Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0707.3229

Leibniz seminorms for "Matrix algebras converge to the sphere"

Marc A. Rieffel
View original source ↗Read source document ↗Export metadata ↓

Abstract

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.

Explore related subjects

math.OAhep-thmath.MG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Marc A. Rieffel. 2010-01-08. Leibniz seminorms for "Matrix algebras converge to the sphere". https://arxiv.org/abs/0707.3229

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Upper and lower limits of scales and the interchange of quantifiers

We define lower and upper limits for a family of structured subobjects of a fixed ambient object, as the direct limit of the meets over tails and the inverse limit of the joins over tails, and construct the canonical comparison morphism between them. The family need not be a diagram: its members are related only through the ambient object. We distinguish two senses in which the comparison may be invertible, before and after forgetting the structure, and prove that they coincide exactly when the forgetful functor reflects isomorphisms; they therefore agree for sets, for vector spaces and for Banach spaces, and separate for locally convex spaces. For countable scales of Banach spaces the separation cannot occur when the upper limit is ultrabornological, so under those hypotheses the obstruction is localised in the topology of that limit. Intermediate conditions correspond to factorisations of the forgetful functor, of which bornological convergence is one. In the cases that matter the comparison is an interchange of quantifiers: membership in the lower limit means that one value of the second index serves every value of the first, and membership in the upper limit that every value of the first is served by some value of the second. We exhibit this for spaces of test functions, for weighted Sobolev spaces, for Laurent series and for the adeles, and discuss three further conditions that suggest themselves.

math.OA↗
arXiv2026

Totally Bounded Elements in W*-probability Spaces

We introduce the notion of a totally ($K$-) bounded element of a $W^*$-probability space $(M, φ)$ and, borrowing ideas of Kadison, give an intrinsic characterization of the $^*$-subalgebra $M_{\operatorname{tb}}$ of totally bounded elements. Namely, we show that $M_{\operatorname{tb}}$ is the unique strongly dense $^*$-subalgebra $M_0$ of totally bounded elements of $M$ for which the collection of totally $1$-bounded elements of $M_0$ is complete with respect to the $\|\cdot\|_φ^\#$-norm and for which $M_0$ is closed under all operators $h_a(\log(Δ))$ for $a \in \mathbb{N}$, where $Δ$ is the modular operator and $h_a(t):=1/\cosh(t-a)$ (see Theorem 4.3). We also prove that totally $K$-bounded elements of an Ocneanu ultraproduct admit representatives with the same total bound using a careful effective estimate of the distance of a given totally bounded element to the totally 1-bounded elements. An alternative proof in the appendix uses an isometric $H^\infty$-lifting theorem for the Ocneanu multiplier quotient, derived from a metric $H^\infty$-lifting theorem for $C^*$-quotients and a $C^*$-algebraic Schur parametrization. We combine these results with Rieffel and Van Daele's bounded operator approach to modular theory to arrive at a new language and axiomatization of $W^*$-probability spaces as metric structures. Previous work of Dabrowski had axiomatized $W^*$-probability spaces using a smeared version of multiplication, but the subalgebra $M_{\operatorname{tb}}$ allows us to give an axiomatization in terms of the original algebra operations. Finally, we prove the (non-)axiomatizability of several classes of $W^*$-probability spaces.

math.OA↗
arXiv2026

Strict comparison and selflessness

It is shown that an infinite dimensional, simple, unital, monotracial C*-algebra with strict comparison with respect to its trace is selfless.

math.OA↗