In this note we include two remarks about bounded ($\underline{not}$ necessarily contractive) linear projections on a von Neumann-algebra. We show that if $M$ is a von Neumann-subalgebra of $B(H)$ which is complemented in B(H) and isomorphic to $M \otimes M$ then $M$ is injective (or equivalently $M$ is contractively complemented). We do not know how to get rid of the second assumption on $M$. In the second part,we show that any complemented reflexive subspace of a $C^*$- algebra is necessarily linearly isomorphic to a Hilbert space.
Let M be a von Neumann algebra of type II_1 which is also a complemented subspace of B(H). We establish an algebraic criterion, which ensures that M is an injective von Neumann algebra. As a corollary we show that if M is a complemented factor of type II_1 on a Hilbert space H, then M is injective if its fundamental group is non-trivial.
If there exists a completely bounded projection of B(H) onto a von Neumann algebra M on H, then M is injective. If there exists a bounded projection and M is properly infinite, the same conclusion holds.
Techniques introduced by G. Pisier in his proof that finite von Neumann factors with property gamma have length at most 5 are modified to prove that the length is 3. It is proved that if such a factor is a complemented subspace of some larger C*-algebra then there exists a projection of norm one from the larger onto the smaller algebra. A new proof of the fact that the second continuous Hochschild cohomology group of such an algebra with coefficients in the algebra vanishes, is also included.
In 1965, Ron Douglas proved that if $X$ is a closed subspace of an $L^1$-space and $X$ is isometric to another $L^1$-space, then $X$ is the range of a contractive projection on the containing $L^1$-space. In 1977 Arazy-Friedman showed that if a subspace $X$ of $C_1$ is isometric to another $C_1$-space (possibly finite dimensional), then there is a contractive projection of $C_1$ onto $X$. In 1993 Kirchberg proved that if a subspace $X$ of the predual of a von Neumann algebra $M$ is isometric to the predual of another von Neumann algebra, then there is a contractive projection of the predual of…
This paper is concerned with derivations in algebras of (unbounded) operators affiliated with a von Neumann algebra $\mathcal{M}$. Let $\mathcal{% A}$ be one of the algebras of measurable operators, locally measurable operators or, $τ$-measurable operators. We present a complete description of von Neumann algebras $\mathcal{M}$ of type $I$ in terms of their central projections such that every derivation in $\mathcal{A}$ is inner. It is also shown that every derivation in the algebra $LS(\mathcal{M})$ of all locally measurable operators with respect to a properly infinite von Neumann algebra $\…
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 define a notion of nonassociative $\mathrm{L}^p$-space associated to a $\mathrm{JBW}^*$-algebra (Jordan von Neumann algebra) equipped with a normal faithful state $φ$. In the particular case of $\mathrm{JW}^*$-algebras underlying von Neumann algebras, we connect these spaces to a complex interpolation theorem of Ricard and Xu on noncommutative $\mathrm{L}^p$-spaces. We also make the link with the nonassociative $\mathrm{L}^p$-spaces of Iochum associated to $\mathrm{JBW}$-algebras and the investigation of contractively complemented subspaces of noncommutative $\mathrm{L}^p$-spaces. More prec…
Recently C. Houdayer and Y. Isono have proved among other things that every biexact group $Γ$ has the property that for any non-singular strongly ergodic action $Γ\curvearrowright (X,μ)$ on a standard measure space the group measure space von Neumann algebra $Γ\ltimes L^\infty(X)$ is full. In this note, we prove the same property for a wider class of groups, notably including $\mathrm{SL}(3,{\mathbb Z})$. We also prove that for any connected simple Lie group $G$ with finite center, any lattice $Γ\le G$, and any closed non-amenable subgroup $H\le G$, the non-singular action $Γ\curvearrowright G…
In his study of amenable unitary representations, M. E. B. Bekka asked if there is an analogue for such representations of the remarkable fixed-point property for amenable groups. In this paper, we prove such a fixed-point theorem in the more general context of a $G$-amenable von Neumann algebra $M$, where $G$ is a locally compact group acting on $M$. The Følner conditions of Connes and Bekka are extended to the case where $M$ is semifinite and admits a faithful, semifinite, normal trace which is invariant under the action of $G$.
Banach non-isomorphism of type III and semifinite noncommutative $L_p$-spaces
Institute for Advanced Study in Mathematics of HIT, Harbin Institute of Technology（哈尔滨工业大学数学学院）；School of Mathematics and Statistics, University of NSW（新南威尔士大学数学与统计学院）
Around 1967, Arveson invented a striking noncommutative generalization of classical $H^\infty$, known as {\em subdiagonal algebras}, which include a wide array of examples of interest to operator theorists. Their theory extends that of the generalized $H^p$ spaces for function algebras from the 1960s, in an extremely remarkable, complete, and literal fashion, but for reasons that are `von Neumann algebraic'. Most of the present paper consists of a survey of our work on Arveson's algebras, and the attendant $H^p$ theory, explaining some of the main ideas in their proofs, and including some impr…
We identify a large class of hyperbolic groups whose von Neumann algebras are not strongly 1-bounded: Sela's hyperbolic towers over $F_2$ subgroups. We also show that any intermediate subalgebra of the diagonal embedding of $L(F_2)$ into its ultrapower doesn't have Property (T).
We initiate a study of maximal subgroups and maximal von Neumann subalgebras which have the Haagerup property. We determine maximal Haagerup subgroups inside $\mathbb{Z}^2 \rtimes SL_2(\mathbb{Z})$ and obtain several explicit instances where maximal Haagerup subgroups yield maximal Haagerup subalgebras. Our techniques are on one hand based on group-theoretic considerations, and on the other on certain results on intermediate von Neumann algebras, in particular these allowing us to deduce that all the intermediate algebras for certain inclusions arise from groups or from group actions. Some rem…
Normal conditional expectations E: M --> N in M of finite index on von Neumann algebras M with discrete center are investigated to find an estimate for the minimal number of generators of M as a Hilbert N-module. Analyzing the case of M being finite type I with discrete center we obtain that these von Neumann algebras M are always finitely generated Hilbert N-modules with a minimal generator set consisting of at most [K(E)]^2 generators, where [.] denotes the integer part of a real number and K(E) = {K: K.E-id_M >= 0}. This result contrasts remarkable examples by P. Jolissaint and S. Popa show…
In a 1991 paper, R. Mercer asserted that a Cartan bimodule isomorphism between Cartan bimodule algebras A_1 and A_2 extends uniquely to a normal *-isomorphism of the von Neumann algebras generated by A_1 and A_2 [13, Corollary 4.3]. Mercer's argument relied upon the Spectral Theorem for Bimodules of Muhly, Saito and Solel [15, Theorem 2.5]. Unfortunately, the arguments in the literature supporting [15, Theorem 2.5] contain gaps, and hence Mercer's proof is incomplete. In this paper, we use the outline in [16, Remark 2.17] to give a proof of Mercer's Theorem under the additional hypothesis that…
The von Neumann algebraic quantum group $\mathrm{SU}_q(1,1)\rtimes \mathbb{Z}_2$ and the DSSYK model
Institute of Physics, University of Amsterdam（阿姆斯特丹大学物理研究所）；School of Mathematics, Trinity College Dublin（都柏林三一学院数学学院）；Korteweg-de Vries Institute for Mathematics, University of Amsterdam（阿姆斯特丹大学科特韦格-德弗里斯数学研究所）
Given locally compact quantum groups $\G_1$ and $\G_2$, we show that if the convolution algebras $L^1(\G_1)$ and $L^1(\G_2)$ are isometrically isomorphic as algebras, then $\G_1$ is isomorphic either to $\G_2$ or the commutant $\G_2'$. Furthermore, given an isometric algebra isomorphism $θ:L^1(\G_2) \rightarrow L^1(\G_1)$, the adjoint is a *-isomorphism between $L^\infty(\G_1)$ and either $L^\infty(\G_2)$ or its commutant, composed with a twist given by a member of the intrinsic group of $L^\infty(\G_2)$. This extends known results for Kac algebras (although our proofs are somewhat different) …
