert space $$|| |A|---|B| ||_{HS} \leqq 2^{1/2} ||A - B||_{HS} $$ . The corresponding inequality for two normal state ϕ and ψ of a von Neumann algebraM is also proved in the following form: $$d(\varphi ,\psi ) \leqq ||\xi (\varphi ) - \xi (\psi )|| \leqq 2^{1/2} d(\varphi ,\psi )$$ . Here ξ(χ) denotes the unique vector representative of a state χ in a natural positive coneP ♯ forM, andd(ϕ, ψ) denotes the Bures distance defined as the infimum (which is also the minimum) of the distance of vector representatives of ϕ and ψ. In particular, $$||\xi (\varphi _1 ) - \xi (\varphi _2 )|| \leqq 2^{1/2} ||\xi _1 - \xi _2 ||$$ for any vector representatives ξ j of ϕ j ,j=1, 2.

Generators of semigroups of completely positive maps

Any generator of a norm continuous semigroup of completely positive normal maps on a von Neumann algebraM can be decomposed into a sum of a completely positive map and a map of the formm→x*m+mx.

Perturbations of flows on Banach spaces and operator algebras

For automorphism groups of operator algebras we show how properties of the difference ‖αt − α't‖ are reflected in relations between the generators δα, δ′α. Indeed for a von Neumann algebraM with separable predual we show that if ‖αt − α't‖ ≦ 0.28 for smallt, then δα = γ0(δ′α+δ′)°γ-1 where γ is an inner automorphism ofM and δ is a bounded derivation ofM. If the difference ‖αt − α't‖=O(t) ast →; 0, then δα = δ′α + δ and if ‖αt − α't‖ ≦ 0.28 for allt then δα=. We prove analogous results for unitary groups on a Hilbert space andC0,C0* groups on a Banach space.