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An Ideal Convergence

Konrad Aguilar
Samantha Brooker
Frédéric Latrémolière
Alejandra López

Communicated by Notices Associate Editor Stephan Ramon Garcia

Article cover

1. Introduction

Quantum physics describes the world using noncommuting variables modeled by operators on a complex Hilbert space. The study of algebras of bounded operators on Hilbert spaces began with the work of J. von Neumann on the foundations of quantum mechanics, and soon found other areas of applications. In particular, I. M. Gelfand introduced C*-algebras, which are, up to an appropriate notion of *-isomorphism, certain algebras of bounded operators on a Hilbert space. These algebras include as important examples: the algebra of -matrices over (denoted ), and the algebra of -valued continuous functions over a compact Hausdorff space (denoted ) with the supremum norm and pointwise conjugation for the adjoint operation. Moreover, C*-algebras naturally appear in representation theory, dynamics, and geometry, in addition to physics Dav96.

We will give a more detailed description of C*-algebras later, but for now, we note that any unital commutative C*-algebra is (up to *-isomorphism) for some compact Hausdorff space . This fact is part of Gelfand’s celebrated contravariant duality between the category of unital commutative C*-algebras and the category of compact Hausdorff spaces, which sparked a new area of research in functional and geometric analysis focused on developing the analogues of various topological and geometric concepts for noncommutative C*-algebras Dav96. A. Connes’ noncommutative geometry is a prime example of this development, and, in particular, Connes opened the path for the study of quantum analogues of compact metric spaces Con94. The theory of compact quantum metric spaces began in earnest with the work of M. A. Rieffel Rie98, with a focus on constructing a topology on the class of compact quantum metric spaces to provide a formal framework for approximations of classical and quantum spaces found in the mathematical physics literature Rie04.

This formal framework begins with tools from classical metric geometry. Indeed, given a metric space , the Hausdorff distance provides a way to approximate subsets by other subsets, and is defined for any two closed and bounded subsets and of by

where for any subset of . For instance, the circle can be approximated by the groups of th roots of unity for the Hausdorff distance induced by the Euclidean distance on the plane (see Figure 1).

Figure 1.

Convergence of to .

Furthermore, this example of Hausdorff convergence provides motivation for an important example coming from physics that we will address later (see the discussion at equation 1).

Rieffel used the Hausdorff distance to develop a generalization of the Gromov-Hausdorff distance Gro81 to noncommutative analogues of compact metric spaces, in order to formalize statements from the high-energy physics literature about approximations of (possibly quantum) spaces by matrix algebras Rie04, which spawned a new area of study called noncommutative metric geometry. This article aims to provide a glimpse of this area by introducing some of the main structures involved and examples stemming from physics and C*-algebra theory. In fact, the opening graphic comes from an intriguing connection between noncommutative metric geometry and C*-algebra theory with a focus on ideals of C*-algebras (which we can represent with graphs for certain classes of C*-algebras) and a notion of convergence of ideals. Moreover, the graphs of the ideals of the particular example we focus on (the Boca-Mundici algebra Boc08) are given by the continued fraction expansions of irrational numbers. Continued fractions have been used several other times in the study of C*-algebras to build some fascinating structures and prove some spectacular results (see the construction of the Effros-Shen algebras in Dav96, Section VI.3 and the proof of the “main step” in the classification of a fundamental class of C*-algebras in EE93), which are related to the structures we present in this paper. To end this article, we list some future work motivated, in part, by the work we highlighted in this article.

We also note that, although we mainly present examples of noncommutative spaces, throughout the paper we visit the commutative case whenever a new concept is discussed. This serves two purposes. First, it allows us to provide a connection to classical structures such as the Cantor set (see Figure 2 and Theorems 3.6 and 3.7). Second, we use the commutative case as a more familiar setting to describe the intuition for some of the approaches in the noncommutative setting (see Theorems 3.12 and 3.13 for connections to convergence of ideals and the Hausdorff distance).

2. Compact Quantum Metric Spaces and the Gromov-Hausdorff Propinquity

To begin our journey of defining the core objects of noncommutative metric geometry, we must begin with C*-algebras. We started this article with bounded operators on a Hilbert space. Therefore, once we define these terms, we move onto a useful and fundamental abstract definition of C*-algebras due to Gelfand, M. Naimark, and I. Segal. From this, we are then in a position to discuss further structures related to C*-algebras (e.g., states) that allow us to tie the theory of metric spaces to the study of C*-algebras by way of commutative C*-algebras. This then gives us the foundation we need to define the notion of a noncommutative/quantum metric space and the tools to generalize the Gromov-Hausdorff distance, while also describing how this provides a framework to answer questions about approximations of spaces by matrix algebras. All the facts we present about C*-algebras can be found in Dav96.

Thus, we begin this journey with: a Hilbert space is a vector space (which we always take over the field of complex numbers) which is complete for the norm induced by an inner product . An inner product on a vector space is a function such that:

(1)

,

(2)

,

(3)

,

(4)

.

An inner product on a vector space defines a norm by setting for all . An example of a Hilbert space is given by for its usual inner product. An infinite-dimensional example of a Hilbert space is given by for the inner product defined by .

A bounded operator on a Hilbert space is a continuous linear map from to itself. The term bounded refers to the property that a linear map on any normed vector space is continuous if and only if it is bounded on the set of vectors of norm . In particular, for any continuous linear map , we define the operator norm of as .

As the composition of bounded operators on a Hilbert space is again a bounded operator on , the space of all bounded operators on a Hilbert space is an associative algebra. There is one additional operation on , called the adjoint, and denoted by a superscript , which is uniquely defined for any by the relation

The resulting structure on is an example of a C*-algebra, a type of operator algebra introduced by Gelfand. Indeed, we can abstract the properties of as follows, which we present now along with subobjects and morphisms.

Definition 2.1.

A Banach algebra is a normed algebra with norm that is complete with respect to the norm-induced metric, and such that for all

Definition 2.2.

A C*-algebra is a Banach algebra equipped with a map called the adjoint satisfying for all , :

(1)

, , ,

(2)

(C*-identity)

We say that is unital if it has a multiplicative identity, . A subalgebra is a C*-subalgebra if it is norm-closed and self-adjoint ().

If is a C*-algebra, then is a *-morphism if it is an adjoint-preserving multiplicative linear map, and is a *-isomorphism if it is a bijective *-morphism. We say is *-isomorphic to if there exists a *-isomorphism between and and denote this by .

Gelfand proved that is a C*-algebra if and only if there exists a Hilbert space , and an injective *-morphism from into . Notably, *-morphisms are continuous; moreover injective *-morphisms are isometric. So every C*-algebra is *-isomorphic to a norm-closed, adjoint-closed algebra of bounded operators on a Hilbert space.

Given a compact Hausdorff space , the algebra of -valued continuous functions over , with the usual pointwise addition and multiplication operation, the supremum norm , and the adjoint given by pointwise conjugation, is a unital commutative C*-algebra. Remarkably, Gelfand proved that any unital commutative C*-algebra is *-isomorphic to the C*-algebra for some compact Hausdorff space, which is uniquely determined by , up to a homeomorphism.

This correspondence between unital commutative C*-algebras and compact Hausdorff spaces can actually be understood as a duality of categories, between the category of unital commutative C*-algebras and their *-morphisms, and the category of compact Hausdorff spaces and their continuous functions. For our purpose, it is interesting to give a description of this correspondence at the level of the objects.

An element of a C*-algebra is self-adjoint when . Since, for any element , we have , we see that the self-adjoint elements of span . The space of self-adjoint elements of a C*-algebra is an ordered vector space, where we define a positive element as any element of the form for some . This order generalizes the order on Hermitian matrices. With this order, we can define a class of positive linear functionals over a C*-algebra. Specifically, a state of a unital C*-algebra is a linear functional from to such that if and is positive, then , and maps the unit of to . A state of a C*-algebra is always continuous, and thus an element of the dual (Banach space) of . The normalized trace on () is a standard example of a state. The state space of is the set of all states over . It is a convex, weak*-compact subset of the dual of .

Among all the states of a unital C*-algebra are the characters of , which are unit-preserving *-morphisms from to . However, for a general C*-algebra , there may be no characters at all! When is a unital commutative C*-algebra, however, the subset of all the characters of is a weak*-closed subset of —hence, a compact Hausdorff space—such that is indeed *-isomorphic to . With this identification, by the Radon-Riesz Theorem, the state space is the space of integrals against Radon probability measures over . Thus, characters of a unital C*-algebra can be seen as (a generalization of) points.

When working with a general (i.e., not necessarily commutative) unital C*-algebra , there may be no character, hence no point, but there still are states. In fact, the state space of a unital C*-algebra spans the dual of the C*-algebra. Thus, while we lose the notion of a point, we keep the notion of a “probability.” This structure is a simple example of a deeper idea. Indeed, the study of C*-algebras, in the noncommutative geometry of Connes and the noncommutative metric geometry of Rieffel, provides noncommutative analogues of classical analytical structures to answer various profound questions.

To introduce the subject of noncommutative metric geometry, we begin with a common special case of the general problem in physics to find approximations of infinite-dimensional problems by finite-dimensional ones. The clock and shift matrices are a standard example from quantum dynamics for finite-dimensional systems. These matrices are famously given for all by

and

where . The smallest C*-algebra containing and is denoted by , which is the operator norm closure of finite linear combinations of finite products of and and their conjugate transposes. We note that , and that both and are unitary matrices. In fact, the map which sends any function to the diagonal matrix of entries is a *-isomorphism from to such that , where . Thus is also what is known as a C*-crossed product of by the action by translation by itself (C*-crossed products are C*-algebraic analogues to semidirect products of groups). When working with finite-dimensional models of quantum mechanics, and are seen as finite-dimensional analogues of the position and momentum observables. In a different direction, they are seen as a matrix model for a certain compactification in string theory. Yet, in a different direction, t’Hooft suggests such matrix models as starting points for a new analysis of quantum mechanics. See Lat16b for references.

Since , it is a common heuristics that is a noncommutative, finite-dimensional approximation for the C*-algebra of -valued continuous functions over the -torus , seen as the universal C*-algebra generated by two commuting unitaries. Is there indeed a way to make this intuition formal?

An approach to this question is motivated by the observation that it is possible to make rigorous sense of the notion that the groups of th roots of unity in converge to a circle as goes to for the Hausdorff distance induced by the Euclidean distance on the plane (see Figure 1). In contrast, the sequence , whose th entry is homeomorphic to for each , converges to for the Hausdorff distance on the real line. Thus, metrics enable us to formally discuss approximations of infinite compact sets by finite sets.

With this motivation in mind, Rieffel proposed to use a suggestion of Connes to define an analogue of a metric for unital C*-algebras that may not be commutative. To this end, we begin with the following question: if is a compact metric space, then how do we encode the metric on the C*-algebra ? Of course, the metric induces the Lipschitz seminorm on by setting

which is only when is constant. Moreover, is dense. Furthermore, an observation of L. Kantorovich is that if, for any two Radon probability measures and on , we define by

then is a metric which induces the weak* topology on the state space of Radon probability measures over . Moreover, identifying points in with their associated Dirac point measure gives an isometry from into . Thus, the Lipschitz seminorm encodes the metric at the level of the C*-algebra , and some of its properties can be made sense of for noncommutative C*-algebras. Rieffel used this as his starting point. Many small variations in the definition of compact quantum metric spaces arose after Rieffel first introduced them in Rie98. However, we present the following list of properties which are mostly standard now and are shared by the usual Lipschitz seminorm .

Definition 2.3 (Rie98Rie04Lat16aLat15Lat16b).

An ordered pair of a unital C*-algebra and a seminorm defined on a dense subspace of is a -Leibniz compact quantum metric space for some and when:

(1)

,

(2)

the Monge-Kantorovich metric , defined between any two states by the number given by

induces the weak* topology on the state space of ,

(3)

for all

(4)

is closed in .

In 3, the usual Leibniz rule is given by and , but as we shall see later (Theorem 3.5), examples of compact quantum metric spaces with different values occur, and without allowing for these different values, we will not be able to produce many convergence results.

The noncommutative C*-algebra can be endowed with a quantum metric structure. To this end, let us fix any continuous length function on . For instance, we can choose to be defined by setting for all as

The function is thus a metric on the -torus , which induces a Lipschitz seminorm on . On the other hand, Rieffel defined in Rie98, for any :

where is the so-called dual action of on uniquely defined by and . Then is a compact quantum metric space.

To discuss the convergence of to , we generalize the intrinsic version of the Hausdorff distance introduced by M. Gromov Gro81 (see the following definition), since, of course, noncommutative algebras are not quotients of commutative ones.

Definition 2.4.

The Gromov-Hausdorff distance between any two compact metric spaces and is

We note that if , where is a compact metric space, then

Now, the Gromov-Hausdorff distance is a distance on the class of all compact metric spaces, up to isometry—meaning that if and only if and are isometric. This distance has also found applications in Riemannian geometry Gro81.

Rieffel used Gromov’s idea to define a first noncommutative analogue of the Gromov-Hausdorff distance between compact quantum metric spaces. Rieffel’s quantum Gromov-Hausdorff distance is a complete pseudometric, which is zero exactly between quantum compact metric spaces whose state spaces are isometric, with an isometry implemented by the dual of a linear map. However, this does not imply that the underlying C*-algebras are *-isomorphic. Variations of Rieffel’s construction were offered (by D. Kerr and H. Li—see KL09 and the references therein—and the third author, to name a few) to address this coincidence property issue. A variation due to the third author solved this problem by accounting for the Leibniz property, used as a means to connect the algebraic structure and the quantum metric structure, and a property which gained importance as research in noncommutative metric geometry progressed.

Rieffel indeed observed in Rie04, Proposition 3.1, as an essential component of his construction, that if is a unit-preserving surjective *-morphism, where and are compact quantum metric spaces, then the dual map induced by between the state spaces of and is an isometry from to if and only if for all in the domain of . We are led to the following definition.

Definition 2.5 (Lat15Lat16b).

A quantum isometry is a surjective unital *-morphism such that

A full quantum isometry is a *-isomorphism (bijective *-morphism) such that .

Following Gromov’s and Rieffel’s prescription with the Leibniz property in mind and the above definition of a quantum isometry, the third author introduced in Lat16aLat15 a new analogue of the Gromov-Hausdorff distance called the propinquity, described as follows. Fix . A tunnel between two -Leibniz compact quantum metric spaces and is given by a -Leibniz compact quantum metric space and two quantum isometries and . The extent of the tunnel is then given by

The third author then introduced the Gromov- Hausdorff propinquity as follows.

Definition 2.6 (Lat15Lat16b).

Fix . The Gromov-Hausdorff propinquity between two -Leibniz compact quantum metric spaces and is defined by

The third author proved that is a complete metric up to full quantum isometry on the class of -Leibniz compact quantum metric spaces. Thus,