Leibniz Seminorms and Best Approximation from C*-subalgebras

We show that if B is a C*-subalgebra of a C*-algebra A such that B contains a bounded approximate identity for A, and if L is the pull-back to A of the quotient norm on A/B, then L is strongly Leibniz. In connection with this situation we study certain aspects of best approximation of elements of a unital C*-algebra by elements of a unital C*-subalgebra.Comment: 24 pages. Intended for the proceedings of the conference "Operator Algebras and Related Topics". v2: added a corollary to the main theorem, plus several minor improvements v3: much simplified proof of a key lemma, corollary to main theorem added v4: Many minor improvements. Section numbers increased by

Leibniz seminorms for "Matrix algebras converge to the sphere"

In an earlier paper of mine relating vector bundles and Gromov-Hausdorff distance for ordinary compact metric spaces, it was crucial that the Lipschitz seminorms from the metrics satisfy a strong Leibniz property. In the present paper, for the now non-commutative situation of matrix algebras converging to the sphere (or to other spaces) for quantum Gromov-Hausdorff distance, we show how to construct suitable seminorms that also satisfy the strong Leibniz property. This is in preparation for making precise certain statements in the literature of high-energy physics concerning "vector bundles" over matrix algebras that "correspond" to monopole bundles over the sphere. We show that a fairly general source of seminorms that satisfy the strong Leibniz property consists of derivations into normed bimodules. For matrix algebras our main technical tools are coherent states and Berezin symbols.Comment: 46 pages. Scattered very minor improvement

The Quantum Gromov-Hausdorff Propinquity

We introduce the quantum Gromov-Hausdorff propinquity, a new distance between quantum compact metric spaces, which extends the Gromov-Hausdorff distance to noncommutative geometry and strengthens Rieffel's quantum Gromov-Hausdorff distance and Rieffel's proximity by making *-isomorphism a necessary condition for distance zero, while being well adapted to Leibniz seminorms. This work offers a natural solution to the long-standing problem of finding a framework for the development of a theory of Leibniz Lip-norms over C*-algebras.Comment: 49 Pages. This is the first half of 1302.4058v2, which has been accepted in Trans. Amer. Math. Soc. The second half is now a different paper entitled "Convergence of Fuzzy Tori and Quantum Tori for the quantum Gromov-Hausdorff Propinquity: an explicit approach

The Dual Gromov-Hausdorff Propinquity

Motivated by the quest for an analogue of the Gromov-Hausdorff distance in noncommutative geometry which is well-behaved with respect to C*-algebraic structures, we propose a complete metric on the class of Leibniz quantum compact metric spaces, named the dual Gromov-Hausdorff propinquity. This metric resolves several important issues raised by recent research in noncommutative metric geometry: it makes *-isomorphism a necessary condition for distance zero, it is well-adapted to Leibniz seminorms, and --- very importantly --- is complete, unlike the quantum propinquity which we introduced earlier. Thus our new metric provides a natural tool for noncommutative metric geometry, designed to allow for the generalizations of techniques from metric geometry to C*-algebra theory.Comment: 42 pages in elsarticle 3p format. This third version has many small typos corrections and small clarifications included. Intended form for publicatio

Leibniz seminorms in probability spaces

In this paper we study the (strong) Leibniz property of centered moments of bounded random variables. We shall answer a question raised by M. Rieffel on the non-commutative standard deviation

Contents

A. Rieffel Abstract. We show that standard deviation σ satisfies the Leibniz inequality σ(fg) ≤ σ(f)‖g ‖ + ‖f‖σ(g) for bounded functions f, g on a probability space, where the norm is the supremum norm. A related inequality that we refer to as “strong ” is also shown to hold. We show that these in fact hold also for noncommutative probability spaces. We extend this to the case of matricial seminorms on a unital C*-algebra, which leads us to treat also the case of a conditional expectation fro