Let $E=E(0,\infty )$ be a symmetric function space and $E(\mathcal{M},\tau )$ be the noncommutative symmetric space corresponding to $E(0,\infty )$ associated with a von Neumann algebra with a ...faithful normal semifinite trace. Our main result identifies the class of spaces $E$ for which every derivation $\delta :\mathcal{A}\rightarrow E(\mathcal{M},\tau )$ is necessarily inner for each $C^*$-subalgebra $\mathcal{A}$ in the class of all semifinite von Neumann algebras $\mathcal{M}$ as those with the Levi property.
For a compact quantum group \mathbb{G} of Kac type, we study the existence of a Haar trace-preserving embedding of the von Neumann algebra L^\infty (\mathbb{G}) into an ultrapower of the hyperfinite ...II _1-factor (the Connes embedding property for L^\infty (\mathbb{G})). We establish a connection between the Connes embedding property for L^\infty (\mathbb{G}) and the structure of certain quantum subgroups of \mathbb{G} and use this to prove that the II _1-factors L^\infty (O_N^+) and L^\infty (U_N^+) associated to the free orthogonal and free unitary quantum groups have the Connes embedding property for all N \ge 4. As an application, we deduce that the free entropy dimension of the standard generators of L^\infty (O_N^+) equals 1 for all N \ge 4. We also mention an application of our work to the problem of classifying the quantum subgroups of O_N^+.
We consider converses to the density theorem for square-integrable, irreducible, projective, unitary group representations restricted to lattices using the dimension theory of Hilbert modules over ...twisted group von Neumann algebras. We show that the restriction of such a σ-projective unitary representation π of a unimodular, second-countable group G to a lattice Γ extends to a Hilbert module over the twisted group von Neumann algebra of (Γ,σ). We then compute the center-valued von Neumann dimension of this Hilbert module. For abelian groups with 2-cocycle satisfying Kleppner's condition, we show that the center-valued von Neumann dimension reduces to the scalar value dπvol(G/Γ), where dπ is the formal dimension of π and vol(G/Γ) is the covolume of Γ in G. We apply our results to characterize the existence of multiwindow super frames and Riesz sequences associated to π and Γ. In particular, we characterize when a lattice in the time-frequency plane of a second-countable, locally compact abelian group admits a Gabor frame or Gabor Riesz sequence.
Let M be a type II1 von Neumann factor and let S(M) be the associated Murray-von Neumann algebra of all measurable operators affiliated to M. We extend a result of Kadison and Liu 30 by showing that ...any derivation from S(M) into an M-bimodule B⊊S(M) is trivial. In the special case, when M is the hyperfinite type II1-factor R, we introduce the algebra AD(R), a noncommutative analogue of the algebra of all almost everywhere approximately differentiable functions on 0,1 and show that it is a proper subalgebra of S(R). This algebra is strictly larger than the corresponding ring of continuous geometry introduced by von Neumann. Further, we establish that the classical approximate derivative on (classes of) Lebesgue measurable functions on 0,1 admits an extension to a derivation from AD(R) into S(R), which fails to be spatial. Finally, we show that for a Cartan masa A in a hyperfinite II1-factor R there exists a derivation δ from A into S(A) which does not admit an extension up to a derivation from R to S(R).
A bijective map between specific structures in operator algebras is called a piecewise isomorphism if it preserves products of commuting elements in both directions. We shall show that any ...bicontinuous piecewise isomorphism between positive cones of invertible elements in von Neumann algebras or between unitary groups of von Neumann algebras can be described in terms of the following parameters: (i) Jordan *-isomorphism between given algebras (ii) one fixed central element in domain (or range) algebra (iii) a hermitian linear map from one algebra into the center of the other one. Especially, in case of factors any piecewise isomorphism between unitary groups is a Jordan *-isomorphism or Jordan *-isomorphism composed with inversion. This extends hitherto known results from von Neumann factors to general von Neumann algebras and brings new Jordan invariants of operator structures.
Let
be a factor von Neumann algebra with
For any
define
and
for all integers
In this article, we prove that a map
satisfies
for all
if and only if L is an additive *-derivation.
Let
M
be a finite von Neumann algebra with no central summands of type I
1
. We show that each nonlinear 2-local Lie n-derivation
δ
:
M
→
M
with
n
≥
3
is of the form d + h, where
d
:
M
→
M
is a ...linear derivation and h is a homogeneous central-valued mapping which annihilates each
(
n
−
1
)
-th commutator of
M
.
We prove that the normalizer of any diffuse amenable subalgebra of a free group factor L(𝔽 r ) generates an amenable von Neumann subalgebra. Moreover, any II₁ factor of the form $Q\overline{\otimes ...}L({\Bbb F}_{r})$ , with Q an arbitrary subfactor of a tensor product of free group factors, has no Cartan subalgebras. We also prove that if a free ergodic measure-preserving action of a free group 𝔽 r , 2 ≤ r ≤ ∞, on a probability space (X, μ) is profinite then the group measure space factor L ∞ (X) ⋊ 𝔽 r has unique Cartan subalgebra, up to unitary conjugacy.
We consider the space of odd spinors on the circle, and a decomposition into spinors supported on either the top or on the bottom half of the circle. If an operator preserves this decomposition, and ...acts on the bottom half in the same way as a second operator acts on the top half, then the fusion of both operators is a third operator acting on the top half like the first, and on the bottom half like the second. Fusion restricts to the Banach-Lie group of restricted orthogonal operators, which supports a central extension of implementers on a Fock space. In this article, we construct a lift of fusion to this central extension. Our construction uses Tomita-Takesaki theory for the Clifford-von Neumann algebras of the decomposed space of spinors. Our motivation is to obtain an operator-algebraic model for the basic central extension of the loop group of the spin group, on which the fusion of implementers induces a fusion product in the sense considered in the context of transgression and string geometry. In upcoming work we will use this model to construct a fusion product on a spinor bundle on the loop space of a string manifold, completing a construction proposed by Stolz and Teichner.
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