The generator rank of C-algebras

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Abstract

We show that every AF-algebra is generated by a single operator. This was previously unclear, since the invariant that assigns to a C-algebra its minimal number of generators lacks natural permanence properties. In particular, it may increase when passing to ideals or inductive limits.

To obtain a better behaved theory, we not only ask if a C-algebra is generated by n elements, but also if generating n-tuples are dense. This defines the generator rank, which we show has many natural permanence properties: it does not increase when passing to ideals, quotients or inductive limits.

Introduction

The generator problem for C-algebras asks to determine which C-algebras are singly generated. More generally, for a given C-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 C-algebra A, let us denote by gen(A) the minimal number of self-adjoint generators for A, and set gen(A)= 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-C-algebra as the element a+ib. In particular, A is singly generated if and only if gen(A)2. For a compact, metric space X, it is easy to see that gen(C(X))k if and only if X can be embedded into Rk.

It is also easy to see that gen(F)2 for every finite-dimensional C-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 XR2 be the topologists sine-curve given by:X={0}×[1,1]{(t,sin(t1)):t(0,1/2π]}. Then X can be embedded into R2 but not into R1, and therefore gen(C(X))=2. However, X is an inverse limit of spaces Xn that are each homeomorphic to the interval. Therefore C(X)limnC(Xn), with gen(C(X))=2, while on the other hand gen(C(Xn))=gen(C([0,1]))=1 for all n. The spaces X and X1,X2,X3 are shown below.

By considering the spaces X×[0,1] and Xn×[0,1], one obtains a sequence of singly generated C-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 C-algebra A, let Genn(A)saAsan 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 gr(A)n, if Genn+1(A)sa is dense in Asan+1; 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 n+1; similarly, a unital C-algebra has real rank at most n if every tuple of n+1 self-adjoint elements can be approximated by a selfadjoint tuple that generates the C-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 rr(A)gr(A).

To handle nonunital C-algebras, we first develop a precursor to the generator rank, denoted gr0, and then define gr(A):=gr0(A˜), where A˜ is the minimal unitization of A. To also handle nonseparable C-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-C-algebra generated by b; see Definition 2.1.

A C-algebra has generator rank zero if and only if it is commutative with totally disconnected spectrum; see Proposition 5.7. A unital, separable C-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 C-algebras:

Theorem B 5.6

Let X be a locally compact, Hausdorff space. Thengr(C0(X))=locdim(X×X), 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 C-algebras, ‘gen(A)n+1’ records that Genn+1(A)sa is nonempty, while ‘gr(A)n’ records that Genn+1(A)sa 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 C-algebra and let IA be a closed, two-sided ideal. Thenmax{gr(I),gr(A/I)}gr(A)gr(I)+gr(A/I)+1. Further, if A=limλAλ is an inductive limit, thengr(A)liminfλgr(Aλ).

By showing that finite-dimensional C-algebras have generator rank at most one, we deduce:

Theorem D 7.3

Let A be a separable AF-algebra. Then gr(A)1, 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 C-algebras. We obtain in particular that every Z-stable approximately subhomogeneous (ASH) C-algebra has generator rank one. In further work, [18], we show that Z-stable C-algebras of real rank zero have generator rank one. It follows that every classifiable, simple C-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 C-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 N:={0,1,2}. Given a C-algebra A, we use Asa and A+ to denote the set of self-adjoint and positive elements in A, respectively. We denote by A˜ and A+ the minimal and forced unitization of A, respectively. By an ideal in a C-algebra we always mean a closed, two-sided ideal.

Given a,bA, and ε>0, we write a=εb if ab<ε. Given aA and GA, we write aεG if there exists bG with a=εb. We use bold letters to denote tuples of elements, for example a=(a1,,an)An. Given a,bAn, we write a=εb if aj=εbj for j=1,,n. We denote by C(a) the sub-C-algebra of A generated by the elements of a. We write Asan for (Asa)n, 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 gr0 for C-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 C-algebra A as gr0(A˜) and in Section 6 we will see that gr satisfies the same permanence properties as gr0.

Definition 2.1

Let A be a C-algebra. We define gr0(A) as the smallest integer n0 such that for every a0,,anAsa, ε>0 and cA, there exist b0,,bnAsa such thatbjaj<ε for j=0,,n, and cεC

The generator rank

We define the generator rank of a C-algebra A as the invariant gr0 developed in Section 2 applied to the minimal unitization A˜; see Definition 3.1. It follows that gr0(A) and gr(A) 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 C-algebra A is the smallest integer n such that the self-adjoint (n+1)-tuples in A that generate A as a C-algebra are dense in Asan+1; see Theorem 3.4. This is similar

Reduction to the separable case

In this short section, we recall some notions from model theory of C-algebras that will be used in Section 5 to reduce some proofs to the case of separable C-algebras. For details we refer to [10] and [9].

4.1

Given a C-algebra A, we let Subsep(A) denote the collection of separable sub-C-algebras of A. A family SSubsep(A) is σ-complete if for every countable, directed subfamily TS we have {B:BT}S. Further, S is cofinal if for every B0Subsep(A) there exists BS such that B0B.

We say that a

Generator rank of commutative C-algebras

In this section, we compute the generator rank of commutative C-algebras; see Theorem 5.6. We first consider commutative C-algebras that are unital and separable. We then generalize to the unital, nonseparable case, and finally to the case of arbitrary commutative C-algebras.

For topological spaces X and Y we denote by E(X,Y) the space of injective, continuous maps XY.

Let X be a compact, metric space and nN. We seek to characterize when gr(C(X))n. Given aC(X)san+1 and xX, set a(x):=(a0(x)

Permanence properties

In this section, we show that the generator rank enjoys the same permanence properties as its precursor gr0.

Recall that A+ denotes the forced unitization of a C-algebra A.

Lemma 6.1

Let A be a C-algebra. Then gr(A+)=gr(A).

Proof

If A is nonunital, then A+=A˜ and the result follows from the definition. If A is unital, then A+=AC, and by Proposition 5.12 we havegr(A+)=gr(AC)=max{gr(A),gr(C)}=gr(A). 

Theorem 6.2

Let A be a C-algebra, and let IA be an ideal. Then:max{gr(I),gr(A/I)}gr(A)gr(I)+gr(A/I)+1.

Proof

It is known

Generator rank of AF-algebras

In this section, we show that every finite-dimensional C-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 A,B be C-algebras of real rank zero. Then gr(AB)=max{gr(A),gr(B)}.

Proof

Since A and B are quotients of AB, it follows from Theorem 6.2 that gr(AB)max{gr(A),gr(B)}. 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).

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