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Register a new userInclusions of ternary rings of operators and conditional expectations
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It is shown that if T is a ternary ring of operators (TRO), X is a nondegenerate sub-TRO of T and there exists a contractive idempotent surjective map P:T-->X, then P has a unique, explicitly described extension to a conditional expectation between the associated linking algebras. A version of the result for W*-TROs is also presented and some applications mentioned.
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Actions of locally compact groups and quantum groups on W*-ternary rings of operators are discussed and related crossed products introduced. The results generalise those for von Neumann algebraic actions with proofs based mostly on passing to the linking von Neumann algebra. They are motivated by the study of fixed point spaces for convolution operators generated by contractive, non-necessarily positive measures, both in the classical and in the quantum context.
Anzai skew-products are shown to be uniquely ergodic with respect to the fixed-point subalgebra if and only if there is a unique conditional expectation onto such a subalgebra which is invariant under the dynamics. For the particular case of skew-products, this solves a question raised by B. Abadie and K. Dykema in the wider context of $C^*$-dynamical systems.
A matrix convex set is a set of the form $mathcal{S} = cup_{ngeq 1}mathcal{S}_n$ (where each $mathcal{S}_n$ is a set of $d$-tuples of $n times n$ matrices) that is invariant under UCP maps from $M_n$ to $M_k$ and under formation of direct sums. We study the geometry of matrix convex sets and their relationship to completely positive maps and dilation theory. Key ingredients in our approach are polar duality in the sense of Effros and Winkler, matrix ranges in the sense of Arveson, and concrete constructions of scaled commuting normal dilation for tuples of self-adjoint operators, in the sense of Helton, Klep, McCullough and Schweighofer. Given two matrix convex sets $mathcal{S} = cup_{n geq 1} mathcal{S}_n,$ and $mathcal{T} = cup_{n geq 1} mathcal{T}_n$, we find geometric conditions on $mathcal{S}$ or on $mathcal{T}$, such that $mathcal{S}_1 subseteq mathcal{T}_1$ implies that $mathcal{S} subseteq Cmathcal{S}$ for some constant $C$. For instance, under various symmetry conditions on $mathcal{S}$, we can show that $C$ above can be chosen to equal $d$, the number of variables, and in some cases this is sharp. We also find an essentially unique self-dual matrix convex set $mathcal{D}$, the self-dual matrix ball, for which corresponding inclusion and dilation results hold with constant $C=sqrt{d}$. Our results have immediate implications to spectrahedral inclusion problems studied recently by Helton, Klep, McCullough and Schweighofer. Our constants do not depend on the ranks of the pencils determining the free spectrahedra in question, but rather on the number of variables $d$. There are also implications to the problem of existence of (unital) completely positive maps with prescribed values on a set of operators.
We make a careful study of one-parameter isometry groups on Banach spaces, and their associated analytic generators, as first studied by Cioranescu and Zsido. We pay particular attention to various, subtly different, constructions which have appeared in the literature, and check that all give the same notion of generator. We give an exposition of the smearing technique, checking that ideas of Masuda, Nakagami and Woronowicz hold also in the weak$^*$-setting. We are primarily interested in the case of one-parameter automorphism groups of operator algebras, and we present many applications of the machinery, making the argument that taking a structured, abstract approach can pay dividends. A motivating example is the scaling group of a locally compact quantum group $mathbb G$ and the fact that the inclusion $C_0(mathbb G) rightarrow L^infty(mathbb G)$ intertwines the relevant scaling groups. Under this general setup, of an inclusion of a $C^*$-algebra into a von Neumann algebra intertwining automorphism groups, we show that the graphs of the analytic generators, despite being only non-self-adjoint operator algebras, satisfy a Kaplansky Density style result. The dual picture is the inclusion $L^1(mathbb G)rightarrow M(mathbb G)$, and we prove an automatic normality result under this general setup. The Kaplansky Density result proves more elusive, as does a general study of quotient spaces, but we make progress under additional hypotheses.
The notion of index for inclusions of von Neumann algebras goes back to a seminal work of Jones on subfactors of type ${I!I}_1$. In the absence of a trace, one can still define the index of a conditional expectation associated to a subfactor and look for expectations that minimize the index. This value is called the minimal index of the subfactor. We report on our analysis, contained in [GL19], of the minimal index for inclusions of arbitrary von Neumann algebras (not necessarily finite, nor factorial) with finite-dimensional centers. Our results generalize some aspects of the Jones index for multi-matrix inclusions (finite direct sums of matrix algebras), e.g., the minimal index always equals the squared norm of a matrix, that we call emph{matrix dimension}, as it is the case for multi-matrices with respect to the Bratteli inclusion matrix. We also mention how the theory of minimal index can be formulated in the purely algebraic context of rigid 2-$C^*$-categories.
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