The generator rank of C⁎-algebras☆
Introduction
The generator problem for -algebras asks to determine which -algebras are singly generated. More generally, for a given -algebra A one wants to compute the minimal number of generators. For a more detailed discussion of the generator problem, we refer to [19].
Given a -algebra A, let us denote by the minimal number of self-adjoint generators for A, and set if A is not finitely generated; see [12]. The restriction to self-adjoint elements is mainly for convenience. It only leads to a minor variation of the original generator problem, since two self-adjoint elements a and b generate the same sub--algebra as the element . In particular, A is singly generated if and only if . For a compact, metric space X, it is easy to see that if and only if X can be embedded into .
It is also easy to see that for every finite-dimensional -algebra F. However, this does not readily show that every AF-algebras is singly generated, since the minimal number of (self-adjoint) generators may increase when passing to inductive limits: Example A Let be the topologists sine-curve given by: Then X can be embedded into but not into , and therefore . However, X is an inverse limit of spaces that are each homeomorphic to the interval. Therefore , with , while on the other hand for all n. The spaces X and are shown below. By considering the spaces and , one obtains a sequence of singly generated -algebras whose inductive limit is not singly generated.
To get a better behaved theory, instead of counting the minimal number of generators, we count the minimal number of ‘stable’ generators. More precisely, given a unital, separable -algebra A, let be the subset of self-adjoint n-tuples that generate A; see Notation 3.2. Then A has generator rank at most n, denoted by , if is dense in ; see Theorem 3.4. The index shift follows the usual convention in (noncommutative) dimension theory: For instance, a topological space has covering dimension at most n if every finite cover can be refined by a cover with index ; similarly, a unital -algebra has real rank at most n if every tuple of self-adjoint elements can be approximated by a selfadjoint tuple that generates the -algebra as a left ideal (see Paragraph 3.8). Using the index shift also leads to simpler formulas that involve the generator rank. For example, by Proposition 3.10, we always have .
To handle nonunital -algebras, we first develop a precursor to the generator rank, denoted , and then define , where is the minimal unitization of A. To also handle nonseparable -algebras, we consider a relative version of generation: given an element c, we ask if a given self-adjoint tuple a can be approximated by tuples b such that c is (approximately) contained in the sub--algebra generated by b; see Definition 2.1.
A -algebra has generator rank zero if and only if it is commutative with totally disconnected spectrum; see Proposition 5.7. A unital, separable -algebra A has generator rank at most one if and only if a generic element of A is a generator; see Remark 3.7. We compute the generator rank of commutative -algebras: Theorem B 5.6 Let X be a locally compact, Hausdorff space. Then where locdim denotes the local dimension, which is defined as the supremum of the covering dimension of all compact subsets (see Paragraph 5.5).
For unital, separable -algebras, ‘’ records that is nonempty, while ‘’ records that is dense. Thus, the generator rank is often much larger than the minimal number of self-adjoint generators. The payoff, however, is that the generator rank has nicer permanence properties: Theorem C 6.2, 6.3 Let A be a -algebra and let be a closed, two-sided ideal. Then Further, if is an inductive limit, then
By showing that finite-dimensional -algebras have generator rank at most one, we deduce: Theorem D 7.3 Let A be a separable AF-algebra. Then , and so a generic element of A is a generator. In particular, A is singly generated.
In subsequent work, [17], we compute the generator rank of subhomogeneous -algebras. We obtain in particular that every -stable approximately subhomogeneous (ASH) -algebra has generator rank one. In further work, [18], we show that -stable -algebras of real rank zero have generator rank one. It follows that every classifiable, simple -algebra has generator rank one.
Acknowledgments
The author thanks James Gabe and Mikael Rørdam for valuable comments and feedback. I also want to thank the anonymous referee whose suggestions helped to greatly improve the paper.
This paper grew out of joint work with Karen Strung, Aaron Tikuisis, Joav Orovitz and Stuart White that started at the workshop ‘Set theory and -algebras’ at the AIM in Palo Alto, January 2012. In particular, the key Proposition 2.8 was obtained in that joint work. The author benefited from many fruitful discussions with Strung, Tikuisis, Orovitz and White, and he wants to thank them for their support.
Notation
We set . Given a -algebra A, we use and to denote the set of self-adjoint and positive elements in A, respectively. We denote by and the minimal and forced unitization of A, respectively. By an ideal in a -algebra we always mean a closed, two-sided ideal.
Given , and , we write if . Given and , we write if there exists with . We use bold letters to denote tuples of elements, for example . Given , we write if for . We denote by the sub--algebra of A generated by the elements of a. We write for , the space of n-tuples of self-adjoint elements.
By an nc-polynomial we mean a polynomial in noncommuting variables.
Section snippets
A precursor of the generator rank
In this section, we introduce the invariant for -algebras and we show that it behaves well when passing to ideals, quotients, inductive limits and extensions. In Section 3, we define the generator rank of a -algebra A as and in Section 6 we will see that gr satisfies the same permanence properties as .
Definition 2.1 Let A be a -algebra. We define as the smallest integer such that for every , and , there exist such that
The generator rank
We define the generator rank of a -algebra A as the invariant developed in Section 2 applied to the minimal unitization ; see Definition 3.1. It follows that and are closely related, and in many cases we know that they agree; see Proposition 3.12 and Paragraph 3.15.
The generator rank of a separable, unital -algebra A is the smallest integer n such that the self-adjoint -tuples in A that generate A as a -algebra are dense in ; see Theorem 3.4. This is similar
Reduction to the separable case
In this short section, we recall some notions from model theory of -algebras that will be used in Section 5 to reduce some proofs to the case of separable -algebras. For details we refer to [10] and [9].
4.1 Given a -algebra A, we let denote the collection of separable sub--algebras of A. A family is σ-complete if for every countable, directed subfamily we have . Further, is cofinal if for every there exists such that . We say that a
Generator rank of commutative -algebras
In this section, we compute the generator rank of commutative -algebras; see Theorem 5.6. We first consider commutative -algebras that are unital and separable. We then generalize to the unital, nonseparable case, and finally to the case of arbitrary commutative -algebras.
For topological spaces X and Y we denote by the space of injective, continuous maps .
Let X be a compact, metric space and . We seek to characterize when . Given and , set
Permanence properties
In this section, we show that the generator rank enjoys the same permanence properties as its precursor .
Recall that denotes the forced unitization of a -algebra A.
Lemma 6.1 Let A be a -algebra. Then . Proof If A is nonunital, then and the result follows from the definition. If A is unital, then , and by Proposition 5.12 we have □
Theorem 6.2 Let A be a -algebra, and let be an ideal. Then: Proof It is known
Generator rank of AF-algebras
In this section, we show that every finite-dimensional -algebra has generator rank at most one; see Lemma 7.2. Using that the generator rank behaves well with respect to inductive limits, we deduce the main result: AF-algebras have generator rank at most one; see Theorem 7.3.
Lemma 7.1 Let be -algebras of real rank zero. Then . Proof Since A and B are quotients of , it follows from Theorem 6.2 that . It remains to verify the converse inequality. Since
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The author was partially supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation, and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under the SFB 878 (Groups, Geometry & Actions) and under Germany's Excellence Strategy EXC 2044-390685587 (Mathematics Münster: Dynamics-Geometry-Structure).