On bounded coordinates in bimodules

Almost uniform convergence on the predual of a von Neumann algebra and an ergodic theorem

Let A A be a von Neumann algebra with a faithful normal state ϕ \phi . It is shown that if a sequence ( x n ) ({x_n}) in A A is orthogonal relative to ϕ \phi and satisfies the condition \[ ∑ k ϕ ( | x k | 2 ) ( log ⁡ k k ) 2 > ∞ , \sum \limits _{k} {\phi (|{x_k}{|^2}){{\left ( {\frac {{\log k}}{k}} \right )}^2} > \infty ,} \] then n 1 ∑ k = 1 n x k → 0 {}_n^1\sum \nolimits _{k = 1}^n {{x_k} \to 0} almost uniformly in A A . Some other results related to this theorem are also discussed.

Let M be a von Neumann algebra, let $\varphi $ be a normal faithful state on M and let $L^p(M,\varphi )$ be the associated Haagerup noncommutative $L^p$ -spaces, for $1\leq p\leq \infty $ . Let $D\in L^1(M,\varphi )$ be the density of $\varphi $ . Given a positive map $T\colon M\to M$ such that $\varphi \circ T\leq C_1\varphi $ for some $C_1\geq 0$ , we study the boundedness of the $L^p$ -extension $T_{p,\theta }\colon D^{\frac {1-\theta }{p}} MD^{\frac {\theta }{p}}\to L^p(M,\varphi )$ which maps $D^{\frac {1-\theta }{p}} x D^{\frac {\theta }{p}}$ to $D^{\frac {1-\theta }{p}} T(x) D^{\frac {\theta }{p}}$ for all $x\in M$ . Haagerup–Junge–Xu showed that $T_{p,{\frac {1}{2}}}$ is always bounded and left open the question whether $T_{p,\theta }$ is bounded for $\theta \not ={\frac {1}{2}}$ . We show that for any $1\leq p<2$ and any $\theta \in [0, 2^{-1}(1-\sqrt {p-1})]\cup [2^{-1}(1+\sqrt {p-1}), 1]$ , there exists a completely positive T such that $T_{p,\theta }$ is unbounded. We also show that if T is $2$ -positive, then $T_{p,\theta }$ is bounded provided that $p\geq 2$ or $1\leq p<2$ and $\theta \in [1-p/2,p/2]$ .

Let (M, ϕ) = (M1, ϕ1) * (M2, ϕ2) be the free product of any σ-finite von Neumann algebras endowed with any faithful normal states. We show that whenever Q ⊂ M is a von Neumann subalgebra with separable predual such that both Q and Q ∩ M1 are the ranges of faithful normal conditional expectations and such that both the intersection Q ∩ M1 and the central sequence algebra Q′ ∩ Mω are diffuse (e.g. Q is amenable), then Q must sit inside M1. This result generalizes the previous results of the first named author in [Ho14] and moreover completely settles the questions of maximal amenability and maximal property Gamma of the inclusion M1 ⊂ M in arbitrary free product von Neumann algebras.

In 2015, Yanni Chen, Don Hadwin and Junhao Shen proved a noncommutative version of Beurling’s theorem for a continuous unitarily invariant norm α on a tracial von Neumann algebra (M, τ) such that α is one dominating with respect to τ . The role of H∞ is played by a maximal subdiagonal algebra A. In the talk, we first will show that if α is a continuous normalized unitarily invariant norm on (M, τ), then there exists a faithful normal tracial state ρ on M and a constant c > 0 such that α is a c times one norm-dominating norm on (M,ρ). Moreover, ρ(x) = τ(xg), where x ∈M , g is positive in L1(Z, τ), where Z is the center of M . Here c and ρ are not unique. However, if there is a c and ρ so that the Fuglede-Kadison determinant of g is positive, then Beurling-Chen-Hadwin-Shen theorem holds for L(α)(M, τ). The key ingredients in the proof of our result include a factorization theorem and a density theorem for for L(α)(M,ρ). Talk time: 2016-07-19 03:00 PM— 2016-07-19 03:20 PM Talk location: Cupples I Room 207

Orthonormal expansions of invariant densities for expanding maps

We introduce a notion of universal preparability for a state of a system, more precisely: for a normal state on a von Neumann algebra. It describes a situation where from an arbitrary initial state it is possible to prepare a target state with arbitrary precision by a repeated interaction with a sequence of copies of another system. For {mathcal{B}(mathcal{H})} we give criteria sufficient to ensure that all normal states are universally preparable which can be verified for a class of non-commutative birth and death processes realized, in particular, by the interaction of a micromaser with a stream of atoms. As a tool, the theory of tight sequences of states and of stationary states is further developed and we show that in the presence of stationary faithful normal states universal preparability of all normal states is equivalent to asymptotic completeness, a notion studied earlier in connection with the scattering theory of non-commutative Markov processes.

We show that any finite dimensional von Neumann algebra admits an orthonormal unitary basis with respect to its standard trace. We also show that a finite dimensional von Neumann subalgebra of M n ( C ) M_n(\mathbb {C}) admits an orthonormal unitary basis under normalized matrix trace if and only if the normalized matrix trace and standard trace of the von Neumann subalgebra coincide. As an application, we verify a recent conjecture of Bakshi-Gupta [Glasg. Math. J. 64 (2022), pp. 586–602], showing that any finite-index regular inclusion N ⊆ M N\subseteq M of I I 1 II_1 -factors admits an orthonormal unitary Pimsner-Popa basis.

We investigate martingale inequalities\nin noncommutative $L^p$-spaces associated with a von Neumann\nalgebra equipped with a faithful normal state. We prove the\nnoncommutative analogue of the classical Burkholder inequality\non the conditioned (or little) square function and extend the\nnoncommutative Burkholder--Gundy inequalities from Comm. Math. Phys.\n189 (1997) 667--698 to this\nnontracial setting. We include several related results.

We give a necessary and sufficient condition for a sesquilinear form to be integrable with respect to a faithful normal state on a von Neumann algebra.

We introduce two notions of the pressure in operator algebras, one is the pressure Pα(π, T) for an automorphism α of a unital exact C*-algebra Open image in new window at a self-adjoint element T in Open image in new window with respect to a faithful unital *-representation π, the other is the pressure Pτ,α(T) for an automorphism α of a hyperfinite von Neumann algebra \( M \) at a self-adjoint element T in \( M \) with respect to a faithful normal α-invariant state τ. We give some properties of the pressure, show that it is a conjugate invariant, and also prove that the pressure of the implementing inner automorphism of a crossed product Open image in new window ×α ℤ at a self-adjoint operator T in Open image in new window equals that of α at T.

We consider quantum information tasks in an operator algebraic setting, where we consider normal states on von Neumann algebras. In particular, we consider subfactors N ⊂ M \mathfrak {N} \subset \mathfrak {M} , that is, unital inclusions of von Neumann algebras with trivial center. One can ask the following question: given a normal state ω \omega on M \mathfrak {M} , how much can one learn by only doing measurements from N \mathfrak {N} ? We argue how the Jones index [ M : N ] [\mathfrak {M}:\mathfrak {N}] can be used to give a quantitative answer to this, showing how the rich theory of subfactors can be used in a quantum information context. As an example we discuss how the Jones index can be used in the context of wiretap channels. Subfactors also occur naturally in physics. Here we discuss two examples: rational conformal field theories and Kitaev’s toric code on the plane, a prototypical example of a topologically ordered model. There we can directly relate aspects of the general setting to physical properties such as the quantum dimension of the excitations. In the example of the toric code we also show how we can calculate the index via an approximation with finite dimensional systems. This explicit construction sheds more light on the connection between topological order and the Jones index.

We prove a generalization of N. Ozawa’s Kurosh-type theorem to the setting of free products of semiexact II 1 \text {II}_1 factors with respect to arbitrary (non-tracial) faithful normal states. We are thus able to distinguish certain resulting type III factors. For example, if M = L F n ⊗ L F m M = LF_n \otimes LF_m and { φ i } \{\varphi _i\} is any sequence of faithful normal states on M M , then the l l -various ( M , φ 1 ) ∗ . . . ∗ ( M , φ l ) (M,\varphi _1) * ... * (M,\varphi _l) are all mutually non-isomorphic.

Implementable positive maps on standard forms

Let M be a factor of type III with separable predual and with normal states phi_1,...,phi_k, omega with omega faithful. Let A be a finite dimensional C*-subalgebra of M. Then it is shown that there is a unitary operator u in M such that phi_i Ad u =omega on A for i=1,...,k. This follows from an embedding result of a finite dimensional C*-algebra with a faithful state into M with finitely many given states. We also give similar embedding results of C*-algebras and von Neumann algebras with faithful states into M. Another similar result for a factor of type II_1 instead of type III holds.