1 Introduction
The generator rank of a unital, separable $C^*\!$ algebra A is the smallest integer $n\geq 0$ such that the selfadjoint $(n+1)$ tuples that generate A as a $C^*\!$ algebra are dense in $A^{n+1}_{\mathrm {sa}}$ (see Definition 2.1 for the nonunital and nonseparable case). This invariant was introduced in [Reference Thiel and WinterThi21] to study the generator problem, which asks to determine the minimal number of (selfadjoint) generators for a given $C^*\!$ algebra.
One difficulty when studying the generator problem is that the minimal number of generators for a $C^*\!$ algebra can increase when passing to ideals or inductive limits. The main advantage of the generator rank is that it enjoys nice permanence properties: it does not increase when passing to ideals, quotients, or inductive limits (see Section 2).
For example, using these permanence properties, one can easily show that approximately finitedimensional $C^*\!$ algebras (AFalgebras) have generator rank at most one. In particular, every AFalgebra is generated by two selfadjoint elements, which solves the generator problem for this class of algebras (see [Reference Thiel and WinterThi21, Theorem 7.3]).
In this paper, we compute the generator rank of subhomogeneous $C^*\!$ algebras. Recall that a $C^*\!$ algebra is said to be dhomogeneous (dsubhomogeneous) if each of its irreducible representations has dimension (at most) d. The typical example of a dhomogeneous $C^*\!$ algebra is $C_0(X,M_d)$ for a locally compact Hausdorff space X. Furthermore, a $C^*\!$ algebra is subhomogeneous if and only if it is a sub $C^*\!$ algebra of $C_0(X,M_d)$ for some X and some d (see, for example, [Reference BlackadarBla06, Proposition IV.1.4.3]).
Subhomogeneous $C^*\!$ algebras and their inductive limits (called approximately subhomogeneous algebras [ASHalgebras]) play an important role in the structure and classification theory of $C^*\!$ algebras since the algebras covered by the Elliott program are either purely infinite or approximately subhomogeneous. To be precise, let us say that a $C^*\!$ algebra is classifiable if it is unital, separable, simple, nuclear, and $\mathcal {Z}$ stable (that is, it tensorially absorbs the Jiang–Su algebra $\mathcal {Z}$ ) and satisfies the Universal Coefficient Theorem (UCT). By the recent breakthrough in the Elliott classification program [Reference Elliott, Gong, Lin and NiuEGLN15, Reference Gong, Lin and NiuGLN20, Reference van MillTWW17], two classifiable $C^*\!$ algebras are isomorphic if and only if their Elliott invariants (Ktheoretic and tracial data) are isomorphic.
Classifiable $C^*\!$ algebras come in two flavors: stably finite and purely infinite. Every stably finite, classifiable $C^*\!$ algebra is automatically an ASHalgebra. A major application of our results is that every $\mathcal {Z}$ stable ASHalgebra has generator rank one (see Corollary C). In [Reference ThielThi20], we show that every $\mathcal {Z}$ stable $C^*\!$ algebra of real rank zero has generator rank one. This includes all purely infinite, classifiable $C^*\!$ algebras. It follows that every classifiable $C^*\!$ algebra has generator rank one and therefore contains a dense $G_\delta $ subset of generators (see Corollary E).
One important aspect of the generator problem is to determine if every separable, simple $C^*\!$ algebra is generated by a single operator (equivalently, by two selfadjoint elements). While this remains unclear, we can refute the possibility that every separable, simple $C^*\!$ algebra contains a dense set of generators: Villadsen constructed examples of separable, simple, approximately homogeneous $C^*\!$ algebras (AHalgebras) of arbitrarily high real rank (see [Reference VilladsenVil99]). Let A be such an AHalgebra with $\operatorname {\mathrm {rr}}(A)=\infty $ . By [Reference Thiel and WinterThi21, Proposition 3.10] (see Proposition 2.4), the real rank is dominated by the generator rank, whence $\operatorname {\mathrm {gr}}(A)=\infty $ . In particular, for every n, the generating selfadjoint ntuples (if there are any) are not dense in $A^{n}_{\mathrm {sa}}$ .
In [Reference Tikuisis, White and WinterTW14, Theorem 3.8], the author and Winter showed that every unital, separable, $\mathcal {Z}$ stable $C^*\!$ algebra is singly generated. The results of this paper and of [Reference ThielThi20] show that under additional assumptions, a (unital) separable, $\mathcal {Z}$ stable $C^*\!$ algebra even contains a dense set of generators. This raises the natural question if every $\mathcal {Z}$ stable $C^*\!$ algebra has generator rank one (see [Reference ThielThi20, Remarks 5.8(2)]).
Given a locally compact Hausdorff space X, the local dimension $\operatorname {\mathrm {locdim}}(X)$ is defined as the supremum of the covering dimension of all compact subsets, with the convention that $\operatorname {\mathrm {locdim}}(\emptyset )=1$ . For $\sigma $ compact (in particular, second countable), locally compact Hausdorff spaces, the local dimension agrees with the usual covering dimension (in general they differ). In Section 4, we compute the generator rank of arbitrary homogeneous $C^*\!$ algebras.
Theorem A (4.17)
Let A be a dhomogeneous $C^*\!$ algebra. Set $X:=\operatorname {\mathrm {Prim}}(A)$ . If $d=1$ , then $\operatorname {\mathrm {gr}}(A) = \operatorname {\mathrm {locdim}}(X\times X)$ . If $d\geq 2$ , then
In particular, $\operatorname {\mathrm {gr}}(C(X,M_d)) = \left \lceil \tfrac {\dim (X)+1}{2d2}\right \rceil $ if X is a compact Hausdorff space and $d\geq 2$ . To prove Theorem A, we first show a Stone–Weierstraßtype result that characterizes when a tuple generates $C(X,M_d)$ : the tuple has to generate $M_d$ pointwise, and it has to suitably separate the points in X (see Proposition 4.1). This indicates the general strategy to determine when generating ntuples in $C(X,M_d)$ are dense: first, we need to characterize when every tuple can be approximated by tuples that generate $M_d$ pointwise, and second, we need to characterize when a pointwise generating tuple can be approximated by tuples that separate the points. To address the first point, we compute the codimension of the manifold of generating ntuples of selfadjoint dmatrices (see Lemma 4.11). For the second point, we use known results characterizing when continuous maps to a manifold can be approximated by embeddings, in conjunction with a suitable version of the homotopy extension lifting property.
In Section 5, we compute the generator rank of dsubhomogeneous $C^*\!$ algebras by induction over d. Given a dsubhomogeneous $C^*\!$ algebra A, we consider the ideal $I\subseteq A$ corresponding to irreducible representations of dimension d. Then $A/I$ is $(d1)$ subhomogeneous. Using Theorem A and the assumption of the induction, we know the generator rank of I and $A/I$ . The crucial result to compute the generator rank of the extension is the following proposition, which we also expect to have further applications in the future.
Proposition B (5.3)
Let A be a separable $C^*\!$ algebra, and let $(I_k)_{k\in \mathbb {N}}$ be a decreasing sequence of ideals satisfying $\bigcup _k \operatorname {\mathrm {hull}}(I_k)=\operatorname {\mathrm {Prim}}(A)$ . Then,
The main result of this paper is the following theorem.
Theorem C (5.5)
Let A be a subhomogeneous $C^*\!$ algebra. For each $d\geq 1$ , set $X_d:=\operatorname {\mathrm {Prim}}_d(A)$ , the subset of the primitive ideal space of A corresponding to ddimensional irreducible representations. Then,
The main application is the following corollary.
Corollary D (5.10)
Let A be a nonzero, separable, $\mathcal {Z}$ stable ASHalgebra. Then, $\operatorname {\mathrm {gr}}(A)=1$ , and so a generic element of A is a generator.
It was shown in [Reference Tikuisis, White and WinterTW14, Theorem 3.8] that every unital, separable, $\mathcal {Z}$ stable $C^*\!$ algebra is singly generated. We note that Corollary D does not require unitality. In particular, Corollary D implies that certain $C^*\!$ algebras are singly generated that were not considered in [Reference Tikuisis, White and WinterTW14].
Together with the main result of [Reference ThielThi20], we obtain the following consequence.
Corollary E [Reference ThielThi20, Corollary 5.7]
Let A be a unital, separable, simple, nuclear, $\mathcal {Z}$ stable $C^*\!$ algebra satisfying the UCT. Then, A has generator rank one. In particular, a generic element in A is a generator.
Notation We set $\mathbb {N}:=\{0,1,2,\ldots \}$ . Given a $C^*\!$ algebra A, we use $A_{\mathrm {sa}}$ to denote the set of selfadjoint elements in A. We denote by $\widetilde {A}$ the minimal unitization of A. By an ideal in a $C^*\!$ algebra, we mean a closed, twosided ideal. We write $M_d$ for the $C^*\!$ algebra of dbyd matrices $M_d(\mathbb {C})$ .
Given $a,b\in A$ , and $\varepsilon>0$ , we write $a=_\varepsilon b$ if $\ab\<\varepsilon $ . Given $a\in A$ and $G\subseteq A$ , we write $a\in _\varepsilon G$ if there exists $b\in G$ with $a=_\varepsilon b$ . We use bold letters to denote tuples, for example, $\mathbf {a}=(a_1,\ldots ,a_n)\in A^n$ . Given $\mathbf {a},\mathbf {b}\in A^n$ , we write $\mathbf {a}=_\varepsilon \mathbf {b}$ if $a_j=_\varepsilon b_j$ for $j=1,\ldots ,n$ . We use $C^*(\mathbf {a})$ to denote the sub $C^*\!$ algebra of A generated by the elements of $\mathbf {a}$ . We write $A^n_{\mathrm {sa}}$ for $(A_{\mathrm {sa}})^n$ , the space of ntuples of selfadjoint elements in A.
2 The generator rank and its precursor
In this section, we briefly recall the definition and basic properties of the generator rank $\operatorname {\mathrm {gr}}$ and its predecessor $\operatorname {\mathrm {gr}}_0$ from [Reference Thiel and WinterThi21].
Definition 2.1 [Reference Thiel and WinterThi21, Definitions 2.1 and 3.1]
Let A be a $C^*\!$ algebra. We define $\operatorname {\mathrm {gr}}_0(A)$ as the smallest integer $n\geq 0$ such that for every $\mathbf {a}\in A^{n+1}_{\mathrm {sa}}$ , $\varepsilon>0$ , and $c\in A$ , there exists $\mathbf {b}\in A^{n+1}_{\mathrm {sa}}$ such that
If no such n exists, we set $\operatorname {\mathrm {gr}}_0(A)=\infty $ . The generator rank of A is $\operatorname {\mathrm {gr}}(A):=\operatorname {\mathrm {gr}}_0(\widetilde {A})$ .
We use $\operatorname {\mathrm {Gen}}_{n}(A)_{\mathrm {sa}}$ to denote the set of tuples $\mathbf {a}\in A^n_{\mathrm {sa}}$ that generate A as a $C^*\!$ algebra. For separable $C^*\!$ algebras, the generator rank and its predecessor can be described by the denseness of such tuples.
Theorem 2.2 [Reference Thiel and WinterThi21, Theorem 3.4]
Let A be a separable $C^*\!$ algebra and $n\in \mathbb {N}$ . Then:

(1) $\operatorname {\mathrm {gr}}_0(A)\leq n$ if and only if $\operatorname {\mathrm {Gen}}_{n+1}(A)_{\mathrm {sa}}\subseteq A^{n+1}_{\mathrm {sa}}$ is a dense $G_\delta $ subset.

(2) $\operatorname {\mathrm {gr}}(A)\leq n$ if and only if $\operatorname {\mathrm {Gen}}_{n+1}(\widetilde {A})_{\mathrm {sa}}\subseteq \widetilde {A}^{n+1}_{\mathrm {sa}}$ is a dense $G_\delta $ subset.
Remark 2.3 Let A be a separable $C^*\!$ algebra. If A has generator rank at most one, then the set of (nonselfadjoint) generators in A is a dense $G_\delta $ subset (see [Reference Thiel and WinterThi21, Remark 3.7]). If A is unital, then the converse also holds: we have $\operatorname {\mathrm {gr}}(A)\leq 1$ if and only if a generic element in A is a generator.
The connection between $\operatorname {\mathrm {gr}}_0, \operatorname {\mathrm {gr}}$ and the real rank is summarized by the next result, which combines Proposition 3.12 and Theorem 3.13 in [Reference Thiel and WinterThi21]. In Theorem 5.5, we show that $\operatorname {\mathrm {gr}}_0$ and $\operatorname {\mathrm {gr}}$ agree for subhomogeneous $C^*\!$ algebras. In general, however, it is unclear if $\operatorname {\mathrm {gr}}_0=\operatorname {\mathrm {gr}}$ (see [Reference Thiel and WinterThi21, Question 3.16]).
Proposition 2.4 Let A be a $C^*\!$ algebra. Then,
We will frequently use the following permanence properties of $\operatorname {\mathrm {gr}}_0$ and $\operatorname {\mathrm {gr}}$ , which were shown in Propositions 2.2, 2.7, and 2.9 and Theorem 6.2 in [Reference Thiel and WinterThi21].
Theorem 2.5 Let A be a $C^*\!$ algebra, and let $I\subseteq A$ be an ideal. Then,
and
Recall that a $C^*\!$ algebra A is said to be approximated by sub $C^*\!$ algebras $A_\lambda \subseteq A$ if, for every finite subset $F\subseteq A$ and $\varepsilon>0$ , there is $\lambda $ such that $a\in _\varepsilon A_\lambda $ for each $a\in F$ . We do not require the subalgebras to be nested. Thus, while $\bigcup _\lambda A_\lambda $ is a dense subset of A, it is not necessarily a subalgebra. The next result combines Propositions 2.3 and 2.4 and Theorem 6.3 in [Reference Thiel and WinterThi21].
Theorem 2.6 Let A be a $C^*\!$ algebra that is approximated by sub $C^*\!$ algebras $A_\lambda \subseteq A$ , and let $n\in \mathbb {N}$ . If $\operatorname {\mathrm {gr}}_0(A_\lambda )\leq n$ for each $\lambda $ , then $\operatorname {\mathrm {gr}}_0(A)\leq n$ . Analogously, if $\operatorname {\mathrm {gr}}(A_\lambda )\leq n$ for each $\lambda $ , then $\operatorname {\mathrm {gr}}(A)\leq n$ .
Moreover, if $A=\varinjlim _{j} A_j$ is an inductive limit, then
Theorem 2.7 [Reference Thiel and WinterThi21, Theorem 5.6]
Let X be a locally compact Hausdorff space. Then,
3 Reduction to the separable case
Let us recall a few concepts from model theory that allow us to reduce some proofs in the following sections to the case of separable $C^*\!$ algebras. We refer to [Reference Farah, Hart, Lupini, Robert, Tikuisis, Vignati and WinterFHL+21, Reference Farah and KatsuraFK10] for details.
3.1. Let A be a $C^*\!$ algebra. We use $\operatorname {\mathrm {Sub}}_{\mathrm {sep}}(A)$ to denote the set of separable sub $C^*\!$ algebras of A. A collection $\mathcal {S}\subseteq \operatorname {\mathrm {Sub}}_{\mathrm {sep}}(A)$ is said to be $\sigma $ complete if we have $\overline {\bigcup \{B:B\in \mathcal {T}\}}\in \mathcal {S}$ for every countable directed subcollection $\mathcal {T}\subseteq \mathcal {S}$ . Furthermore, $\mathcal {S}$ is said to be cofinal if, for every $B_0\in \operatorname {\mathrm {Sub}}_{\mathrm {sep}}(A)$ , there is $B\in \mathcal {S}$ such that $B_0\subseteq B$ . It is well known that the intersection of countably many $\sigma $ complete, cofinal collections is again $\sigma $ complete and cofinal.
In [Reference ThielThi13, Definition 1], I formalized the notion of a noncommutative dimension theory as an assignment that to each $C^*\!$ algebra A associates a number $d(A)\in \{0,1,2,\ldots ,\infty \}$ such that six axioms are satisfied. Axioms (D1)–(D4) describe compatibility with passing to ideals, quotients, directs sums, and unitizations. The other axioms are:

(D5) If $n\in \mathbb {N}$ and if A is a $C^*\!$ algebra that is approximated by sub $C^*\!$ algebras $A_\lambda \subseteq A$ (as in Theorem 2.6) such that $d(A_\lambda )\leq n$ for each $\lambda $ , then $d(A)\leq n$ .

(D6) If A is a $C^*\!$ algebra and $B_0\subseteq A$ is a separable sub $C^*\!$ algebra, then there is a separable sub $C^*\!$ algebra $B\subseteq A$ such that $B_0\subseteq B$ and $d(B)\leq d(A)$ .
It was noted in [Reference Thiel and WinterThi21, Paragraph 4.1] that if d is an assignment from $C^*\!$ algebras to $\{0,1,\ldots ,\infty \}$ that satisfies (D5) and (D6), then for each $n\in \mathbb {N}$ and each $C^*\!$ algebra A satisfying $d(A)\leq n$ , the collection
is $\sigma $ complete and cofinal. It was shown in [Reference Thiel and WinterThi21] that $\operatorname {\mathrm {gr}}_0$ and $\operatorname {\mathrm {gr}}$ satisfy (D5) and (D6).
Lemma 3.2 Let A be a $C^*\!$ algebra, and let $I\subseteq A$ be an ideal. We have:

(1) Let $\mathcal {S}\subseteq \operatorname {\mathrm {Sub}}_{\mathrm {sep}}(I)$ be a $\sigma $ complete and cofinal subcollection. Then, the family $\{B\in \operatorname {\mathrm {Sub}}_{\mathrm {sep}}(A):B\cap I\in \mathcal {S}\}$ is $\sigma $ complete and cofinal.

(2) Let $\mathcal {S}\subseteq \operatorname {\mathrm {Sub}}_{\mathrm {sep}}(A/I)$ be a $\sigma $ complete and cofinal subcollection. Then, the family $\{B\in \operatorname {\mathrm {Sub}}_{\mathrm {sep}}(A):B/(B\cap I)\in \mathcal {S}\}$ is $\sigma $ complete and cofinal.
Proof (1): Set $\mathcal {T} := \{ B \in \operatorname {\mathrm {Sub}}_{\mathrm {sep}}(A) : B\cap I \in \mathcal {S} \}$ . It is easy to see that $\mathcal {T}$ is $\sigma $ complete. To show that it is cofinal, let $B_0\in \operatorname {\mathrm {Sub}}_{\mathrm {sep}}(A)$ . We will inductively find increasing sequences $(I_k)_k$ in $\mathcal {S}$ and $(B_k)_k$ in $\operatorname {\mathrm {Sub}}_{\mathrm {sep}}(A)$ such that
Assume that we have obtained $B_k$ for some $k\in \mathbb {N}$ . Then, $B_k\cap I\in \operatorname {\mathrm {Sub}}_{\mathrm {sep}}(I)$ , and since $\mathcal {S}$ is cofinal in $\operatorname {\mathrm {Sub}}_{\mathrm {sep}}(I)$ , we obtain $I_k\in \mathcal {S}$ such that $B_k\cap I\subseteq I_k$ . Then, let $B_{k+1}$ be the sub $C^*\!$ algebra of A generated by $B_k$ and $I_k$ .
Set $B:=\overline {\bigcup _k B_k}$ , which belongs to $\operatorname {\mathrm {Sub}}_{\mathrm {sep}}(A)$ and contains $B_0$ . We have $B\cap I = \overline {\bigcup _k I_k}$ . Since $\mathcal {S}$ is $\sigma $ complete, $B\cap I$ belongs to $\mathcal {S}$ . Thus, B belongs to $\mathcal {T}$ , as desired.
Statement (2) is shown similarly.
3.3. Recall that a $C^*\!$ algebra is called dhomogeneous (for some $d\geq 1$ ) if all its irreducible representations are ddimensional, and it is called homogeneous if it is dhomogeneous for some d (see [Reference BlackadarBla06, Definition IV.1.4.1, p. 330]).
Let A be a dhomogeneous $C^*\!$ algebra, and set $X:=\operatorname {\mathrm {Prim}}(A)$ , the primitive ideal space of A. Then, X is a locally compact Hausdorff space, and there exists a locally trivial bundle over X with fiber $M_d$ such that A is canonically isomorphic to the algebra of continuous cross sections vanishing at infinity, with pointwise operations (see [Reference FellFel61, Theorem 3.2]).
It follows that the center of A is canonically isomorphic to $C_0(X)$ , and this gives A the structure of a continuous $C_0(X)$ algebra, with each fiber isomorphic to $M_d$ . For the definition and results of $C_{0}(X)$ algebras, we refer the reader to Section 2 of [Reference DadarlatDad09]. Given a $C_0(X)$ algebra A and a closed subset $Y\subseteq X$ , we let $A(Y)$ denote the quotient of A corresponding to Y. The fiber of A at $x\in X$ is $A(x):=A(\{x\})$ . Given $a\in A$ and $x\in X$ , we write $a(x)$ for the image of a in the quotient $A(x)$ . Given $\mathbf {a}=(a_0,\ldots ,a_n)\in A^{n+1}$ , we set $\mathbf {a}(x):=(a_0(x),\ldots ,a_n(x))\in A(x)^{n+1}$ .
Given a locally compact Hausdorff space X, the local dimension of X is
with the convention that $\operatorname {\mathrm {locdim}}(\emptyset )=1$ . As noted in [Reference Thiel and WinterThi21, Paragraph 5.5], if X is nonempty, then $\operatorname {\mathrm {locdim}}(X)$ agrees with the dimension of the onepoint compactification of X. If X is $\sigma $ compact, then $\dim (X)=\operatorname {\mathrm {locdim}}(X)$ .
Lemma 3.4 Let $d\geq 1$ , $l\in \mathbb {N}$ , and let X be a compact Hausdorff space satisfying $\dim (X)\leq l$ . Set $A:=C(X,M_d)$ . Then,
is $\sigma $ complete and cofinal.
Proof $\sigma $ completeness: Let $\mathcal {T}\subseteq \mathcal {S}$ be a countable directed family, and set $C:=\overline {\bigcup \{B:B\in \mathcal {T}\}}$ . To show that C is dhomogeneous, let $\varrho $ be an irreducible representation of C. Since C is dsubhomogeneous (as a subalgebra of A), the dimension of $\varrho $ is at most d. If $\dim (\varrho )<d$ , then the restriction of $\varrho $ to each $B\in \mathcal {T}$ is zero, whence $\varrho =0$ , a contradiction.
In [Reference Brown and PedersenBP09, Section 2.2], Brown and Pedersen introduce the topological dimension of type $\mathrm {I} C^*\!$ algebras. Given a homogeneous $C^*\!$ algebra D, the topological dimension $\operatorname {\mathrm {topdim}}(D)$ is equal to $\operatorname {\mathrm {locdim}}(\operatorname {\mathrm {Prim}}(D))$ . Hence, each $B\in \mathcal {T}$ satisfies $\operatorname {\mathrm {topdim}}(B)=\operatorname {\mathrm {locdim}}(\operatorname {\mathrm {Prim}}(B))\leq l$ . By [Reference ThielThi13, Lemma 3], a continuous trace $C^*\!$ algebra (in particular, a homogeneous $C^*\!$ algebra) has topological dimension at most l whenever it is approximated by sub $C^*\!$ algebras with topological dimension at most l. Hence,
which verifies that C belongs to $\mathcal {S}$ , as desired.
Cofinality: Let $B_0\subseteq A$ be a separable sub $C^*\!$ algebra. We identify A with $C(X)\otimes M_d$ . Let $e_{jk}\in M_d$ , $j,k=1,\ldots ,d$ , be matrix units. Let $C(Y)\subseteq C(X)$ be a separable, unital sub $C^*\!$ algebra such that $f\in C(X)$ belongs to $C(Y)$ whenever $f\otimes e_{jk}\in B_0$ for some $j,k$ . Using that the real rank satisfies (D6), let $C(Z)\subseteq C(X)$ be a separable sub $C^*\!$ algebra containing $C(Y)$ such that $\operatorname {\mathrm {rr}}(C(Z))\leq \operatorname {\mathrm {rr}}(C(X))$ . Then,
and it follows that $C(Z)\otimes M_d\subseteq C(X)\otimes M_d$ has the desired properties.
Proposition 3.5 Let $d\geq 1$ , $l\in \mathbb {N}$ , and let A be a dhomogeneous $C^*\!$ algebra satisfying $\operatorname {\mathrm {locdim}}(\operatorname {\mathrm {Prim}}(A))\leq l$ . Then,
is $\sigma $ complete and cofinal.
Proof As in the proof of Lemma 3.4, we obtain that $\mathcal {S}$ is $\sigma $ complete.
Cofinality: Let $B_0\subseteq A$ be a separable sub $C^*\!$ algebra. Let $I\subseteq A$ be the ideal generated by $B_0$ . Then, I is dhomogeneous and $X:=\operatorname {\mathrm {Prim}}(I)$ is $\sigma $ compact. We view I as a $C_0(X)$ algebra with all fibers isomorphic to $M_d$ . Since the $M_d$ bundle associated with I is locally trivial, and since X is $\sigma $ compact, we can choose a sequence of compact subsets $X_0,X_1,X_2,\ldots \subseteq X$ that cover X such that $I(X_j)\cong C(X_j)\otimes M_d$ for each $j\in \mathbb {N}$ .
Given j, let $\pi _j\colon I\to C(X_j)\otimes M_d$ be the corresponding quotient map, and set
Applying Lemmas 3.2(2) and 3.4, we obtain that $\mathcal {S}_j$ is $\sigma $ complete and cofinal. It follows that $\mathcal {S}:=\bigcap _{j=0}^\infty \mathcal {S}_j$ is $\sigma $ complete and cofinal as well. Choose $B\in \mathcal {S}$ satisfying $B_0\subseteq B$ .
To verify that B is dhomogeneous, let $\varrho $ be an irreducible representation of B. Since B is dsubhomogeneous, we have $\dim (\varrho )\leq d$ . Extend $\varrho $ to an irreducible representation $\varrho '$ of I (a priori on a possibly larger Hilbert space). Then, there exists $x\in X$ such that $\varrho '$ is isomorphic to the quotient map to the fiber at x. Let $j\in \mathbb {N}$ such that $x\in X_j$ . Since B belongs to $\mathcal {S}_j$ , it exhausts the fiber at x, and we deduce that $\dim (\varrho )\geq d$ .
To see that $\operatorname {\mathrm {locdim}}(\operatorname {\mathrm {Prim}}(B))\leq l$ , let $K\subseteq \operatorname {\mathrm {Prim}}(B)$ be a compact subset. For each j, let $F_j\subseteq \operatorname {\mathrm {Prim}}(B)$ be the closed subset corresponding to the quotient $\pi _j(B)$ of B. Since B belongs to $\mathcal {S}_j$ , we have $\operatorname {\mathrm {locdim}}(F_j)\leq l$ . Hence, $\dim (K\cap F_j)\leq l$ . We have $K=\bigcup _j (K\cap F_j)$ , and therefore
by the Countable Sum Theorem (see [Reference PhillipsPea75, Theorem 3.2.5, p. 125]; see also the introduction to Section 5).
4 Homogeneous $C^*\!$ algebras
In this section, we compute the generator rank of homogeneous $C^*\!$ algebra; (see Theorem 4.17). We first consider the unital separable case (Lemma 4.15), we then generalize to the unital nonseparable case (Proposition 4.16) and finally to the general case. Unlike for commutative $C^*\!$ algebras, the unital separable case is highly nontrivial and it requires a delicate analysis of the codimension of certain submanifolds of $(M_d)^{n+1}_{\mathrm {sa}}$ (Lemma 4.11) in connection with a suitable version of the homotopy extension lifting property (Lemma 4.14).
The next result characterizes generating tuples in separable $C(X)$ algebras with simple fibers, and thus in particular in unital, separable, homogeneous $C^*\!$ algebras. Given a map $\varphi \colon D\to E$ between $C^*\!$ algebras and $\mathbf {a}=(a_0,\ldots ,a_n)\in D^{n+1}$ , we set
Proposition 4.1 Let X be a compact metric space, and let A be a separable $C(X)$ algebra such that all fibers are simple. Let $n\in \mathbb {N}$ and $\mathbf {a}\in A^{n+1}_{\mathrm {sa}}$ . Then, $\mathbf {a}\in \operatorname {\mathrm {Gen}}_{n+1}(A)_{\mathrm {sa}}$ if and only if the following are satisfied:

(a) $\mathbf {a}$ generates each fiber, that is, $\mathbf {a}(x)\in \operatorname {\mathrm {Gen}}_{n+1}(A(x))_{\mathrm {sa}}$ for each $x\in X$ .

(b) $\mathbf {a}$ separates the points of X in the sense that for distinct $x,y\in X$ , there is no isomorphism $\alpha \colon A(x)\to A(y)$ satisfying $\alpha (\mathbf {a}(x))=\mathbf {a}(y)$ .
Proof Let us first assume that $\mathbf {a}\in \operatorname {\mathrm {Gen}}_{n+1}(A)_{\mathrm {sa}}$ . For $x\in X$ , let $\pi _x\colon A\to A(x)$ be the quotient map onto the fiber at x. Since $\pi _x$ is a surjective ${}^*$ homomorphism, it maps $\operatorname {\mathrm {Gen}}_{n+1}(A)_{\mathrm {sa}}$ to $\operatorname {\mathrm {Gen}}_{n+1}(A(x))_{\mathrm {sa}}$ , which verifies (a). Similarly, for distinct points $x,y\in X$ , the map $\pi _x\oplus \pi _y\colon A\to A(x)\oplus A(y)$ is a surjective ${}^*$ homomorphism. It follows that $(\mathbf {a}(x),\mathbf {a}(y))=(\pi _x\oplus \pi _y)(\mathbf {a})\in \operatorname {\mathrm {Gen}}_{n+1}(A(x)\oplus A(y))_{\mathrm {sa}}$ . To verify (b), assume that $\alpha \colon A(x)\to A(y)$ is an isomorphism satisfying $\alpha (\mathbf {a}(x))=\mathbf {a}(y)$ . Then,
which contradicts that $(\mathbf {a}(x),\mathbf {a}(y))$ generates $A(x)\oplus A(y)$ . Thus, no such $\alpha $ exists.
Conversely, let us assume that (a) and (b) are satisfied. Set $B:=C^*(\mathbf {a})$ . We need to prove $B=A$ . This follows from [Reference Tikuisis, White and WinterTW14, Lemma 3.2] once we show that B exhausts the fiber $A(x)$ for each $x\in X$ , and that for distinct $x,y\in X$ , there exists $b\in B$ such that $b(x)$ is full in $A(x)$ and $b(y)=0$ . The exhaustion of fibers follows directly from (a).
Let $x,y\in X$ be distinct, and set $C:=(\pi _x\oplus \pi _y)(B)\subseteq A(x)\oplus A(y)$ . Note that C is the sub $C^*\!$ algebra of $A(x)\oplus A(y)$ generated by $(\mathbf {a}(x),\mathbf {a}(y))$ . If $C\neq A(x)\oplus A(y)$ , using that $A(x)$ and $A(y)$ are simple, it follows from [Reference Thiel and WinterThi21, Lemma 5.10] that there exists an isomorphism $\alpha \colon A(x)\to A(y)$ such that
which implies that $\alpha (\mathbf {a}(x))=\mathbf {a}(y)$ . Since this contradicts (b), we deduce that $C=A(x)\oplus A(y)$ . Hence, there exists $b\in B$ such that $b(x)$ is full in $A(x)$ and $b(y)=0$ .
Notation 4.2 For $d\geq 2$ and $n\in \mathbb {N}$ , we set
Note that $E_d^{n+1}$ is isomorphic to $\mathbb {R}^{(n+1)d^2}$ as topological vector spaces. In particular, $E_d^{n+1}$ is a (real) manifold with $\dim (E_d^{n+1})=(n+1)d^2$ .
We let ${\mathcal {U}}_d$ denote the unitary group of $M_d$ . It is a compact Lie group of dimension $d^2$ . Every automorphism of $M_d$ is inner, and the kernel of ${\mathcal {U}}_d\to \operatorname {\mathrm {Aut}}(M_d)$ is the group of central unitaries $\mathbb {T} 1\subseteq {\mathcal {U}}_d$ . Hence, $\operatorname {\mathrm {Aut}}(M_d)$ is naturally isomorphic to $\mathcal {PU}_d:={\mathcal {U}}_d/(\mathbb {T} 1)$ , the projective unitary group, which is a compact Lie group of dimension $d^21$ . Given $u\in {\mathcal {U}}_d$ , we use $[u]$ to denote its class in $\mathcal {PU}_d$ .
The action $\mathcal {PU}_d\curvearrowright M_d$ induces an action $\mathcal {PU}_d\curvearrowright E_d^{n+1}$ by setting
for $u\in {\mathcal {U}}_d$ and $\mathbf {a}=(a_0,\ldots ,a_n)\in E_d^{n+1}$ .
4.3. Let A be a unital, separable, dhomogeneous $C^*\!$ algebra, and let $n\in \mathbb {N}$ . Set $X:=\operatorname {\mathrm {Prim}}(A)$ . We consider A with its canonical $C(X)$ algebra structure, with each fiber isomorphic to $M_d$ (see Paragraph 3.3). Set
Given $x\in X$ , let $\pi _x\colon A\to A(x)$ denote the map to the fiber at x. This induces a map $A^{n+1}_{\mathrm {sa}}\to (A(x))^{n+1}_{\mathrm {sa}}$ , which we also denote by $\pi _x$ . Choose an isomorphism $A(x)\cong M_d$ , which induces an isomorphism $(A(x))^{n+1}_{\mathrm {sa}} \cong E^{n+1}_d = (M_d)^{n+1}_{\mathrm {sa}}$ . Since the isomorphism $A(x)\cong M_d$ is unique up to an automorphism of $M_d$ , we obtain a canonical homeomorphism $(A(x))^{n+1}_{\mathrm {sa}}/\operatorname {\mathrm {Aut}}(A(x)) \cong E^{n+1}_d / \mathcal {PU}_d$ . We let $\psi _x\colon A^{n+1}_{\mathrm {sa}}\to E^{n+1}_d/\mathcal {PU}_d$ be the resulting natural map.
Given $\mathbf {a}\in A^{n+1}_{\mathrm {sa}}$ , one checks that $\psi _x(\mathbf {a})$ depends continuously on x. This allows us to define $\Psi \colon A^{n+1}_{\mathrm {sa}}\to C(X,E^{n+1}_d/\mathcal {PU}_d)$ by
for $\mathbf {a}\in A^{n+1}_{\mathrm {sa}}$ and $x\in X$ . Restricting $\Psi $ to $\operatorname {\mathrm {Gen}}^{\mathrm {fiber}}_{n+1}(A)_{\mathrm {sa}}$ gives a continuous map
We let $E(X,G^{n+1}_d/\mathcal {PU}_d)$ denote the set of continuous maps $X\to G^{n+1}_d/\mathcal {PU}_d$ that are injective. By Proposition 4.1, a tuple $\mathbf {a}\in A^{n+1}_{\mathrm {sa}}$ belongs to $\operatorname {\mathrm {Gen}}_{n+1}(A)_{\mathrm {sa}}$ if and only if (a): $\mathbf {a}\in \operatorname {\mathrm {Gen}}^{\mathrm {fiber}}_{n+1}(A)_{\mathrm {sa}}$ , and (b): $\Psi (\mathbf {a})\in E(X,G^{n+1}_d/\mathcal {PU}_d)$ . Thus, to determine the generator rank of A, we need to answer the following questions:

(a) When is $\operatorname {\mathrm {Gen}}^{\mathrm {fiber}}_{n+1}(A)_{\mathrm {sa}}$ dense in $A^{n+1}_{\mathrm {sa}}$ ?

(b) When is $E(X,G_d^{n+1}/\mathcal {PU}_d)$ dense in $C(X,G_d^{n+1}/\mathcal {PU}_d)$ ?
Analogous as for the computation of the generator rank for unital, separable, commutative $C^*\!$ algebras in [Reference Thiel and WinterThi21, Section 5], the answer to question (a) is determined by $\dim (X)$ , and the answer to (b) is determined by $\dim (X\times X)$ . However, while in the commutative case the dominating condition was (b) involving $\dim (X\times X)$ , we will see that for dhomogeneous $C^*\!$ algebras with $d\geq 2$ the dominating condition is (a) involving $\dim (X)$ .
To study (a), we will determine the dimension of $E^{n+1}_d\setminus G^{n+1}_d$ . For this, we study the action $\mathcal {PU}_d\curvearrowright E_d^{n+1}$ . We will show that $G^{n+1}_d$ consists precisely of the tuples in $E_d^{n+1}$ with trivial stabilizer subgroup (see Lemma 4.7). This allows us to describe $E^{n+1}_d\setminus G^{n+1}_d$ as the union of the submanifolds corresponding to nontrivial stabilizer subgroups. We then estimate the dimension of these submanifolds (see Lemma 4.11).
To study (b), we show that $G_d^{n+1}$ is an open subset of $E_d^{n+1}$ (see Lemma 4.9). Hence, $G_d^{n+1}$ is a manifold with $\dim (G_d^{n+1})=\dim (E_d^{n+1})=(n+1)d^2$ . We let $G_d^{n+1}/\mathcal {PU}_d$ denote the quotient space. Since $\mathcal {PU}_d$ is a compact Lie group of dimension $d^21$ , it follows that $G_d^{n+1}/\mathcal {PU}_d$ is a manifold of dimension $(n+1)d^2(d^21)=nd^2+1$ . We then use a result of [Reference LuukkainenLuu81] which characterizes when a continuous map to a manifold can be approximated by injective maps.
Finally, we use a version of the homotopy extension lifting property for the projection $G_d^{n+1}\to G_d^{n+1}/\mathcal {PU}_d$ (see Lemma 4.14) to show that a given tuple in $\operatorname {\mathrm {Gen}}^{\mathrm {fiber}}_{n+1}(A)_{\mathrm {sa}}$ can be approximated by tuples that are mapped to $E(X,G_d^{n+1}/\mathcal {PU}_d)$ by $\Psi $ .
4.4. Let G be a compact Lie group, acting smoothly on a connected manifold M. We briefly recall the orbittype decomposition. For details, we refer the reader to [Reference BredonBre72, Reference MeinrekenMei03]. We will later apply this for the action $\mathcal {PU}_d\curvearrowright E^{n+1}_d$ .
The stabilizer subgroup of $m\in M$ is
Two subgroups H and $H'$ of G are conjugate, denoted $H\sim H'$ , if there exists $g\in G$ such that $H=gH'g^{1}$ . We let
denote the collection of all conjugation classes of stabilizer subgroups. Set
for $t\in T$ . We have $\operatorname {\mathrm {Stab}}(g.m)=g\operatorname {\mathrm {Stab}}(m)g^{1}$ for all $g\in G$ and $m\in M$ , which implies that each $M_t$ is Ginvariant.
Let us additionally assume that each $M_t$ is connected. Then, by [Reference MeinrekenMei03, Theorem 1.30], each $M_t$ is a smooth embedded submanifold of M, and M decomposes as a disjoint union $M=\bigcup _{t\in T} M_t$ . (See also [Reference BredonBre72, Theorem IV.3.3, p. 182].) Furthermore, this decomposition satisfies the frontier condition: for all $t',t\in T$ , if $M_{t'}\cap \overline {M_t}\neq \emptyset $ , then $M_{t'}\subseteq \overline {M_t}$ . This defines a partial order on T by setting $t'\leq t$ if $M_{t'}\subseteq \overline {M_t}$ . The depth of $t\in T$ is defined as $\operatorname {\mathrm {depth}}(t)=0$ if t is maximal, and otherwise
In many cases, one knows that T is finite and contains a largest element (see Sections IV.3 and IV.10 of [Reference BredonBre72]).
Set $M_{\mathrm {free}}:=\{m\in M : \operatorname {\mathrm {Stab}}(m)=\{1\}\}$ . If $M_{\mathrm {free}}\neq \emptyset $ , then the conjugacy class of the trivial subgroup is the largest element in T, and $M_{\mathrm {free}}$ is an open submanifold of M. The restriction of the action to $M_{\mathrm {free}}$ is free.
Proposition 4.5 Retain the situation from Paragraph 4.4. Assume that M is metrizable with metric $d_M$ , T is finite, and $M\neq M_{\mathrm {free}}\neq \emptyset $ . Let X be a compact Hausdorff space. Then, the following are equivalent:

(1) $C(X,M_{\mathrm {free}})\subseteq C(X,M)$ is dense with respect to the metric $d(f,g):=\sup _{x\in X}d_M(f(x),g(x))$ , for $f,g\in C(X,M)$ .

(2) $\dim (X)<\dim (M)\dim (M\setminus M_{\mathrm {free}})$ .
Proof Note that T contains exactly one element of depth zero, namely the conjugacy class of $\{1\}$ . Therefore,
and it follows that
To show that (1) implies (2), assume that $\dim (X)\geq \dim (M)\dim (M\setminus M_{\mathrm {free}})$ . Choose $t\in T$ of depth $\geq 1$ such that $\dim (X)\geq \dim (M)\dim (M_t)$ . As noted in [Reference Beggs and EvansBE91, Proposition 1.6], it follows that $C(X,M\setminus M_t)\subseteq C(X,M)$ is not dense, which implies that (1) fails.
Assuming (2), let us prove (1). Let $f\in C(X,M)$ and $\varepsilon>0$ . The proof is similar to that of Theorem 1.3 in [Reference Beggs and EvansBE91]. We inductively change f to avoid each $M_t$ , but instead of proceeding by the (co)dimension of the submanifolds, we use their depths.
It follows from the frontier condition that for each $t\in T$ , the set $\overline {M_t}\setminus M_t$ is contained in the union of submanifolds $M_s$ with $s\in T$ and $\operatorname {\mathrm {depth}}(s)>\operatorname {\mathrm {depth}}(t)$ . Let $t_1,\ldots ,t_K$ be an enumeration of the elements in T with depth $\geq 1$ , such that $\operatorname {\mathrm {depth}}(t_1)\geq \operatorname {\mathrm {depth}}(t_2)\geq \cdots \geq \operatorname {\mathrm {depth}}(t_K)$ . Note that $M_{t_1}$ is a closed submanifold (since $t_1$ has maximal depth and thus $\overline {M_{t_1}}\setminus M_{t_1}=\emptyset $ ), and for each $j\geq 2$ , the set $\overline {M_{t_j}}\setminus M_{t_j}$ is contained in $M_{t_1}\cup \ldots M_{t_{j1}}$ . Furthermore, every $M_{t_j}$ is a submanifold of codimension $\geq \dim (X)+1$ .
By [Reference Beggs and EvansBE91, Lemma 1.4], if $Y\subseteq M$ is submanifold of codimension $\geq \dim (X)+1$ , if $\delta>0$ , and if $g\in C(X, M)$ satisfies $g(X)\cap (\overline {Y}\setminus Y)=\emptyset $ , then there exists $g'\in C(X,M)$ such that $d(g,g')\leq \delta $ and $g'(X)\cap \overline {Y}=\emptyset $ . Set $f_0:=f$ . We will inductively find $f_k\in C(X,M)$ such that, for each $k=1,\ldots ,K$ , we have
First, using that the boundary of $M_{t_1}$ is empty, we can apply [Reference Beggs and EvansBE91, Lemma 1.4] to obtain $f_1\in C(X,M)$ such that
For $k\geq 2$ , assuming that we have chosen $f_{k1}$ , let $\delta _k$ denote the (positive) distance between the compact set $f_{k1}(X)$ and $\overline {M_{t_1}}\cup \cdots \cup \overline {M_{t_{k1}}}$ . Applying [Reference Beggs and EvansBE91, Lemma 1.4], we obtain $f_k\in C(X,M)$ such that
By choice of $\delta _k$ , it follows that $f_k(X)$ is disjoint from $\overline {M_{t_1}}\cup \cdots \cup \overline {M_{t_k}}$ .
Finally, the element $f_K$ belongs to $C(X,M_{\mathrm {free}})$ and satisfies $d(f,f_K)<\varepsilon $ .
4.6. We let $\operatorname {\mathrm {Sub}}_1(M_d)$ denote the collection of sub $C^*\!$ algebras of $M_d$ that contain the unit of $M_d$ . Given $\mathbf {a}\in E^{n+1}_d:=(M_d)^{n+1}_{\mathrm {sa}}$ , we set $C^*_1(\mathbf {a}):=C^*(\mathbf {a},1)\in \operatorname {\mathrm {Sub}}_1(M_d)$ . We let $\mathcal {PU}_d$ act on $\operatorname {\mathrm {Sub}}_1(M_d)$ by $[u].B:=uBu^*$ for $u\in {\mathcal {U}}_d$ and $B\in \operatorname {\mathrm {Sub}}_1(M_d)$ . Given $B_1,B_2\in \operatorname {\mathrm {Sub}}_1(M_d)$ , we write $B_1\sim B_2$ if $B_1$ and $B_2$ lie in the same orbit of this action, that is, if $B_1=uB_2u^*$ for some $u\in {\mathcal {U}}_d$ .
Given $\mathbf {a}\in E^{n+1}_d$ , we have $C^*_1(\mathbf {a})=M_d$ if and only if $C^*(\mathbf {a})$ , and thus
Given a sub $C^*\!$ algebra $B\subseteq M_d$ , we let $B' := \{ c \in M_d : bc=cb \text { for all } b\in B \}$ denote its commutant. We always have $B'\in \operatorname {\mathrm {Sub}}_1(M_d)$ , and by the bicommutant theorem, we have $B"=B$ for all $B\in \operatorname {\mathrm {Sub}}_1(M_d)$ .
Lemma 4.7 Let $\mathbf {a}\in E^{n+1}_d$ . Then,
Furthermore, we have $\mathbf {a}\in G^{n+1}_d$ if and only if $\operatorname {\mathrm {Stab}}(\mathbf {a})=\{[1]\}$ .
Proof Given $u\in {\mathcal {U}}_d$ , we have $[u].\mathbf {a}=\mathbf {a}$ if and only if $uxu^*=x$ for every $x\in C^*(\mathbf {a})$ . This implies the formula for $\operatorname {\mathrm {Stab}}(\mathbf {a})$ .
If $\mathbf {a}\in G^{n+1}_d$ , then $C^*(\mathbf {a})'=\mathbb {C} 1$ , which implies that $\operatorname {\mathrm {Stab}}(\mathbf {a})$ is trivial. Conversely, assuming that $\mathbf {a}\in E_d^{n+1}\setminus G_d^{n+1}$ , let us verify that $\mathbf {a}$ has nontrivial stabilizer subgroup. Since $C^*(\mathbf {a})\neq M_d$ , we also have $C^*_1(\mathbf {a})\neq M_d$ . Using the bicommutant theorem, we deduce that $C^*_1(\mathbf {a})'$ is strictly larger than the center of $M_d$ . Using that $C^*(\mathbf {a})'=C^*_1(\mathbf {a})'$ , we obtain a noncentral unitary in $C^*(\mathbf {a})'$ .
Lemma 4.8 Let $\mathbf {a},\mathbf {b}\in E^{n+1}_d$ . Then, we have $\operatorname {\mathrm {Stab}}(\mathbf {a}) \sim \operatorname {\mathrm {Stab}}(\mathbf {b})$ if and only if $C^*_1(\mathbf {a}) \sim C^*_1(\mathbf {b})$ .
Proof Let $B_1,B_2\in \operatorname {\mathrm {Sub}}_1(M_d)$ . If $u\in {\mathcal {U}}_d$ satisfies $uB_1u^*=B_2$ , then one checks $uB_1'u^*=B_2'$ . Using also that $B_1$ and $B_2$ agree with their bicommutants, we obtain
Using that $C^*_1(\mathbf {a}) = C^*(\mathbf {a})"$ and $C^*_1(\mathbf {a})' = C^*(\mathbf {a})'$ , and similarly $C^*_1(\mathbf {b}) = C^*(\mathbf {b})"$ and $C^*_1(\mathbf {b})' = C^*(\mathbf {b})'$ , we need to show
To prove the forward implication, we assume that $\operatorname {\mathrm {Stab}}(\mathbf {a}) \sim \operatorname {\mathrm {Stab}}(\mathbf {b})$ . Let $v\in {\mathcal {U}}_d$ such that $[v]\operatorname {\mathrm {Stab}}(\mathbf {a})[v]^{1}= \operatorname {\mathrm {Stab}}(\mathbf {b})$ . Given $u\in {\mathcal {U}}(C^*(\mathbf {a})')$ , it follows from Lemma 4.7 that
Using that $\mathbb {T} 1\subseteq {\mathcal {U}}(C^*(\mathbf {b})')$ , we obtain $vuv^*\in {\mathcal {U}}(C^*(\mathbf {b})')$ . Since $C^*(\mathbf {a})'$ is spanned by its unitary elements, we get $vC^*(\mathbf {a})'v^*\subseteq C^*(\mathbf {b})'$ . The reverse inclusion is shown analogously, whence $vC^*(\mathbf {a})'v^*= C^*(\mathbf {b})'$ , that is, $C^*(\mathbf {a})'\sim C^*(\mathbf {b})'$ .
Conversely, if $C^*(\mathbf {a})' \sim C^*(\mathbf {b})'$ , let $v\in {\mathcal {U}}_d$ such that $uC^*(\mathbf {a})'v^*=C^*(\mathbf {b})'$ . Using Lemma 4.7, we get $[v]\operatorname {\mathrm {Stab}}(\mathbf {a})[v]^{1}= \operatorname {\mathrm {Stab}}(\mathbf {b})$ , that is, $\operatorname {\mathrm {Stab}}(\mathbf {a}) \sim \operatorname {\mathrm {Stab}}(\mathbf {b})$ .
Lemma 4.9 Let B be a finitedimensional $C^*\!$ algebra and $n\geq 1$ . Then, the set $\{\mathbf {a}\in B^{n+1}_{\mathrm {sa}} : C^*_1(\mathbf {a})=B\}$ is a pathconnected, dense, open subset of $B^{n+1}_{\mathrm {sa}}$ .
Proof Set $G:=\{\mathbf {a}\in B^{n+1}_{\mathrm {sa}} : C^*_1(\mathbf {a})=B\}$ .
Denseness: By [Reference Thiel and WinterThi21, Lemma 7.2], we have $\operatorname {\mathrm {gr}}(B)\leq 1\leq n$ . Since B is unital and separable, it follows from Theorem 2.2 that $\operatorname {\mathrm {Gen}}_{n+1}(B)_{\mathrm {sa}}\subseteq B^{n+1}_{\mathrm {sa}}$ is dense. Using that $\operatorname {\mathrm {Gen}}_{n+1}(B)_{\mathrm {sa}}\subseteq G$ , we get that G is also dense in $B^{n+1}_{\mathrm {sa}}$ .
Openness: Let $\mathcal {D}$ denote the family of sub $C^*\!$ algebras $D\subseteq B$ such that $D+\mathbb {C} 1_B$ is a proper sub $C^*\!$ algebra of B (that is, $C^*_1(D)\neq B$ ). Then,
Thus, we need to show that $\bigcup _{D\in \mathcal {D}} D^{n+1}_{\mathrm {sa}}$ is a closed subset of $B^{n+1}_{\mathrm {sa}}$ .
We let ${\mathcal {U}}(B)$ denote the unitary group of B. It naturally acts on $\mathcal {D}$ by setting $u.D:=uDu^*$ for $u\in {\mathcal {U}}(B)$ and $D\in \mathcal {D}$ . Since B is finitedimensional, two sub $C^*\!$ algebras $D_1,D_2\subseteq B$ are unitarily equivalent if and only if $D_1\cong D_2$ and the inclusions induce the same maps in ordered $K_0$ theory. It follows that the action ${\mathcal {U}}(B)\curvearrowright \mathcal {D}$ has only finitely many orbits, and we choose representatives $D_1,\ldots ,D_m\in \mathcal {D}$ . Then, $\mathcal {D}=\bigcup _{j=1}^m \bigcup _{u\in {\mathcal {U}}{B}} uD_ju^*$ .
For each j, since $D_j$ is a closed subset of B, it follows that $(D_j)^{n+1}_{\mathrm {sa}}$ is a closed subset of $B^{n+1}_{\mathrm {sa}}$ . Since B is finitedimensional, ${\mathcal {U}}(B)$ is compact, and it follows that
is closed, as desired.
Pathconnectedness: We only sketch the argument for the case $B=M_d$ for some $d\,{\geq}\, 2$ . Let $\mathbf {a}\in \operatorname {\mathrm {Gen}}_{n+1}(M_d)_{\mathrm {sa}}$ . Using that the unitary group of $M_d$ is pathconnected, and that $a_0$ is unitarily equivalent to a diagonal matrix, we find a path in $\operatorname {\mathrm {Gen}}_{n+1}(M_d)_{\mathrm {sa}}$ from $\mathbf {a}$ to some $\mathbf {b}$ such that $b_0$ is diagonal. By splitting multiple eigenvalues of $b_0$ and moving them away from zero, we find a path $(x_t)_{t\in [0,1]}$ inside the selfadjoint, diagonal matrices starting with $x_0=b_0$ and ending with some $x_1$ such that $x_1$ has k distinct, nonzero diagonal entries, and such that $b_0\in C^*(x_t)$ for each $t\in [0,1]$ . Then, $t\mapsto (x_t,b_1,\ldots ,b_n)$ defines a path inside $\operatorname {\mathrm {Gen}}_{n+1}(M_d)_{\mathrm {sa}}$ .
Let S denote the set of selfadjoint matrices in $M_d$ such that every offdiagonal entry is nonzero. Note that S is pathconnected. Next, we let $(y_t)_{t\in [0,1]}$ be a path inside the selfadjoint matrices starting with $y_0=b_1$ , ending with some matrix $y_1$ that has the eigenvalues $1,2,\ldots ,d$ such that $y_t$ belongs to S for every $t\in (0,1]$ . Note that $x_1$ and $y_t$ generated $M_d$ for every $t\in (0,1]$ . It follows that $t\mapsto (x_1,y_t,b_2,\ldots ,b_n)$ defines a path inside $\operatorname {\mathrm {Gen}}_{n+1}(M_d)_{\mathrm {sa}}$ .
Conjugating by a suitable path of unitaries, we find a path in $\operatorname {\mathrm {Gen}}_{n+1}(M_d)_{\mathrm {sa}}$ from $(x_1,y_1,b_2,\ldots ,b_n)$ to some $\mathbf {c}=(c_0,c_1,\ldots ,c_n)$ such that $c_1=\operatorname {\mathrm {diag}}(1,2,\ldots ,d)$ . Arguing as above, we find a path in $\operatorname {\mathrm {Gen}}_{n+1}(M_d)_{\mathrm {sa}}$ that changes $c_0$ to the matrix $\tilde {c_0}$ with all entries $1$ . Then, $\tilde {c_0}$ and $c_1$ generate $M_d$ .
Then, $t\mapsto (\tilde {c_0},c_1,(1t)c_2,\ldots ,(1t)c_n)$ is a path in $\operatorname {\mathrm {Gen}}_{n+1}(M_d)_{\mathrm {sa}}$ connecting to $(\tilde {c_0},c_1,0\ldots ,0)$ . Thus, every $\mathbf {a}\in \operatorname {\mathrm {Gen}}_{n+1}(M_d)_{\mathrm {sa}}$ is pathconnected to the same element.
4.10. Let $d\geq 2$ , and $n\in \mathbb {N}$ . The compact Lie group $\mathcal {PU}_d$ acts smoothly on the manifold $E^{n+1}_d:=(M_d)^{n+1}_{\mathrm {sa}}$ . We will describe the corresponding orbittype decomposition of $E^{n+1}_d$ .
Given $\mathbf {a},\mathbf {b}\in E^{n+1}_d$ , by Lemma 4.8, we have $\operatorname {\mathrm {Stab}}(\mathbf {a})\sim \operatorname {\mathrm {Stab}}(\mathbf {b})$ if and only if $C^*_1(\mathbf {a})\sim C^*_1(\mathbf {b})$ . Moreover, given $B\in \operatorname {\mathrm {Sub}}_1(M_d)$ , there exists $\mathbf {a}\in E^{n+1}_d$ with $B=C^*_1(\mathbf {a})$ . It follows that the orbit types of $\mathcal {PU}_d\curvearrowright E^{n+1}_d$ naturally correspond to the orbit types of the action $\mathcal {PU}_d\curvearrowright \operatorname {\mathrm {Sub}}_1(M_d)$ .
Given $B_1,B_2\in \operatorname {\mathrm {Sub}}_1(M_d)$ , it is well known that $B_1\sim B_2$ if and only if $B_1$ and $B_2$ are isomorphic, that is, $B_1\cong B_2\cong \oplus _{j=1}^L M_{d_j}$ for some $L,d_1,\ldots ,d_L\geq 1$ , and if, for each j, the maps $M_{d_j}\to B_1\to M_d$ and $M_{d_j}\to B_2\to M_d$ have the same multiplicity $m_j$ . Thus, to parametrize the orbit types of $\mathcal {PU}_d\curvearrowright \operatorname {\mathrm {Sub}}_1(M_d)$ , we consider
Given $(\mathbf {d},\mathbf {m})\in T_0$ , we let $B(\mathbf {d},\mathbf {m})\subseteq M_d$ be the sub $C^*\!$ algebra of block diagonal matrices, with $m_1$ equal blocks of size $d_1$ , followed by $m_2$ equal blocks of size $d_2$ , and so on. We point out that the numbers $d_1,\ldots ,d_L$ are not required to be distinct. For example, $B((d),(1))=M_d$ , $B((1),(d))=\mathbb {C} 1$ , and $B((1,\ldots ,1),(1,\ldots ,1))$ is the algebra of diagonal matrices.
We define an equivalence relation on $T_0$ by setting $(\mathbf {d},\mathbf {m})\sim (\mathbf {d}',\mathbf {m}')$ if all tuples $\mathbf {d},\mathbf {m},\mathbf {d}',\mathbf {m}'$ contain the same number of elements, say $L\geq 1$ , and if there is a permutation $\sigma $ of $\{1,\ldots ,L\}$ such that
For example, we have $((2,2),(1,2))\sim ((2,2),(2,1))$ , but $((2,2),(1,2))\nsim ((2),(3))$ .
We have $(\mathbf {d},\mathbf {m})\sim (\mathbf {d}',\mathbf {m}')$ if and only if $B(\mathbf {d},\mathbf {m})\sim B(\mathbf {d}',\mathbf {m}')$ .
Set $T:=T_0/_\sim $ . Given $(\mathbf {d},\mathbf {m})\in T_0$ , we let $[\mathbf {d},\mathbf {m}]$ denote its equivalence class in T. For every $B\in \operatorname {\mathrm {Sub}}_1(M_d)$ , there exists $(\mathbf {d},\mathbf {m})\in T_0$ such that $B\sim B(\mathbf {d},\mathbf {m})$ . It follows that the orbit types of $\mathcal {PU}_d\curvearrowright \operatorname {\mathrm {Sub}}_1(M_d)$ are parametrized by T:
Given $[\mathbf {d},\mathbf {m}]\in T$ , set
Then, $E_{[\mathbf {d},\mathbf {m}]}$ is the submanifold of $E^{n+1}_d$ corresponding to orbit type $[\mathbf {d},\mathbf {m}]$ , and the orbittype decomposition (as described in Paragraph 4.4) for $\mathcal {PU}_d\curvearrowright E^{n+1}_d$ is
By Lemma 4.7, a tuple $\mathbf {a}\in E^{n+1}_d$ has trivial stabilizer group if and only if $\mathbf {a}$ belongs to $G^{n+1}_d$ . It follows that $G^{n+1}_d = E_{[(d),(1)]}$ , and in the notation of Paragraph 4.4, with $M=E^{n+1}_d$ , we have $M_{\mathrm {free}}=G^{n+1}_d$ .
Lemma 4.11 Let $[\mathbf {d},\mathbf {m}]\in T$ with $[\mathbf {d},\mathbf {m}]\neq [(d),(1)]$ . Then, $E_{[\mathbf {d},\mathbf {m}]}$ is a connected submanifold of $E^{n+1}_d$ satisfying
Furthermore, $\dim ( E_{[(d1,1),(1,1)]} ) = (n+1)d^2 2n(d1)$ .
Proof Set $B:=B(\mathbf {d},\mathbf {m})$ . Note that a tuple $\mathbf {a}\in E^{n+1}_d$ belongs to $E_{[\mathbf {d},\mathbf {m}]}$ if and only if $C^*_1(\mathbf {a})\sim B$ . Set
By Lemma 4.9, F is connected. Since every orbit in $E_{[\mathbf {d},\mathbf {m}]}$ meets F, and since $\mathcal {PU}_d$ is connected, it follows that $E_{[\mathbf {d},\mathbf {m}]}$ is connected as well.
By [Reference BredonBre72, Theorem IV.3.8], if a compact Lie group L acts smoothly on a connected manifold M such that all orbits have the same type, then $\dim (M)=\dim (M/L)+\dim (L/K)$ , where K is the stabilizer subgroup of any element in M.
Let $K\subseteq \mathcal {PU}_d$ be the stabilizer subgroup of some element in F. By considering the restricted action $\mathcal {PU}_d\curvearrowright E_{[\mathbf {d},\mathbf