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Register a new userSmooth approximation of Lipschitz projections
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Hanfeng Li
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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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It is shown that for any approximately central (AC) projection $e$ in the Flip orbifold $A_theta^Phi$ (of the irrational rotation C*-algebra $A_theta$), and any modular automorphism $alpha$ (arising from SL$(2,mathbb Z)$), the AC projection $alpha(e)$ is centrally Murray-von Neumann equivalent to one of the projections $e, sigma(e), kappa(e), kappa^2(e),$ $sigmakappa(e), sigmakappa^2(e)$ in the $S_3$-orbit of $e,$ where $sigma, kappa$ are the Fourier and Cubic transforms of $A_theta$. (The equivalence being implemented by an approximately central partial isometry in $A_theta^Phi$.) For smooth automorphisms $alpha,beta$ of the Flip orbifold $A_theta^Phi$, it is also shown that if $alpha_*=beta_*$ on $K_0(A_theta^Phi),$ then $alpha(e)$ and $beta(e)$ are centrally equivalent for each AC projection $e$.
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.
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.
Let $X$ be a $d$-dimensional random vector and $X_theta$ its projection onto the span of a set of orthonormal vectors ${theta_1,...,theta_k}$. Conditions on the distribution of $X$ are given such that if $theta$ is chosen according to Haar measure on the Stiefel manifold, the bounded-Lipschitz distance from $X_theta$ to a Gaussian distribution is concentrated at its expectation; furthermore, an explicit bound is given for the expected distance, in terms of $d$, $k$, and the distribution of $X$, allowing consideration not just of fixed $k$ but of $k$ growing with $d$. The results are applied in the setting of projection pursuit, showing that most $k$-dimensional projections of $n$ data points in $R^d$ are close to Gaussian, when $n$ and $d$ are large and $k=csqrt{log(d)}$ for a small constant $c$.
In this short note, we prove that for a $C^*$-algebra $aa$ generated by $n$ elements, $M_{k}(tilde{aa})$ is generated by $k$ mutually unitarily equivalent and almost mutually orthogonal projections for any $kge de(n)=minbig{kinmathbb N,|,(k-1)(k-2)ge 2nbig}$. Then combining this result with recent works of Nagisa, Thiel and Winter on the generators of $C^*$--algebras, we show that for a $C^*$-algebra $aa$ generated by finite number of elements, there is $dge 3$ such that $M_d(tilde A)$ is generated by three mutually unitarily equivalent and almost mutually orthogonal projections. Furthermore, for certain separable purely infinite simple unital $C^*$--algebras and $AF$--algebras, we give some conditions that make them be generated by three mutually unitarily equivalent and almost mutually orthogonal projections.
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