A Cantor spectrum diagonal in O_2
Philipp Sibbel Wilhelm Winter

§ ABSTRACT 
We prove the existence of a C^*-diagonal in the Cuntz algebra 𝒪_2 with spectrum homeomorphic to the Cantor space.


Speaker and Style Disentanglement of Speech Based on Contrastive Predictive Coding Supported Factorized Variational Autoencoder

    Yuying Xie1, Michael Kuhlmann2, Frederik Rautenberg2 , Zheng-Hua Tan1, Reinhold Haeb-Umbach2

    1. Department of Electronic Systems, Aalborg University, Denmark 

    2. Department of Communications Engineering, Paderborn University, Germany 



    
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In 1986, Kumjian defined the notion of a C^*-diagonal in a C^*-algebra as an analogue to the notion of a Cartan subalgebra in von Neumann algebras (<cit.>). He showed that diagonal C^*-pairs occur as groupoid C^*-algebras of twisted étale equivalence relations. In <cit.>, Renault noted that the notion of a C^*-diagonal is sometimes too restrictive as it does not cover a large number of canonical maximal abelian subalgebras in important C^*-algebras. As an example, he mentioned the Cuntz algebras 𝒪_n and more generally graph algebras, which contain obvious regular maximal abelian subalgebras that are not C^*-diagonals. The difference is that these C^*-algebras can be constructed as groupoid C^*-algebras from groupoids which have some isotropy, more precisely, they are only topologically principal as opposed to principal. As a remedy, Renault defined the weaker notion of a Cartan subalgebra in a C^*-algebra.



Let A be a C^*-algebra. An abelian sub-C^*-algebra D in A is called a Cartan subalgebra if


  (0) D contains an approximate unit of A,

  (1) D is maximal abelian in A,

  (2) D is regular, i.e. the normalisers of D,

    𝒩_A(D) := {n ∈ A | n^*Dn ⊂ D  and  nDn^* ⊂ D},

generate A as a C^*-algebra, and

  (3) there exists a faithful conditional expectation Ψ of A onto D.

D is called a C^*-diagonal if, moreover,


  (4)
the unique extension property of pure states is satisfied, that is, every pure state on D has a unique extension to a (necessarily pure) state on A.




Renault then showed the analogous result to Kumjian's theorem, that is, the reduced C^*-algebra of a second-countable, topologically principal, Hausdorff, étale, twisted groupoid contains a canonical Cartan subalgebra corresponding to the unit space, and every Cartan subalgebra in a separable C^*-algebra uniquely arises in this fashion (<cit.>). A Cartan subalgebra then is a C^*-diagonal if and only if the corresponding groupoid is principal. 


In the following years, motivated by results in the theory of von Neumann algebras, Cartan subalgebras were studied intensively in the C^*-algebraic setting. Several existence and non-uniqueness results for Cartan subalgebras and C^*-diagonals were obtained (e.g. <cit.>, <cit.>, <cit.>, <cit.>, <cit.>), where the interplay between the study of Cartan subalgebras and classification theory of C^*-algebras attracted particular attention. In <cit.>, Li completed the proof that every classifiable (in the sense of Elliott's programme; cf. <cit.>) simple C^*-algebra has a Cartan subalgebra and that a unital, separable, simple C^*-algebra with finite nuclear dimension contains a Cartan subalgebra if and only if it satisfies the Universal Coefficient Theorem (UCT) of Rosenberg and Schochet (also see <cit.>).



In <cit.>, Cuntz introduced an important class of C^*-algebras, which he denoted by 𝒪_n, for 2 ≤ n ≤∞, and which were later called Cuntz algebras. For n finite, 𝒪_n is the universal C^*-algebra generated by n isometries S_1, …, S_n with pairwise orthogonal range projections which add up to the unit. The Cuntz algebra 𝒪_n then contains a subalgebra isomorphic to M_n^∞, the UHF algebra of type n^∞. It is generated by elements of the form S_μ S_ν^* for finite sequences μ, ν∈{1,…,n}^k of the same length, where S_μ = S_μ_1… S_μ_k for μ = (μ_1,…,μ_k). It is well-known that the subalgebra D_n^∞ generated by diagonal matrices in M_n^∞ is a C^*-diagonal. The corresponding subalgebra in the Cuntz algebra 𝒪_n, which we also denote by D_n^∞, is generated by elements of the form S_μ S_μ^* and turns out to be a Cartan subalgebra in 𝒪_n; however, the unique extension property of pure states is not satisfied and therefore it is not a C^*-diagonal in 𝒪_n (cf. <cit.>). In particular, an element in {1,…,n}^, the spectrum of D_n^∞, corresponds to a pure state on D_n^∞ with a unique extension to 𝒪_n if and only if the element is an (eventually) aperiodic sequence (<cit.>, <cit.>). 


The natural question whether 𝒪_n contains a C^*-diagonal is surprisingly difficult to answer. In <cit.>, Hjelmborg presented a construction of 𝒪_2 as the C^*-algebra of a principal groupoid; by Renault's and Kumjian's work (<cit.>, <cit.>), this yields a C^*-diagonal in 𝒪_2. Hjelmborg's argument factorises through Kirchberg–Phillips classification and, in particular, it uses Kirchberg's 𝒪_2-absorption theorem, which states that A ⊗𝒪_2 is isomorphic to 𝒪_2 if and only if A is a simple, separable, unital and nuclear C^*-algebra. The spectrum of Hjelmborg's C^*-diagonal is 𝕋×{1,2}^, and in particular one-dimensional. Via tensoring with itself, one then also obtains C^*-diagonals with higher-dimensional spectra in 𝒪_2. 


Until now it remained unknown whether 𝒪_2 contains a C^*-diagonal with Cantor spectrum. Below we will give an affirmative answer – and here is why we are excited about this: A long-term goal is it to develop a structure theory for Cartan and diagonal sub-C^*-algebras of a given classifiable C^*-algebra. This programme is in its infancy, and it is natural to focus on spectra which are as universal as possible, and in particular robust under taking tensor products. Perhaps more important, we hope to eventually employ K-theoretic tools to build distinguishing invariants, and Cantor spectrum promises the most direct access to such methods.



We write M_n for the n × n - matrices with coefficients in , and 𝒦≅𝒦(ℓ^2()) for the compact operators on an infinite-dimensional separable Hilbert space. For i,j ∈{1,…,n}, let e_ij denote the standard matrix units in M_n (respectively in 𝒦). For a non-unital C^*-algebra A, we write A^+ for its unitisation.



Before giving full details, let us outline the construction. Let X be a Cantor space and fix a minimal homeomorphism on X. Then, the associated -action is free, which implies that 𝒞(X), the C^*-algebra of continuous functions on X, is a C^*-diagonal in the crossed product C^*-algebra 𝒞(X) ⋊_α (see <cit.>), where α denotes the induced action on 𝒞(X). Let v ∈𝒞(X) ⋊_α be the unitary implementing the action, i.e. v h v^* = α (h) for all h ∈𝒞(X).


Let D_2^∞ denote the canonical Cartan subalgebra in the Cuntz algebra 𝒪_2. It is generated by elements of the form S_μ S_μ^* for μ∈{1,2}^k and k ∈. Then,

    D_2^∞⊗𝒞(X) ⊂𝒪_2 ⊗ (𝒞(X) ⋊_α)

is a Cartan subalgebra which is not a C^*-diagonal (see <cit.>). However, we will see that 

    D:= D_2^∞⊗𝒞(X) ⊂C^*(D_2^∞⊗𝒞(X), S_1 ⊗ v, S_2 ⊗ v)=:B

is a C^*-diagonal, and the ambient C^*-algebra B is nuclear and simple; see Lemmas <ref> and <ref>.


But now, the infinite tensor product D^⊗∞⊂ B^⊗∞ is still a C^*-diagonal with Cantor spectrum in a simple nuclear C^*-algebra. B contains a unital copy of the Cuntz algebra 𝒪_2, and since the latter is strongly self-absorbing (see <cit.>), we have that B^⊗∞≅ B^⊗∞⊗𝒪_2; cf. <cit.>. The isomorphism with 𝒪_2 then follows from Kirchberg's 𝒪_2-absorption theorem, and we have our main result:



D^⊗∞ is a C^*-diagonal in B^⊗∞ with spectrum homeomorphic to the Cantor space, and B^⊗∞ is isomorphic to the Cuntz algebra 𝒪_2.


We now carry out the steps above in detail.


D ⊂ B as in (<ref>) is a C^*-diagonal.



Since D_2^∞⊗𝒞(X) is a Cartan subalgebra in 𝒪_2 ⊗ (𝒞(X) ⋊_α) by <cit.>, it is maximal abelian also in B and there is a faithful conditional expectation B → D_2^∞⊗𝒞(X). To see that D_2^∞⊗𝒞(X) is regular in B, it suffices to check that S_1 ⊗ v and S_2⊗ v are normalisers. But for elementary tensors of the form d ⊗ f ∈ D_2^∞⊗𝒞(X), we have

    (S_i ⊗ v) (d ⊗ f)(S_i ⊗ v)^* = (S_i d S_i^* ⊗ vfv^*)

and

    (S_i ⊗ v)^* (d ⊗ f)(S_i ⊗ v) = (S_i^* d S_i ⊗ v^*fv),

which again lie in D_2^∞⊗𝒞(X). Now, since sums of elementary tensors are dense in the closed subalgebra D_2^∞⊗𝒞(X) and since conjugation with S_i ⊗ v and (S_i ⊗ v)^* is continuous, it follows that S_i ⊗ v are indeed normalisers.


Thus, it only remains to show the unique extension property of pure states. For this, consider an arbitrary pure state on D_2^∞⊗𝒞(X). It is of the form ρ⊗σ for a pure state ρ on D_2^∞ and a pure state σ on 𝒞(X). Let ρ⊗σ be an extension to a pure state on 𝒪_2 ⊗ ( 𝒞(X) ⋊_α). Since ρ⊗σ( 1_𝒪_2⊗  .  ) is a state on 𝒞(X) ⋊_α extending σ, it must be the unique extension, denoted by σ, and hence pure. It follows from <cit.> that ρ⊗σ=ρ⊗σ for a state ρ on 𝒪_2, which extends ρ. Note that a priori, ρ does not need to be unique on 𝒪_2 but we claim that on B, ρ⊗σ already is uniquely determined. To see this, we consider an arbitrary product k = (d ⊗ f) k_1  …  k_l for l ∈, where d ∈ D_2^∞ and f ∈𝒞(X), and each k_i is S_1⊗ v, S_1^*⊗ v^*, S_2⊗ v or S_2^*⊗ v^*. As the action on X is free and σ is obtained using the conditional expectation of 𝒞(X) ⋊_α onto 𝒞(X) followed by the point evaluation σ, ρ⊗σ  (k) can only be nonzero if v and v^* occur the same number of times in the product. Therefore, if ρ⊗σ  (k) is nonzero, then k also contains the same number of S_i's and S_i^*'s. Thus, the first tensor factor of k has to be in the subalgebra of 𝒪_2 generated by elements of the form S_μ S_ν^* with ℓ(μ) = ℓ(ν). On this subalgebra, which is isomorphic to the UHF algebra of type 2^∞, ρ also is uniquely determined. Now, since the linear span of elements like k is dense in B, ρ⊗σ is uniquely determined on B.



Our proof that B is nuclear closely follows Cuntz's argument for 𝒪_n. 
Our initial proof of the simplicity of B also followed Cuntz's ideas, but discussions with S. Evington prompted us to use a result by Echterhoff (<cit.>). In order to set up notation, we recall the construction in the case n=2 from <cit.>, where 𝒪_2 was presented as a crossed product cut down by a projection. 


We fix an isomorphism ϕ : 𝒦→𝒦⊗ M_2 such that ϕ(e_11) = e_11⊗ e_11 and let id_M_2^∞: M_2^∞→ M_2^∞ denote the identity map. Then, we can define an automorphism Φ : 𝒦⊗ M_2^∞→𝒦⊗ M_2^∞ via Φ = ϕ⊗id_M_2^∞, that is 

    Φ= ϕ⊗id_M_2^∞: 𝒦⊗ M_2 ⊗…⟶ (𝒦⊗ M_2) ⊗ M_2 ⊗… = 𝒦⊗ M_2^∞.

Now, Φ canonically extends to an automorphism Φ^+ on the unitisation (𝒦⊗ M_2^∞)^+ and we obtain a -action on (𝒦⊗ M_2^∞)^+. Thus, we can consider the crossed product (𝒦⊗ M_2^∞)^+ ⋊_Φ^+. Let u ∈ (𝒦⊗ M_2^∞)^+ ⋊_Φ^+ be the unitary implementing the action, i.e. ux u^*= Φ^+(x) for all x ∈ (𝒦⊗ M_2^∞)^+.


Now, 𝒪_2 is (isomorphic to) the crossed product (𝒦⊗ M_2^∞)^+⋊_Φ^+ cut down by the projection p: = e_11⊗1_M_2^∞∈ (𝒦⊗ M_n^∞)^+, i.e.
    𝒪_2 ≅ p   ((𝒦⊗ M_2^∞)^+⋊_Φ^+) p,

sending the generators S_1 and S_2 to up and (e_11⊗ e_21⊗1_M_2⊗… )  up, respectively (also see <cit.>). Note that under this identification, we have

    S_i_1… S_i_kS_j_k^* …S_j_1^* = e_11⊗ e_i_1 j_1⊗…⊗ e_i_k j_k⊗1_M_2⊗…  ,

so D_2^∞⊂𝒪_2 is identified with e_1 1⊗ D_2^∞⊂ e_11⊗ M_2^∞⊂ p ((𝒦⊗ M_2^∞)^+ ⋊_Φ^+)  p.


We define elements 

    S̃_1    := S_1 ⊗ v = up ⊗ v, 
    S̃_2    := S_2 ⊗ v = (e_11⊗ e_21⊗1_M_2⊗… )   up ⊗ v

in 𝒪_2 ⊗ (𝒞(X) ⋊_α)⊂((𝒦⊗ M_2^∞)^+⋊_Φ^+ )⊗ (𝒞(X) ⋊_α). Then, S̃_1 and S̃_2 also satisfy the Cuntz relations with unit p̃:= p ⊗1_𝒞(X), i.e. S̃_i^* S̃_i = S̃_1 S̃_1^*+S̃_2S̃_2^* = p̃=p̃^*=p̃^2 and p̃S̃_i p̃=S̃_i. Furthermore, we have the identification

    B    = C^*(D_2^∞⊗𝒞(X), S_1 ⊗ v , S_2 ⊗ v) 
       ≅C^*(e_11⊗ D_2^∞⊗𝒞(X), S̃_1 , S̃_2),

when viewing B as a subalgebra of ((𝒦⊗ M_2^∞)^+⋊_Φ^+ )⊗ (𝒞(X) ⋊_α). 



B is nuclear and simple.



First, we note that Φ^+ ⊗α yields a -action on (𝒦⊗ M_2^∞)^+ ⊗𝒞(X) which is implemented by u ⊗ v. Thus, by the universal property of the crossed product (cf. <cit.>), there exists a surjection 

    ((𝒦⊗ M_2^∞)^+ ⊗𝒞(X)) ⋊_Φ^+ ⊗α⟶C^*((𝒦⊗ M_2^∞)^+ ⊗𝒞(X), u ⊗ v) =:E.

We consider E as a subalgebra of ((𝒦⊗ M_2^∞)^+ ⋊_Φ^+ ) ⊗( 𝒞(X) ⋊_α).


As crossed products of nuclear C^*-algebras by amenable groups are nuclear (<cit.>), we find that ((𝒦⊗ M_2^∞)^+ ⊗𝒞(X)) ⋊_Φ^+ ⊗α is nuclear. Now, since E is a quotient of a nuclear C^*-algebra, it is nuclear as well (<cit.>, <cit.>), and so is the hereditary sub-C^*-algebra p̃ E p̃ (<cit.>). 



We now check that inside ((𝒦⊗ M_2^∞)^+ ⋊_Φ^+) ⊗ (𝒞(X) ⋊_α), B coincides with p̃Ep̃, which will conclude the proof of nuclearity. This will be done analogously to the presentation of 𝒪_2 as a crossed product cut down by a projection (see <cit.>).


First, note that since S̃_1 = p̃S̃_1p̃ and S̃_2 = p̃S̃_2p̃, we have B ⊂p̃ Ep̃. For the reverse inclusion, let us consider an a ∈ E ⊂ ((𝒦⊗ M_2^∞)^+ ⋊_Φ^+ ) ⊗( 𝒞(X) ⋊_α) of the form

    a = ∑_i=-N^N x_i u^i ⊗ y_i v^i,

where N ∈ and x_i ∈ (𝒦⊗ M_2^∞)^+, y_i ∈𝒞(X) for all i= -N,…,N. Elements of this form are dense in E. When writing x̃_i = u^-i x_i u^i and ỹ_i = v^-i y_i v^i for i < 0, we obtain 

    a = ∑_i<0 u^i x̃_i ⊗ v^i ỹ_i+ x_0 ⊗ y_0+ ∑_i>0 x_i u^i ⊗ y_i v^i.

Using that up = S_1 = 1_𝒪_2 S_1 = pup, we see that px_i u^i p = (px_ip)(up)^i for i > 0 and pu^ix̃_i p = ((up)^*)^-i p x̃_i p for i<0. This yields

    p̃ a p̃   = ∑_i<0 ((up)^*)^-i p x̃_ip ⊗ v^i ỹ_i+ px_0p ⊗ v_0 + ∑_i>0 (px_ip)(up)^i ⊗ y_i v	^i
       = ∑_i<0 (S_1^*)^-i p x̃_ip ⊗ v^i ỹ_i+ px_0p⊗ y_0+ ∑_i>0 (px_ip)S_1^i ⊗ y_i v^i
       = ∑_i<0 ((S_1⊗ v)^*)^-i (p x̃_ip ⊗ỹ_i)+ px_0p ⊗ y_0+ ∑_i>0 ((px_ip) ⊗ y_i)(S_1 ⊗ v)^i.

Thus, p̃ E p̃ is generated as a C^*-algebra by p  (𝒦⊗ M_2^∞)^+  p ⊗𝒞(X) = e_11⊗ M_2^∞⊗𝒞(X) together with S̃_1. Recall from (<ref>) that 

    S̃_i_1…S̃_i_kS̃_j_k^* …S̃_j_1^* = e_11⊗ e_i_1 j_1⊗…⊗ e_i_k j_k⊗1_M_2⊗…⊗1_𝒞(X),

whence e_11⊗ M_2^∞⊗𝒞(X) is generated by S̃_1, S̃_2 and e_11⊗1_M_2^∞⊗𝒞(X). Therefore, p̃ E p̃ is generated by S̃_1, S̃_2 and e_11⊗1_M_2^∞⊗𝒞(X), which yields that p̃ E p̃ is contained in B.



We further observe that p̃Ep̃ = p̃ F p̃, where F = C^*(𝒦⊗ M_2^∞⊗𝒞(X), u ⊗ v). By the universal property of the crossed product, there exists a ^*-homomorphism

    (𝒦⊗ M_2^∞⊗𝒞(X)) ⋊_Φ⊗α⟶ F,

the image of which contains p̃ F p̃ as a hereditary subalgebra. By <cit.>, the crossed product (𝒦⊗ M_2^∞⊗𝒞(X)) ⋊_Φ⊗α is simple, hence so is its image in F, and therefore also the hereditary subalgebra p̃ F p̃, which in turn is isomorphic to B.




We proceed by taking infinite tensor products and by <cit.>, we again obtain a C^*-diagonal

    D^⊗∞⊂ B^⊗∞.

The spectrum of D^⊗∞ is the product of the individual spectra and therefore again a Cantor space.

Note that B^⊗∞ is still simple, separable, unital and nuclear. Furthermore, 𝒪_2 embeds unitally into increasingly higher tensor factors of B^⊗∞, hence embeds unitally into the central sequence algebra ℓ^∞( , B^⊗∞) / c_0( , B^⊗∞) ∩ (B^⊗∞)^'. This yields B^⊗∞≅ B^⊗∞⊗𝒪_2 (<cit.>, <cit.>), since 𝒪_2 is strongly self-absorbing (cf. <cit.>).


Now by Kirchberg's 𝒪_2-absorption theorem (<cit.>, <cit.>) and since B^⊗∞ is simple, separable, unital and nuclear, we have B^⊗∞≅ B^⊗∞⊗𝒪_2 ≅𝒪_2; cf. <cit.>.


Hence, we have found a C^*-diagonal in 𝒪_2 whose spectrum is a Cantor space and the proof of our theorem is complete. 



We close with a remark and some open questions which are motivated by both our construction and our result.



B cannot coincide with 𝒪_2 and it is therefore necessary to take the infinite tensor product B^⊗∞ in our construction. Indeed, S. Evington pointed out to us that in general B carries interesting and non-trivial K-theoretic information; this will be pursued in subsequent work.






  (i) To what extent does our construction depend on the choice of the Cantor minimal system, and what would be a distinguishing invariant?

  (ii) Do (D ⊂ B) or (D^⊗∞⊂ B^⊗∞) have finite diagonal dimension in the sense of <cit.>?

  (iii) Does the Cuntz algebra 𝒪_n for n ≥ 3, or a general Kirchberg algebra contain a C^*-diagonal (with Cantor spectrum)? 





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