Leibniz seminorms for 'Matrix algebras converge to the sphere'

In a previous paper [29] I showed how to give a precise meaning to statements in the literature of high-energy physics and string theory of the kind "Matrix algebras converge to the sphere". (See [29] for numerous references to the relevant physics literature.) I did this by introducing the concept of "compact quantum metric spaces", in which the metric data is given by a seminorm on the non-commutative "algebra of functions". This seminorm plays the role of the usual Lipschitz seminorm on the algebra of continuous functions on an ordinary compact metric space. However, I was somewhat puzzled by the fact that I needed virtually no algebraic conditions on the seminorm, only an important analytic condition. But when I later began trying to give precise meaning to further statements in the physics literature of the kind "here are the vector bundles over the matrix algebras that correspond to the monopole bundles over the sphere" (see [31] for many references), I found that for ordinary metric spaces a strong form of the Leibniz inequality for the seminorm played a crucial role [31]. (See, for example, the proof of proposition 2.3 of [31].) However, on returning to the non-commutative case of matrix algebras converging to the sphere (or to other spaces), for some time I did not see how to construct useful seminorms that brought the matrix algebras and sphere close together while also having the strong Leibniz property. The main purpose of this paper is to show how to construct such seminorms. As in the earlier paper [29], the setting is that of coadjoint orbits of compact semisimple Lie groups, of which the 2-sphere is the simplest example. The main technical tools continue to be coherent states and Berezin symbols.

In the first four sections of this paper we show that a fairly general setting for obtaining seminorms that possess the strong Leibniz property that we need consists of derivations into normed bimodules, and we examine various aspects of this topic. The strong Leibniz property for a seminorm L on a normed unital algebra A consists of the usual Leibniz inequality together with the inequality

whenever a is invertible in A. I have not seen this latter inequality discussed in the literature. In Section 4 we put together the various conditions that we have found to be important, and there-by give a tentative definition for a “compact C * -metric space”.

In Section 5 we examine the use of seminorms with the strong Leibniz property in connection with quantum Gromov-Hausdorff distance. (I expect that many of the ideas and techniques developed in this paper will apply to many other classes of examples beyond “Matrix algebras converge to the sphere”.) In Section 6 we extend to the case of strongly Leibniz seminorms the construction technique introduced in [28] that we called “bridges”. Sections 7 and 8 contain those pieces of our development that can be carried out for certain homogeneous spaces of any compact group (including finite ones). Section 9 gives the statement of our main theorem for coadjoint orbits, while Sections 10 through 13 contain the detailed technical development needed to prove our main theorem. Finally, in Section 14 we relate our results to other variants of quantum Gromov-Hausdorff distance that have been developed by David Kerr, Hanfeng Li, and Wei Wu [13,14,17,18,38,39,40].

We can describe our basic setup and our main theorem somewhat more specifically as follows, where definitions for various terms are given in later sections. Let G be a compact semisimple Lie group, let (U, H) be an irreducible unitary representation of G, and let P be the rank-one projection along a highest weight vector for (U, H). Let α be the action of G on L(H) by conjugation by U, and let H be the α-stability group of P . Let A = C(G/H). Let ω be the highest weight for U, and for each n ∈ Z >0 let (U n , H n ) be the irreducible representation of G of highest weight nω. Let α also denote the action of G on B n = L(H n ) by conjugation by U n .

Choose on G a continuous length-function ℓ. Then ℓ and the translation action of G on A, as well as the actions α of G on each B n , determine seminorms L A on A and L B n on B n that make (A, L A ) and each (B n , L B n ) into compact C * -metric spaces.

Main Theorem (sketchy statement of Theorem 9.1). For any ε > 0 there exists an N such that for any n ≥ N we can explicitly construct a strongly Leibniz seminorm, L n , on A ⊕ B n making A ⊕ B n into a compact C * -metric space, such that the quotients of L n on A and B n are L A and L B n , and for which the quantum Gromov-Hausdorff distance between A and B n is no greater than ε.

I plan to apply the results of this paper in a future paper to discuss vector bundles over non-commutative spaces (e.g., monopole bundles), along the lines used for ordinary spaces in [31].

I developed part of the material presented here during a ten-week visit at the Isaac Newton Institute in Cambridge, England, in the Fall of 2006. I

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