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Register a new userRanges of bimodule projections and reflexivity
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We develop a general framework for reflexivity in dual Banach spaces, motivated by the question of when the weak* closed linear span of two reflexive masa-bimodules is automatically reflexive. We establish an affirmative answer to this question in a number of cases by examining two new classes of masa-bimodules, defined in terms of ranges of masa-bimodule projections. We give a number of corollaries of our results concerning operator and spectral synthesis, and show that the classes of masa-bimodules we study are operator synthetic if and only if they are strong operator Ditkin.
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We develop a symbol calculus for normal bimodule maps over a masa that is the natural analogue of the Schur product theory. Using this calculus we are able to easily give a complete description of the ranges of contractive normal bimodule idempotents that avoids the theory of J*-algebras. We prove that if $P$ is a normal bimodule idempotent and $|P| < 2/sqrt{3}$ then $P$ is a contraction. We finish with some attempts at extending the symbol calculus to non-normal maps.
We study containment and uniqueness problems concerning matrix convex sets. First, to what extent is a matrix convex set determined by its first level? Our results in this direction quantify the disparity between two product operations, namely the product of the smallest matrix convex sets over $K_i subseteq mathbb{C}^d$, and the smallest matrix convex set over the product of $K_i$. Second, if a matrix convex set is given as the matrix range of an operator tuple $T$, when is $T$ determined uniquely? We provide counterexamples to results in the literature, showing that a compact tuple meeting a minimality condition need not be determined uniquely, even if its matrix range is a particularly friendly set. Finally, our results may be used to improve dilation scales, such as the norm bound on the dilation of (non self-adjoint) contractions to commuting normal operators, both concretely and abstractly.
We study w*-semicrossed products over actions of the free semigroup and the free abelian semigroup on (possibly non-selfadjoint) w*-closed algebras. We show that they are reflexive when the dynamics are implemented by uniformly bounded families of invertible row operators. Combining with results of Helmer we derive that w*-semicrossed products over factors (on a separable Hilbert space) are reflexive. Furthermore we show that w*-semicrossed products of automorphic actions on maximal abelian selfadjoint algebras are reflexive. In all cases we prove that the w*-semicrossed products have the bicommutant property if and only if so does the ambient algebra of the dynamics.
In deformation-rigidity theory it is often important to know whether certain bimodules are weakly contained in the coarse bimodule. Consider a bimodule $H$ over the group algebra $mathbb{C}[Gamma]$, with $Gamma$ a discrete group. The starting point of this paper is that if a dense set of the so-called coefficients of $H$ is contained in the Schatten $mathcal{S}_p$ class $p in [2, infty)$ then the $n$-fold tensor power $H^{otimes n}_Gamma$ for $n geq p/2$ is quasi-contained in the coarse bimodule. We apply this to gradient bimodules associated with the carre du champ of a symmetric quantum Markov semi-group. For Coxeter groups we give a number of characterizations of having coefficients in $mathcal{S}_p$ for the gradient bimodule constructed from the word length function. We get equivalence of: (1) the gradient-$mathcal{S}_p$ property introduced by the second named author, (2) smallness at infinity of a natural compactification of the Coxeter group, and for a large class of Coxeter groups: (3) walks in the Coxeter diagram called parity paths. We derive three strong solidity results: two are known, one is new. The first result is a concise proof of a result by T. Sinclair for discrete groups admitting a proper cocycle into a $p$-integrable representation. The second result is strong solidity for hyperbolic right-angled Coxeter groups. The final -- and new -- result extends current strong solidity results for right-angled Hecke von Neumann algebras beyond right-angled Coxeter groups that are small at infinity.
We show that any Lipschitz projection-valued function p on a connected closed Riemannian manifold can be approximated uniformly by smooth projection-valued functions q with Lipschitz constant close to that of p. This answers a question of Rieffel.
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