We obtain a Mikhlin multiplier theory for the nonabelian free groups. Let F∞ be a free group on infinite many generators {g1,g2,⋯}. Given d≥1 and a bounded symbol m on Zd satisfying the classical ...Mikhlin condition, the linear map Mm:CF∞→CF∞ defined by λ(g)↦m(k1,⋯,kd)λ(g) for g=gi1k1⋯ginkn∈F∞ in reduced form (with kl=0 in m(k1,⋯,kd) for l>n), extends to a completely bounded map on Lp(Fˆ∞) for all 1<p<∞, where Fˆ∞ is the group von Neumann algebra of F∞. In the process, we establish a platform to transfer Lp-completely bounded maps on tensor products of von Neumann algebras to Lp-completely bounded maps on the corresponding amalgamated free products. A similar result holds for any free product of discrete groups.
We introduce a free probabilistic quantity called free Stein irregularity, which is defined in terms of free Stein discrepancies. It turns out that this quantity is related via a simple formula to ...the Murray-von Neumann dimension of the closure of the domain of the adjoint of the non-commutative Jacobian associated to Voiculescu's free difference quotients. We call this dimension the free Stein dimension, and show that it is a ∗-algebra invariant. We relate these quantities to the free Fisher information, the non-microstates free entropy, and the non-microstates free entropy dimension. In the one-variable case, we show that the free Stein dimension agrees with the free entropy dimension, and in the multivariable case compute it in a number of examples.
We study the approximation numbers of weighted composition operators f↦w⋅(f∘φ) on the Hardy space H2 on the unit disc. For general classes of such operators, upper and lower bounds on their ...approximation numbers are derived. For the special class of weighted lens map composition operators with specific weights, we show how much the weight w can improve the decay rate of the approximation numbers, and give sharp upper and lower bounds. These examples are motivated from applications to the analysis of relative commutants of special inclusions of von Neumann algebras appearing in quantum field theory (Borchers triples).
We extend Gour et al.'s characterization of quantum majorization via conditional min-entropy to the context of semi-finite von Neumann algebras. Our method relies on a connection between conditional ...min-entropy and the operator space projective tensor norm for injective von Neumann algebras. We then use this approach to generalize the tracial Hahn-Banach theorem of Helton, Klep and McCullough to vector-valued noncommutative L1-spaces.
We characterize orthonormal bases, Riesz bases and frames which arise from the action of a countable discrete group Γ on a single element ψ of a given Hilbert space H. As Γ might not be abelian, this ...is done in terms of a bracket map taking values in the L1-space associated to the group von Neumann algebra of Γ. Our result generalizes recent work for LCA groups in 26. In many cases, the bracket map can be computed in terms of a noncommutative form of the Zak transform.
We prove a Beurling-type theorem for H∞-invariant spaces of Lα(M,τ), where α is a unitarily invariant, locally ∥⋅∥1-dominating, mutually continuous norm with respect to τ, where M is a von Neumann ...algebra with a faithful, normal, semifinite tracial weight τ, and H∞ is an extension of Arveson's noncommutative Hardy space. We use our main result to characterize the H∞-invariant subspaces of a noncommutative Banach function space I(τ) with the norm ∥⋅∥E on M, the crossed product of a semifinite von Neumann algebra by an action β, and B(H) for a separable Hilbert space H.
Let 1≤p<∞$1\le p<\infty$ and let T:Lp(M)→Lp(N)$T: L^p({\mathcal {M}})\rightarrow L^p(\mbox{${\mathcal {N}}$})$ be a bounded map between noncommutative Lp$L^p$‐spaces. If T is bijective and ...separating, we prove the existence of decompositions M=M1⊕∞M2${\mathcal {M}}={\mathcal {M}}_1\mathop {\oplus }\limits ^\infty {\mathcal {M}}_2$, N=N1⊕∞N2$\mbox{${\mathcal {N}}$}=\mbox{${\mathcal {N}}$}_1 \mathop {\oplus }\limits ^\infty \mbox{${\mathcal {N}}$}_2$ and maps
T1:Lp(M1)→Lp(N1),T2:Lp(M2)→Lp(N2),\begin{equation*}\hskip7pc T_1: L^p\big ({\mathcal {M}}_1\big )\rightarrow L^p\big (\mbox{${\mathcal {N}}$}_1\big ), \quad T_2: L^p\big ({\mathcal {M}}_2\big )\rightarrow L^p\big (\mbox{${\mathcal {N}}$}_2\big ),\hskip-7pc \end{equation*}such that T=T1+T2$T=T_1+T_2$, T1 has a direct Yeadon type factorisation and T2 has an anti‐direct Yeadon type factorisation. We further show that T−1$T^{-1}$ is separating in this case. Next we prove that for any 1≤p<∞$1\le p<\infty$ (resp. any 1≤p≠2<∞$1\le p\not=2<\infty$), a surjective separating map T:Lp(M)→Lp(N)$T: L^p({\mathcal {M}})\rightarrow L^p(\mbox{${\mathcal {N}}$})$ is S1‐bounded (resp. completely bounded) if and only if there exists a decomposition M=M1⊕∞M2${\mathcal {M}}={\mathcal {M}}_1 \mathop {\oplus }\limits ^\infty {\mathcal {M}}_2$ such that T|Lp(M1)$T|_{L^p({{\mathcal {M}}_1})}$ has a direct Yeadon type factorisation and M2${\mathcal {M}}_2$ is subhomogeneous.
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