Cartan subalgebras in dimension drop algebras

We completely classify Cartan subalgebras of dimension drop algebras with coprime parameters. More generally, we classify Cartan subalgebras of arbitrary stabilised dimension drop algebras that are non-degenerate in the sense that the dimensions of their fibres in the endpoints are maximal. Conjugacy classes by an automorphism are parametrised by certain congruence classes of matrices over the natural numbers with prescribed row and column sums. In particular, each dimension drop algebra admits only finitely many non-degenerate Cartan subalgebras up to conjugacy. As a consequence of this parametrisation, we can provide examples of subhomogeneous C*-algebras with exactly n Cartan subalgebras up to conjugacy. Moreover, we show that in many dimension drop algebras, Cartan subalgebras are conjugate if and only if their spectrum is homeomorphic.


Introduction
Cartan subalgebras constitute a centrepiece of modern structure theory of von Neumann algebras. They were introduced by Dixmier in [10] and related to measurable group theory and ergodic theory by Singer in [37]. Thanks to Popa's intertwining by bimodule techniques introduced in [29, Theorem A.1] and [30,Theorem 2.1], which are particularly well compatible with Cartan subalgebras, best possible classification results for Cartan subalgebras in certain crossed product von Neumann algebras could be obtained [32,33]. C * -algebraic Cartan subalgebras find their origin in the notion of groupoid C * -algebras [34] and were introduced much later than their von Neumann algebraic counterparts in [35] by Renault, building on Kumjian's work on C * -diagonals [20]. In analogy with Feldman-Moore's work on measured equivalence relations and von Neumann algebraic Cartan subalgebras [12,13], Cartan subalgebras of C * -algebras connect to topological dynamics [22] and geometric group theory [23,39,43] and therefore provide structure that has similar potential as Cartan subalgebras in von Neumann algebras. Besides classical applications of groupoid C * -algebras, which take a slightly different perspective, the potential of Cartan subalgebras in C * -algebras is visible through a characterisation of a positive solution of the infamous UCT problem in [2,3]. Moreover, (non-)existence and (non-)uniqueness results are discussed in [24], where among others a classification of Cartan subalgebras in homogeneous C * -algebras in terms of principal bundles is provided. Recently in [18] yet another connection with C * -algebraic Cartan subalgebras was found. We say that two Cartan subalgebras in a C * -algebra B 1 , B 2 ⊂ A are conjugate, if there is an automorphism α ∈ Aut(A) such that α(B 1 ) = B 2 . The C * -superrigidity problem for torsion-free virtually abelian groups -i.e., to recover the group from its group C * -algebras -can be solved assuming a classification up to conjugacy for Cartan subalgebras in certain subhomogeneous C * -algebras. This parallels the importance of classification results for Cartan subalgebras in proofs of W * -superrigidity [5,15], although taking very different forms. Work in [18] strongly motivates us to study Cartan subalgebras in subhomogeneous C * -algebras.
Further motivation for the study of Cartan subalgebras in C * -algebras comes from the recent breakthrough results in the structure and classification theory of simple nuclear C * -algebras achieved by many hands; see among others [11,14,25,26,36,42]. Every simple nuclear C * -algebras that is classifiable in the sense of the Elliott program has a Cartan subalgebra. For UCT Kirchberg algebras this follows either by work of Spielberg [41] or by combining results of Katsura [17] and Yeend [44,45], while for classifiable finite C * -algebras this was proven by Li [21]. While existence of Cartan algebras is settled, their classification is a wide open problem. Even for the Jiang-Su algebra Z, [16], which plays a key role in the structure and classification theory of simple nuclear C * -algebras and which can be constructed as an inductive limit of prime dimension drop algebras, we only know that uniqueness of Cartan subalgebras does not hold [9,21].
The purpose of this article is to understand Cartan subalgebras of stabilised dimension drop algebras up to conjugacy by an automorphism. For this, we take a purely C * -algebraic approach and classify a large class of Cartan subalgebras of dimension drop algebras and their stabilisations. Dimension drop algebras are arguably among the most simple and at the same time most important subhomogeneous C * -algebras, so that they form a natural starting point for a systematic study of Cartan subalgebras in general subhomogeneous C * -algebras. We adopt the following notation from [16] for stabilised dimension drop algebras throughout the article, denoting the algebra of matrices of size m × m by M m , Our first main result provides a parametrisation of conjugacy classes of non-degenerate Cartan subalgebras in terms of classical combinatorial objects explained below. We emphasise that we classify Cartan subalgebras up to conjugacy by an automorphism. This is in contrast to the usual conjugation by unitary results in von Neumann algebras -see [38,40] for an account on the model theoretic complexity of 'conjugacy by an automorphism' and 'conjugacy by a unitary'. As a consequence, our strategy for the classification in Theorem B is substantially different from the strategy employed in a von Neumann algebraic setup. We first provide a list of Cartan subalgebras in dimension drop algebras which are explicitly described. Within this class we provide a classification result. Only then we prove that every non-degenerate Cartan subalgebra of a dimension drop algebra is conjugate to one coming from our list.
Theorem B (See Corollary 6.10). Conjugacy classes of non-degenerate Cartan subalgebras in I m,n,o are parametrised by congruence classes of matrices in M(mo, n, no, m).
Here M(a, b, c, d) denotes the set of matrices of size a × c with entries in the natural numbers such that each of the a rows sums to b and each of the c columns sums to d. Note that in order to obtain a non-empty set, the compatibility condition ab = cd is required, see Notation 6.8. Two matrices A, B ∈ M m,n (N) are congruent if there are permutation matrices ρ 1 ∈ Sym(m) and ρ 2 ∈ Sym(n) such that A = ρ 1 Bρ 2 or, in case m = n, A t = ρ 1 Bρ 2 , see Definition 6.6. Thanks to the parametrisation provided by Theorem B, we are able to count Cartan subalgebras in some families of dimension drop algebras. This allows us to solve the C * -algebraic analogue of a well-known open problem in von Neumann algebras, namely to find for each n ∈ N some I I 1 factor with exactly n Cartan subalgebras up to (unitary) conjugacy. In the von Neumann algebraic setting only partial results addressing this problem are available [8,19,27,31,40]. In the C * -algebraic context, Li-Renault [24] translate conjugacy of Cartan subalgebras of homogeneous C * -algebras into a problem about principal bundles and then make use of known results on principle bundles to provide examples of homogeneous C * -algebras with exactly p(n) Cartan subalgebras up to conjugacy for any n ∈ N 1 , where p(n) denotes the number of partitions of n.
Theorem C (See Corollary 6.14). For every n ∈ N there is a subhomogeneous C * -algebra that has exactly n Cartan subalgebras up to conjugacy.
In Remark 6.15, we show that Theorem C also allows one to construct C * -algebras with exactly continuum many Cartan subalgebras up to conjugacy.
Comparing von Neumann algebraic with C * -algebraic Cartan subalgebras, it becomes apparent that in the latter context there is one fundamental obstruction to conjugacy. While all separable abelian diffuse (i.e., without minimal projections) von Neumann algebras are pairwise isomorphic, separable abelian C * -algebras are classified by the homeomorphism type of their spectrum. In [18], it was already observed that plain uniqueness of Cartan subalgebra results are too much to be expected in the setting provided by C * -superrigidity of virtually abelian groups. The second most optimistic approach tries to prove that two Cartan subalgebras of a subhomogeneous C * -algebra are conjugate if and only if their spectra are homeomorphic -this statement would suffice to prove C * -superrigidity of virtually abelian groups based on results in [18]. Surprisingly, this statement holds true in all dimension drop algebras and most stabilised dimension drop algebras, when one restricts to non-degenerate Cartan subalgebras.
Theorem D (See Theorem 7.8). Let I m,n,o be a stabilised dimension drop algebra such that either (m, n) = (2, 2) or o = 1. Then two non-degenerate Cartan subalgebras of I m,n,o are conjugate by an automorphism if and only if their spectra are homeomorphic.
In Proposition 7.9, we prove that the excluded cases I 2,2,o , for o 2, do not obey the conclusion of Theorem D, hence rendering a general classification of Cartan subalgebras by their spectrum too optimistic. Nevertheless, Theorem D is enough evidence to justify the following question.
Question. In which subhomogeneous C * -algebras are two Cartan algebras conjugate if and only if their spectra are homeomorphic?
This article has six sections. After the introduction and some preliminaries, we study in § 3 Cartan subalgebras of one-sided dimension drop algebras I 1,n,m . It is at this point where we introduce the notion of non-degenerate Cartan subalgebras in stabilised dimension drop algebras, and then prove that one-sided dimension drop algebras have a unique non-degenerate Cartan subalgebra up to conjugacy. This will be an important tool in the remaining sections. Section 4 is devoted to the study of non-degenerate Cartan subalgebras in stabilised dimension drop algebras, which culminates in Theorem A. In § 5, we introduce a class of Cartan subalgebras of dimension drop algebras that we term twisted standard Cartan subalgebras and classify them up to conjugacy. We conclude that section by proving that every non-degenerate Cartan subalgebra of a stabilised dimension drop algebra is conjugate to a twisted standard Cartan subalgebra. In § 6, we provide a parametrisation of conjugacy classes of twisted standard Cartan subalgebras by congruence classes of matrices as described above. This leads to the proof of Theorem B. Furthermore, it allows us to deduce some explicit counting results. In particular, we obtain Theorem C. In § 7, we study the spectra of twisted standard Cartan subalgebras and prove Theorem D.

Preliminaries
In this section, we fix some notation used throughout this work and recall some important definitions. For a natural number n 1, we write n = {1, . . . , n}. If X is any set, we denote by Sym(X ) the set of all bijections of X . For short, Sym(n) = Sym(n). Identifying a permutation σ ∈ Sym(n) with its permutation matrix satisfying we obtain an embedding Sym(n) → U(n). We moreover denote by e i j ∈ M n the matrix whose i jth entry is 1 and whose other entries are all zero. Similarly, e i ∈ C denotes the ith standard vector.
Given an inclusion of C * -algebras B ⊂ A, the normaliser of B in A is is a closed subset of A that is also closed under multiplication and the * -operation, and the unitary normaliser We also denote by Aut(B ⊂ A) the set of all * -automorphisms of A that preserve B setwise, that is, α ∈ Aut(A) belongs to Aut(B ⊂ A) exactly if α(B) = B. Furthermore, we call B ⊂ A a MASA if B is a maximal abelian C * -subalgebra of A.  Recall that a C * -algebra A is called n-subhomogeneous for n ∈ N, if all of its irreducible representations have dimension at most n. We say that A is subhomogeneous if it is n-subhomogeneous for some n ∈ N. Furthermore, A is called homogeneous if there exists some n ∈ N such that all irreducible representations of A have dimension n. Subhomogeneous C * -algebras are exactly the C * -subalgebras of homogeneous C * -algebras; see [6,IV.1.4.3]. Definition 2.3 (Cf. [16]). The dimension drop algebra with parameters m, n ∈ N 1 is More generally, a stabilised dimension drop algebra is for parameters m, n, o ∈ N 1 .
Let X ⊂ C([0, 1], M n ) be a subset. For a subinterval I ⊂ [0, 1], we define We also write X {t} = X t . For x ∈ X , we denote by We conclude this section with a few words about Cartan subalgebras in homogeneous C * -algebras over an interval. In [24], the relation between Cartan subalgebras in homogeneous C * -algebras over a space X and principal Aut(D n ⊂ M n )-bundles over X was pointed out. In particular, in homogeneous C * -algebras over contractible spaces, there is a unique Cartan subalgebra. Theorem 2.4 (Cf. [24, § 2]). Let A be a homogeneous C * -algebra over a contractible space. Then A admits a unique Cartan subalgebra up to conjugacy.
We need a stronger result for the special case of homogeneous C * -algebras over an interval, which immediately follows from Li-Renault's. Proof. Let B ⊂ A be a Cartan subalgebra. Since the interval I is contractible, we can apply Theorem 2.4 to obtain an automorphism α ∈ Aut(A) such that α(B) is the standard Cartan subalgebra C = C 0 (I, D m ). Composing α with an automorphism in Aut(C ⊂ A), we may assume that α| C 0 (I ) = id. We can then consider α as a continuous map I → Aut(M m ) ∼ = U(m)/T. Let (I n ) n∈N be an ascending sequence of closed intervals in I such that n I n = I . Since U(m) U(m)/T has the structure of a principal T-bundle, it has the homotopy lifting property, so that we find for every n ∈ N a lift u n ∈ C(I n , U(m)) for α| I n . In other words, α| I n = Ad u n . Now define v 1 = u 1 . If v n is defined, consider f n := (u n+1 | I n ) * v n ∈ C(I n , T). Extend f n to an element g n ∈ C(I n+1 , T) and set v n+1 = u n+1 g n . We obtain a family (v n ) n of unitaries with v n ∈ C(I n , U(m)) satisfying v n+1 | I n = v n and Ad v n = α| I n ∈ Aut(C(I n , D m )). We can now define v ∈ C b (I, M m ) by the requirement v| I n = v n for all n ∈ N and obtain a unitary satisfying Ad v = α. This proves that B is unitarily conjugate to the standard Cartan subalgebra C ⊂ A via the unitary v ∈ M(A).

One-sided dimension drop algebras
In this section, we are going to investigate uniqueness of Cartan subalgebras in one of the easiest possible subhomogeneous C * -algebras, which we call here one-sided dimension drop algebras.
the standard Cartan subalgebra of J m,n . It is a Cartan subalgebra indeed, as the following argument shows.
First we show that C m,n ⊂ J m,n is a MASA. This follows from the fact that for every t ∈ (0, 1], the fibre D m ⊗ D n = (C m,n ) t ⊂ (J m,n ) t = M m ⊗ M n is a MASA. So if f ∈ C m,n ∩ J m,n , then f (t) ∈ D m ⊗ D n for all t ∈ (0, 1] and hence This shows that f ∈ C m,n . Second, the natural conditional expectation C([0, 1], M m ⊗ M n ) → C([0, 1], D m ⊗ D n ) restricts to a faithful conditional expectation J m,n → C m,n . It remains to show that C m,n is regular in J m,n .
Let f ∈ J m,n and ε > 0. Since f is continuous, there is 1] ) and numbers denotes the set of elementary tensors. Now, extend eachf i to the unique element f i ∈ J m,n that is affine on We conclude that each f i ∈ N J m,n (C m,n ), which finishes the proof that C m,n is a Cartan subalgebra of J m,n .
We first prove a uniqueness of Cartan subalgebras result in one-sided dimension drop algebras of the form J 1,n . For this, we introduce the following notation. If u ∈ C b ((0, 1], M n ) is a unitary, then Ad u induces a unique automorphism of J 1,n . Indeed, if f ∈ J 1,n and u ∈ C b ((0, 1], U(n)), then f (t) → λ ∈ C as t → 0 implies that also u t f (t)u * t → λ as t → 1.

Lemma 3.4. For every Cartan subalgebra
such that (Ad u)(B) = C 1,n . In particular, J 1,n has a unique Cartan subalgebra up to conjugacy by an automorphism.
Proof. A similar argument as in Example 3.3 shows that is a Cartan subalgebra. Hence, Corollary 2.5 provides us with a unitary u ∈ C b ((0, 1], M n ) such that u B (0,1] u * = C 0 ((0, 1], D n ). As discussed above, Ad u induces a unique automorphism of J 1,n , which then satisfies (Ad u)(B) = C 1,n .
We would like to reduce considerations about Cartan subalgebras in general one-sided dimension drop algebras, to the case considered in Lemma 3.4. We can do so for the class of so-called non-degenerate Cartan subalgebras in stabilised dimension drop algebras described in the next definition, which will be treated in the rest of the article. Gábor Szabó kindly pointed out to us that non-degenerate Cartan subalgebras in I m,n,o are exactly the C * -diagonals in the sense of Kumjian [20], which are precisely the Cartan subalgebras with the unique extension property in the sense of Anderson [1]. In stabilised dimension drop algebras with coprime parameters, every Cartan subalgebra is non-degenerate, as we see in Theorem 4.5.
Before proving uniqueness of non-degenerate Cartan subalgebras in arbitrary stabilised dimension drop algebras, let us analyse their normalisers. Proof. Let i, j ∈ {1, . . . , m} and i = j. Since span N J m,n (B) ⊂ J m,n is dense, span(e ii ⊗ 1)N J m,n (B)(e j j ⊗ 1) is dense in (e ii ⊗ 1)J m,n (e j j ⊗ 1). It follows that is closed under scalar multiplication, we can assume that c = 1.
Since e ii ⊗ 1, e j j ⊗ 1 ∈ B ⊂ N J m,n (B), we actually have x ∈ N J m,n (B). In particular, x * x ∈ B and hence |x| ∈ B. Note that actually |x| ∈ (e j j ⊗ 1)B(e j j ⊗ 1). By continuity, we find some ε > 0 such that |x| t − (e j j ⊗ 1) . This finishes the proof of the lemma. Proof. Let B ⊂ J m,n be a non-degenerate Cartan subalgebra. We find a unitary u ∈ J m,n such that (u Bu . We can extend v to a unitary in J m,n such that vu| [2ε,1] = 1. Hence, from now on we may assume that there is ε ∈ (0, 1] such that B [0,ε] contains the constant function with value e ii ⊗ 1 for all i ∈ {1, . . . , m}. As (e 11 ⊗ 1)(J m,n ) [0,ε] (e 11 ⊗ 1) ∼ = J 1,n , Lemma 3.4 applies to its Cartan subalgebra (e 11 ⊗ 1)B [0,ε] . We thus obtain a partial isometry u 1 ∈ C b ((0, ε], M m ⊗ M n ) whose support and range projection equal e 11 ⊗ 1, which normalises (J m,n ) [0,ε] and satisfies Put . By construction, Ad u defines an automorphism of (J m,n ) [0,ε] . By making ε > 0 smaller if necessary, we may assume by Lemma 3.6 that e i1 ⊗ 1 and e 1i ⊗ 1 are elements of We can now extend Ad u to an automorphism of J m,n which is the identity on [2ε, 1]. Conjugating B by this automorphism, we may assume that Hence, multiplying u with some unitary from C([δ, 1], N M m ⊗M n (D m ⊗ D n )), we may assume that u| [δ,ε] ≡ 1. Then u extends to a unitary in J m,n that satisfies u| [0,ε] ≡ 1. This unitary conjugates B onto the standard Cartan subalgebra of J m,n .

Non-degenerate Cartan subalgebras
The main purpose of this section is to prove Theorem 4.5 characterising those stabilised dimension drop algebras, in which every Cartan subalgebra is non-degenerate. We start with a proposition characterising non-degenerate Cartan subalgebras as C * -diagonals in the sense of Kumjian [20]. Proof. Let E : J m,n → B be the unique conditional expectation onto B. Using Theorem 2.4, one checks that Since B ⊂ J m,n is a Cartan subalgebra, its normaliser spans J m,n . In particular, Write H = C m ⊗ C n and consider the natural action of M m ⊗ M n on H . Let {ξ i j | 1 i m, 1 j n} be an orthonormal basis of H whose associated orthogonal projections p i j generate the MASA C. Define linear subspaces Observe that for each u ∈ G and i, j there are some k, l such that (u ⊗ 1)ξ i j ∈ Tξ kl . Assume that for fixed i, j there are u 1 , In conclusion, F(u ⊗ 1) ∈ T so that u ∈ T follows from faithfulness of F. This shows that Further, the sum of all H i j coincides with H . Observe also that the equality implies that two subspaces H i j and H kl are either equal or orthogonal. This shows that m 2 divides mn and hence m divides n.
We continue with the prototypical example of a degenerate Cartan subalgebra in J m,m , which will be an important ingredient in the proof of Theorem 4.5.
• In order to show that C([0, 1], B) ∩ J m,m is regular in J m,m , we first observe that its restriction to (0, 1] is a standard Cartan subalgebra, which is hence regular. It thus suffices to show that every element Let us now construct B. We write H = M m for the m × m-matrices endowed with the scalar product T ,Ŝ = tr(S * T ). Let π : M m → B(H ) be the * -representation induced by left multiplication. Observe that under the isomorphism between H and C m ⊗ C m sending e i j to e i ⊗ e j , π corresponds to the first tensor factor embedding into M m ⊗ M m ∼ = B(H ).
Let λ ∈ T be a primitive mth root of unity and consider the element U = diag(1, λ, λ 2 , . . . , λ m−1 ) ∈ M m . Further let V be defined as the permutation matrix associated to the cycle (1 2 · · · m) ∈ Sym(m). Observe the commutation relation Note also that elements of the form U i V j for i, j ∈ {1, . . . , m} form an orthonormal basis of H . We consider the rank one projections p i j : H → C U i V j and let B be the subalgebra of B(H ) generated by them. Note that B is an abelian subalgebra of dimension m 2 and hence a MASA. For x ∈ M m andŷ ∈ H it holds that Hence, the conditional expectation E B : B(H ) → B restricted to π(M m ) is calculated as showing that E B | π(M m ) = tr. Next note that M m = span{U i V j | i, j ∈ {1, . . . , m}} and the commutation relation U V = λV U yields that π(U i V j ) ∈ N B(H ) (B) as π(U i V j ) p kl π(U i V j ) * ∈ C p (k+i),(l+ j) , where the indices are considered mod m.
n is a degenerate Cartan subalgebra, since its dimension at 0 equals mn d 2 .
Theorem 4.5. The following three statements are equivalent for a stabilised dimension drop algebra I m,n,o .
• Every Cartan subalgebra in I m,n,o is non-degenerate.
• Every Cartan subalgebra of I m,n,o is a C * -diagonal, that is, it has the unique extension property.
Proof. By Proposition 4.1, a Cartan subalgebra of I m,n,o is non-degenerate if and only it is a C * -diagonal. It hence suffices to prove the equivalence between the first two statements of the theorem. Assume that I m,n,o contains a degenerate Cartan subalgebra B. Possibly replacing I m,n,o by I n,m,o , we may assume that dim B 0 = mo. Restricting to the interval [0, 1 2 ], we obtain a degenerate Cartan subalgebra of J mo,n . Considering the case of one-sided dimension drop algebras, we show that mo and n are not coprime, which implies that either m and n or o and n are not coprime.
We let m, n ∈ N be arbitrary and assume that B ⊂ J m,n is a degenerate Cartan subalgebra. Our aim is to show that m, n are not coprime. Since B is degenerate, there is a minimal projection p 0 ∈ B 0 with rank strictly bigger than one. Let p 0 , . . . , p K ∈ B 0 be the minimal projections and k ∈ {1, . . . , K }. Since the span of N J m,n (B) is dense in J m,n , it follows that there is a non-zero element x ∈ p 0 N M m ⊗1 (B 0 ) p k . It follows that x * x ∈ p k B 0 p k = C p k and x x * ∈ p 0 B 0 p 0 = C p 0 are non-zero elements. So rank( p k ) = rank( p 0 ). From it follows that rank( p 0 ) divides m. Let us write d = rank( p 0 ).
Extending p 0 to an element p ∈ B such that p| [0,ε] is a projection, we consider ( p Bp) [0,ε] ⊂ ( p J m,n p) [0,ε] , which is isomorphic with a Cartan subalgebra C ⊂ J d,n satisfying C 0 = C1. So Lemma 4.2 applies to show that d divides n. Since d = 1, this shows that m and n have a non-trivial common divisor.
Let us now assume that m, n, o are not pairwise coprime. Possibly changing the role of m and n, we may assume without loss of generality that there is some d = 1 that divides mo and n. Example 4.4 provides us with a degenerate Cartan subalgebra is a degenerate Cartan subalgebra in I m,n,o .

Twisted standard Cartan subalgebras
In this section, we parametrise non-degenerate t } is a Cartan subalgebra of I m,m -this is proven verbatim as is Proposition 5.1. Now u(e 11 ⊗ 1)u * ∈ B is a non-trivial projection.
We are now going to expand the idea of Example 5.3 in order to obtain a finite family of Cartan subalgebras of I m,n,o parametrised by elements of Sym(m × n × o). In Theorem 5.10, we classify these up to conjugacy by an automorphism and in Theorem 5.11 we show that every non-degenerate Cartan subalgebra of I m,n,o is conjugate to one of the following kinds.  Next, we argue that the choice of 1 3 and 2 3 is arbitrary. For every 0 < t 1 < t 2 < 1, there is an orientation preserving homeomorphism of [0, 1] satisfying 1 3 → t 1 and 2 3 → t 2 . It extends to an automorphism α of I m,n,o , since it must fix the endpoints of [0, 1]. Replacing 1 3 by t 1 and 2 3 by t 2 , we can apply the same construction as above to obtain a Cartan subalgebra which is conjugate to B σ by the automorphism α.
Since α is an automorphism and B σ is a MASA, this finishes the proof of the lemma.
Before we proceed to prove our next theorem, let us fix some notation for wreath products.

S. Barlak and S. Raum
Notation 5.9. If F is some group, then F ∼ Sym(n) = n i=1 F Sym(n) denotes the wreath product, where Sym(n) acts by permuting the copies of F. We also use the notation Sym(n) Fixing m, n, o ∈ N 1 , we obtain an embedding Sym(m × o) ∼ Sym(n) Sym(m × n × o). Concretely, an element of Sym(m × o) ∼ Sym(n) can be described as a pair (σ 1 , σ 2 ) with σ 1 ∈ Sym(m × o) acting on the first and last coordinate of m × n × o and σ 2 : m × n × o → n is a map such that σ 2 (i, · , k) ∈ Sym(n) for all i ∈ m, k ∈ o. This way, σ 2 acts on the second coordinate of m × n × o. Similarly, we obtain an embedding Sym(m) We remark that considering permutations as unitaries, we have the equalities and similarly

741
Let now α ∈ Aut(I m,n,o ) be an orientation preserving automorphism in the sense of Notation 5.7. Assume that α satisfies α(B σ ) = B π . We can apply Lemma 5.6 and assume that α| C([0,1]) = id. Let us consider the elements u σ , u π ∈ C([ In particular, its permutational part is constant. Let us denote it by [α t ], t ∈ [ 1 3 , 2 3 ]. Evaluating at 1 3 and 2 3 , we obtain the equality The definitions of u σ and u π give then π −1 Since and thus its permutational part is constant, say [α (0,1/3] ] ≡ ρ 1 . Now α (0,1/3] is the restriction of an automorphism of I m,n,o , so that S. Barlak Multiplying u with an element from C( , we may hence assume that u 2 3 = 1. Similarly, we obtain Since T m × T n × T o is connected, we may multiply u with a function from and that B is the twisted standard Cartan subalgebra associated with π −1 . This completes the proof of the theorem.

Counting Cartan subalgebras
Let us start this section by counting conjugacy classes of non-degenerate Cartan subalgebras in the example of I 2,2 , which is the easiest non-trivial case of all dimension drop algebras.
Example 6.1. I 2,2 has at least 2 non-degenerate Cartan subalgebras up to conjugacy by an automorphism. These are the standard Cartan subalgebra and the Cartan subalgebra constructed in Example 5.3. Indeed, both are not conjugate to each other, since their spectra are not homeomorphic. Theorem 5.11 implies that these are all non-degenerate Cartan subalgebras of I 2,2 up to conjugacy. For this, we have to prove that has cardinality 2. This follows from a counting argument. The group has cardinality 2 · 2 · 2 = 8, while Sym(2 × 2) has cardinality 4 · 3 · 2 = 24. It follows that the quotient Sym(2 × 2)/Sym(2) ∼ Sym(2) has cardinality 3. Since Sym(2) ∼ Sym(2) acts non-trivially on this set, we infer that the orbit space has cardinality at most 2. Combined with the initial observation that I 2,2 has at least 2 non-conjugate Cartan subalgebras, this shows that I 2,2 has exactly 2 non-conjugate non-degenerate Cartan subalgebras.

Matrix combinatorics and Cartan subalgebras
In order to count conjugacy classes of non-degenerate Cartan subalgebras in any other -more complicated -case than the one described in Example 6.1, it is necessary to improve on the inexplicit parametrisation provided by double cosets of permutations. It turns out to be advantageous to replace permutations by their permutation matrices and then compress their information. Indeed, Theorem 6.7 will describe conjugacy classes of non-degenerate Cartan subalgebras of dimension drop algebras in terms of certain congruence classes of matrices resulting from the following procedure. This in turn will lead to the desired parametrisation in Theorem 6.10.
We start with a lemma that will be used in the proof of key Proposition 6.4. Lemma 6.3. Let C, D ∈ M m,n ({0, 1}) be matrices such that in each row and each column there is at most one non-zero entry. If i∈m, j∈n C i j = i∈m, j∈n D i j , then there are permutation matrices ρ 1 ∈ Sym(m), ρ 2 ∈ Sym(n) such that C = ρ 1 Dρ 2 .
Proof. First note that the assumptions imply that C and D have the same number of non-zero rows and non-zero columns. We can hence find a permutation matrix ρ 2 ∈ Sym(n) such that C and Dρ 2 have the same non-zero columns. Further, we can find a permutation matrix ρ 1 ∈ Sym(m) such that C and ρ 1 Dρ 2 have the same non-zero rows. Since the non-zero columns of ρ 1 Dρ 2 and Dρ 2 are the same, we infer that C and ρ 1 Dρ 2 have the same non-zero rows and non-zero columns. Restricting to these non-zero rows and columns we obtain two {0, 1}-valued matrices with exactly one non-zero entry in each row and each column -these are permutation matrices. So we can replace ρ 1 with some other ρ 1 ∈ Sym(m) and obtain C = ρ 1 Dρ 2 . Proposition 6.4. Let σ, π ∈ Sym(m × n × o). Let A, B ∈ M m×o, n×o (N) be the reduced matrices of σ and π , respectively. Then the following statements are equivalent. (ii) There are elements ρ 1 ∈ Sym(m × o), ρ 2 ∈ Sym(n × o) such that ρ 1 Aρ 2 = B.
Proof. We first show that (i) implies (ii). Let and denote by C the reduced matrix of σρ. Consider ρ as an element in Sym(m × n × o) and denote by ρ 1 : m × n × o → m and ρ 2 ∈ Sym(n × o) the first and second coordinate of ρ as described in Notation 5.9. Recall that for fixed ( j, k) ∈ n × o, the map ρ 1 (·, j, k) : m → m is a bijection. For all i ∈ m, j ∈ n, k, k ∈ o (multiplication with permutation matrices).
So C = Aρ 2 . Similarly for ρ ∈ Sym(m × o) ∼ Sym(n) written in coordinates ρ 1 ∈ Sym(m × o) and ρ 2 : m × o × n → n, we obtain that the associated matrix of ρσ is ρ 1 A. This shows the first implication.
Let us now prove that (ii) implies (i). To this end, take σ, π ∈ Sym(m × n × o) with reduced matrices A and B, respectively and assume that there are ρ 1 ∈ Sym(m × o) and ρ 2 ∈ Sym(n × o) such that ρ 1 Aρ 2 = B. By the previous paragraph, we can replace π by ρ −1 1 πρ −1 2 and assume that A = B. By the definition of reduced matrices, for all (i, k) ∈ m × o and all ( j , k ) ∈ n × o we have i ∈m, j∈n σ (i, j,k)(i , j ,k ) = i ∈m, j∈n π (i, j,k)(i , j ,k ) .
Put differently, the matrices C, D ∈ M n, m ({0, 1}) defined by C ji = σ (i, j,k)(i , j ,k ) and D ji = π (i, j,k)(i , j ,k ) have the same sum of their entries. C and D are matrices with entries in {0, 1} and on each row and on each column there is at most one non-zero entry. Hence, Lemma 6.3 provides us with permutations ρ 1 (i, ·, k) ∈ Sym(n) and ρ 2 (·, j , k ) ∈ Sym(m) such that C = ρ 1 (i, ·, k)Dρ 2 (·, j , k ). Since (i, k) ∈ m × o and ( j , k ) ∈ n × o were arbitrary, we can put these permutations together and obtain elements ρ 1 ∈ m×o Sym(n) and ρ 2 ∈ n×o Sym(m) such that σ = ρ 1 πρ 2 . This finishes the proof of the lemma. So the reduced matrix of νσ −1 ν satisfies j,k∈m which proves the lemma.
Let us give a name to the relation on M m,n (N) described by Proposition 6.4 and Lemma 6.5. It does resemble, but is not exactly the same as usual notions of congruence: while on non-square matrices it agrees with a common notion of matrix congruence, we additionally call a square matrix and its transpose congruent. Proof. This follows from combining Theorem 5.10 with Proposition 6.4 and Lemma 6.5.
In order to obtain a useful parametrisation of non-degenerate Cartan subalgebras, we have to identify those matrices that can arise as the reduced matrix of a permutation. The following notation from matrix combinatorics allows us to describe them concisely.

S. Barlak and S. Raum
The (i, k), ( j , k )th entry ofÃ is now given as where (1 2 · · · m) ∈ Sym(m) and (1 2 · · · n) ∈ Sym(n) are the respective full shift permutation matrices. Since the sum in every row of A is n and the sum of every column of A is m, it follows that every row and every column ofÃ, when interpreted as a matrix in M m×n×o ({0, 1}), has exactly one non-zero entry. In other words,Ã is a permutation matrix and thus corresponds to some π ∈ Sym(m × n × o). By construction, A is the reduced matrix of π . This concludes the proof. Proof. This follows from Theorems 5.11 and 6.7 together with Proposition 6.9.

The asymptotic number of Cartan subalgebras
Theorem 6.10 in principle allows to apply results from enumerative combinatorics providing asymptotic formulae for the cardinality of M(a, b, c, d) (see Notation 6.8). We refer to [7] and references therein for the reader who wants to know more about this topic. However, there is no exact formula for the number of such matrices. Further, the congruence relation introduced in Definition 6.6 does not make part of the combinatorics literature, which obstructs a direct application. Crude lower bounds on the number of congruence classes in M(mo, n, no, m) can for example be given by appealing to the possible entries of a matrix in M(mo, n, no, m) as a subset of {1, . . . , n}. Despite this lower bound, it appears to be an interesting combinatorial problem to derive asymptotic formula for congruence classes in M(a, b, c, d).

Explicit formula
In Example 6.1, we saw that the dimension drop algebra I 2,2 has exactly 2 Cartan subalgebras up to conjugacy. This is the base case for two one-parameter series of dimension drop algebras that admit an explicit formula for the number of their Cartan subalgebras. Both results have interesting consequences.
The next proposition counts the number of Cartan subalgebras in stabilisations I 2,2,o of I 2,2 , making use of Theorem 6.10. In § 7, these results will provide examples of stabilised dimension drop algebras for which we will not be able to recover Cartan subalgebras up to conjugacy from the homeomophism type of their spectra. We denote by p the partition function, which counts the number of partitions of a non-negative integer. Proposition 6.11. In I 2,2,o there are p(2o) non-degenerate Cartan subalgebras up to conjugacy.
Proof. By Theorem 6.10, it suffices to classify congruence classes of matrices in M(2o, 2, 2o, 2). We provide a symmetric normal form for congruence classes in M(2o, 2, 2o, 2) only using the Sym(2o) × Sym(2o) action. A normal form is given by block diagonal matrices with the following blocks ordered by increasing size. Fix o ∈ N and assume that we have normal forms in M(2o , 2, 2o , 2) for o < o. Let A ∈ M(2o, 2, 2o, 2). We may replace A by a congruent matrix, so that A 11 = 0 holds. If A 11 = 2, then it is the only non-zero entry in the first row and the first column of A, so A is block diagonal and we may apply the induction hypothesis to the matrix obtained from A by erasing the first row and the first column in order to obtain a normal form for A. If A 11 = 1, then we may further arrange for A 12 = 1 = A 21 by passing to a congruent matrix. Assume now that we arrived at a matrix congruent to A which satisfies A 11 = A k,k−1 = A k−1,k = 1 for all k k 0 for some k 0 2. If A k 0 ,k 0 = 1, then A is block diagonal and we obtain a normal form as before. Otherwise, we can replace A by a congruent matrix that satisfies A k 0 ,k 0 +1 = A k 0 +1,k 0 = 1. This algorithm terminates and proves that A is congruent to a normal form as described before.
Next we show that the provided normal forms are pairwise non-congruent. To this end, associate with a matrix in M(2o, 2, 2o, 2) the graph whose vertices are indexed by 2o × 2o and whose edges are given by the rule (i, j) ∼ (i , j ) if and only if the following two conditions are satisfied: A i j = 1 = A i j and at the same time i = i or j = j . Then the multiset of the size of connected components of this graph is a congruence invariant of A. It distinguishes the normal forms provided before, because the block of size k × k, k 2 described above produces a single non-trivial connected component with 2k vertices. Since the size of the blocks of our normal forms runs through all positive natural numbers and they have to fill the diagonal of a 2o × 2o matrix, we conclude that there are p(2o) congruence classes in M(2o, 2, 2o, 2). This finishes the proof of the proposition. Remark 6.12. Proposition 6.11 provides the exceptional examples of stabilised dimension drop algebras in which isomorphism and conjugacy of non-degenerate Cartan subalgebras is not the same. This is formally stated in Proposition 7.9, which stands in contrast to the positive result provided by Theorem 7.8.
The following proposition treats the other class of dimension drop algebras admitting an explicit computation of the number of their Cartan subalgebras. Proposition 6.13. The dimension drop algebra I 2,n has exactly n 2 + 1 non-degenerate Cartan subalgebras up to conjugacy. In particular, if n is odd, then I 2,n has n+1 2 Cartan subalgebras up to conjugacy.
Proof. Example 6.1 showed that there are exactly 2 non-degenerate Cartan subalgebras in I 2,2 , so that we may assume n 3. By Theorem 6.10, we have to count congruence classes in M(2, n, n, 2). We provide a normal form. Let A ∈ M(2, n, n, 2). Let k = |{ j ∈ n | A 1 j = 2}|. Replacing A by a congruent matrix, we may assume that A 11 , A 12 , . . . , A 1k = 2.
Since each column of A sums to 2, all entries of A are elements of {0, 1, 2}. Moreover, each row of A sums to n, so that we have k = |{ j ∈ n | A 1 j = 0}|. Replacing A by a congruent matrix, we may assume that A 1,k+1 , A 1,k+2 , . . . , A 1,2k = 0. Note that A 1,2k+1 , . . . , A 1,n = 1 follows. Moreover, A 2 j = 2 − A 1 j for all j ∈ n. This is our normal form for A. Two different normal forms are distinguished by |{(i, j) ∈ 2 × n | A i j = 2}| ∈ 2Z, which hence is a complete invariant for congruence classes in M(2, n, n, 2). So indeed there are exactly n 2 + 1 congruence classes. Let us now consider the case when n is odd. In this case, Theorem 4.5 shows that all Cartan subalgebras in I 2,n are non-degenerate. Further, which finishes the proof of the proposition.
Corollary 6.14. For every n ∈ N there is a subhomogeneous C * -algebra that has exactly n Cartan subalgebras up to conjugacy.
Proof. In [24, § 2.2], examples of homogeneous C * -algebras without any Cartan subalgebras are presented. Furthermore, every homogeneous C * -algebras over a contractible space has a unique Cartan subalgebra up to conjugacy; see Theorem 2.4. For n 1, the dimension drop algebra I 2,2n+1 has exactly n + 1 Cartan subalgebras up to conjugacy by Proposition 6.13.
Remark 6.15. Based on Corollary 6.14 it is possible to provide examples of C * -algebras with exactly continuum many Cartan subalgebras up to conjugacy. Let A = n 2 I 2,2n+1 . Every automorphism α ∈ Aut(A) satisfies α(I 2,2n+1 ) = I 2,2n+1 for all n 2, since I 2,2n+1 is 2n + 1-subhomogeneous. Further, it is easy to check that a Cartan subalgebra of A is a direct sum of Cartan subalgebras of I 2,2n+1 for n 2. Combining these two observations with Proposition 6.13, we find that Cartan subalgebras of A are parametrised by the product set n 2 1 + n whose cardinality is the continuum.

Isomorphism and conjugacy
Our next aim is to show that although dimension drop algebras do not have a unique Cartan subalgebra up to conjugacy, often the next to best possible result is true. Often Cartan subalgebras in dimension drop algebras are classified by their spectrum -which is their only intrinsic invariant. We start by giving a concrete model for the spectrum of the twisted standard Cartan subalgebras considered in Example 5.4 where the equivalence relation ∼ is given by the following three types of identifications.
(i, j, k, 0) ∼ (i, j , k, 0) for all i ∈ m, j, j ∈ n and k ∈ o, the valency of v ∈ V( ). A graph all of whose vertices have the same valency n ∈ N is called n-regular.
We say that two vertices v 1 , v 2 of are adjacent if there is an edge e of such that a(e) = {v 1 , v 2 }. A bi-partite graph is a graph admitting a partition V( ) = V 1 V 2 such that no vertex from V i is adjacent to a vertex of V i , i ∈ {1, 2}. If all vertices from V 1 have valency m and all vertices from V 2 have valency n, we call an (m, n)-semi-regular bi-partite graph.
If is a finite bi-partite graph and v 1 , v 2 , . . . , v m , w 1 , . . . , w n is an enumeration of the vertices such that the sets {v 1 , . . . , v m } and {w 1 , . . . , w n } witness the fact that is bi-partite, then the adjacency matrix of is the m × n matrix whose i, jth entry is the number of edges connecting v i and w j . Vice versa, if A ∈ M m,n (N), then the bi-partite graph associated with A has vertices indexed by (m × {r}) ∪ (n × {c}) and edges indexed by (i, j, k) with k ∈ A i j and a(i, j, k) = {(i, r), ( j, c)}.
Notation 7.4. Let be a graph as described in the formalism of Notation 7.3. We adopt the following notation for the geometric realisation | | of . It is the unique up to homeomorphism 1-dimensional CW-complex whose 0-cells are indexed by V( ) and whose 1-cells are indexed by E( ), with the 1-cell of e ∈ E( ) attached to a(e). Note that the latter could be a one-point set, in which case the 1-cell is glued to this single 0-cell, giving rise to a loop. Proposition 7.5. Let σ ∈ Sym(m × n × o). Let be the bi-partite graph associated with the reduced matrix of σ . Then the spectrum of B σ is homeomorphic with the geometric realisation of .
Proof. We write A for the reduced matrix of σ . Denote by X the spectrum of B σ as described in Proposition 7.1 and by Y the geometric realisation of . In order to show that X and Y are homeomorphic, it suffices to obtain a description of X as a CW-complex combinatorially isomorphic to the CW-complex described in Notation 7.4.
To define the 1-cells of X , recall that X is a quotient of with respect to an equivalence relation that identifies in particular the points (σ (i, j, k), 1 − ) and (i, j, k, 1 + ) for all i ∈ m, j ∈ n and k ∈ o. We take the 1-cells of X to be the image of {σ (i, j, k)} × [0, 1 − ] ∪ {(i, j, k)} × [1 + , 2] in X . This way, 1-cells of X are naturally indexed by m × n × o. Further, the 1-cell indexed by (i, j, k) is glued to the 0-cells (σ (i, j, k) 1 , * , σ (i, j, k) 3 ) and ( * , j, k).
Recall also the CW-structure on Y . Its 0-cells are indexed by {(i, k, r) | i ∈ m, k ∈ o} ∪ {( j, k, c) | j ∈ n, k ∈ o}, coming from the rows and columns of A. There are A (i,k)( j ,k ) 1-cells glued between the 0-cells (i, k, r) and ( j , k , c).
We can now establish a combinatorial isomorphism between the CW-complexes underlying X and Y . Fixing the natural bijection between the 0-cells of X {(i, * , k, 0) ∈ X | i ∈ m, k ∈ o} ∪ {( * , j, k, 2) ∈ X | j ∈ n, k ∈ o} and those of Y {(i, k, r) | i ∈ m, k ∈ o} ∪ {( j, k, c) | j ∈ n, k ∈ o}, it suffices to show that the number of 1-cells glued between two 0-cells is preserved by this bijection.
For i ∈ m, j ∈ n and k, k ∈ o, the 1-cells between (i, * , k, 0) and ( * , j , k , 2) is the cardinality of the set {(a, b, c) ∈ m × n × o | σ (a, b, c) 1 = i, σ (a, b, c) 3 = k, b = j , c = k }. This cardinality is calculated as i ∈m δ σ (i , j ,k ) 1 ,i δ σ (i , j ,k ) 3 ,k = i ∈m, j∈n δ σ (i , j ,k ),(i, j,k) = i ∈m, j∈n where we adopted the permutation matrix notation from Definition 6.2. So the number of 1-cells between (i, * , k, 0) and ( * , j , k , 2) equals the number A (i,k)( j ,k ) of 1-cells between (i, k, r) and ( j , k , c). This is what we had to show. Proposition 7.6. Let , be graphs with geometric realisations X and Y , respectively. Assume that either • all vertices of and have valency at least 3, or • both and are bi-partite and (2, n)-semi-regular for some n 3.
Proof. We consider the space X with its structure of a 1-dimensional CW-complex. Set where a branch point is a point without any neighbourhood locally homeomorphic to an interval and π 0 denotes the set of connected components of a space. If all vertices of have valency at least 3, then V consists of the 0-cells of X . Further for each connected component e ∈ π 0 (X \ V ), the closure e is a 1-cell of X and each 1-cell of X arises uniquely in this way. Hence, the homeomorphism type of X recovers its CW-structure and therefore the isomorphism type of . If thus vertices of are also assumed to have valency at least 3, then X ∼ = Y implies ∼ = .
Assume now that and are bi-partite and (2, n)-semi-regular for some n 3. We associate with and the unique n-regular graphs˜ and˜ whose barycentric