Certain class of higher-dimensional simplicial complexes and universal C^∗-algebras

Abstract

In this article we introduce a universal C^∗-algebras associated to certain simplicial flag complexes. We denote it by ${\mathcal{C}}_{{\mathrm{\Gamma }}^n}$it is a subalgebra of the noncommutative n-sphere which introduced by J.Cuntz. We present a technical lemma to determine the quotient of the skeleton filtration of a general universal C^∗-algebra associated to a simplicial flag complex. We examine the K-theory of this algebra. Moreover we prove that any such algebra divided by the ideal I₂ is commutative.

2000 AMS

19 K 46

Introduction

In this section, we give a survey of some basic definitions and properties of the universal C^∗-algebra associated to a certain flag complex which we will use in the sequel. Such algebras in general was introduced first by Cuntz (2002) and studied by Omran (2005, 2013).

Definition 1

A simplicial complex Σ consists of a set of vertices V _Σ and a set of non-empty subsets of V _Σ , the simplexes in Σ, such that:

• If s ∈ V _Σ , then {s} ∈ Σ.

• If F ∈ Σ and ∅ ≠ E ⊂ F then E ∈ Σ.

A simplicial complex Σ is called flag or full, if it is determined by its 1-simplexes in the sense that {s₀, …, s _n } ∈ Σ ⇔ {s _i , s _j } ∈ Σ for all 0 ≤ i < j ≤ n.

Σ is called locally finite if every vertex of Σ is contained in only finitely many simplexes of Σ, and finite-dimensional (of dimension ≤n) if it contains no simplexes with more than n+1-vertices. For a simplicial complex Σ one can define the topological space |Σ| associated to this complex. It is called the “geometric realization” of the complex and can be defined as the space of maps f:V _Σ  → [0, 1] such that $\sum _{s\in V_{\mathrm{\Sigma }}}f\left(s\right)=1$ and f(s₀) ..... f(s _i ) = 0 whenever {s₀, …, s _i } ∉ Σ. If Σ is locally finite, then |Σ| is locally compact.

Let Σ be a locally finite flag simplicial complex. Denote by V _Σ the set of its vertices. Define ${\mathcal{C}}_{\mathrm{\Sigma }}$ as the universal C^∗-algebra with positive generators h _s , s ∈ V, satisfying the relations

$$h_{s_0}{h}_{s_1}\dots h_{s_n}=0\text{whenever}\{s_0,s_1,\dots ,s_n\}\notin V_{\mathrm{\Sigma }},$$

$$\sum _{s\in V_{\mathrm{\Sigma }}}{h}_s{h}_t=h_t\forall t\in V_{\mathrm{\Sigma }}.$$

Here the sum is finite, because Σ is locally finite.

${\mathcal{C}}_{\mathrm{\Sigma }}^{\mathit{\text{ab}}}$ is the abelian version of the universal C^∗-algebra above, i.e. satisfying in addition h _s h _t  = h _t h _s forall s, t ∈ V _Σ . Denote by I _k the ideal in ${\mathcal{C}}_{\mathrm{\Sigma }}$ generated by products containing at least n + 1 different generators. The filtration (I _k ) of ${\mathcal{C}}_{\mathrm{\Sigma }}$ is called the skeleton filtration.

Let

$$\mathrm{\Delta }:=\left\{(s_0,\dots ,s_n)\in {\mathbb{R}}^{n+1}|0\le s_i\le 1,\sum _{i=1}^n{s}_i=1\right\}$$

be the standard n-simplex. Denote by ${\mathcal{C}}_{\mathrm{\Delta }}$ the associated universal C^∗-algebra with generators h _s , s ∈ {s₀, …, s _n }, such that h _s  ≥ 0 and $\sum _s{h}_s=1$. Denote by ${\mathcal{I}}_{\mathrm{\Delta }}$ the ideal in ${\mathcal{C}}_{\mathrm{\Delta }}$ generated by products of generators containing all the $h_{s_i},i=0,\dots ,n$. For each k, denote by I _k the ideal in ${\mathcal{C}}_{\mathrm{\Delta }}$ generated by all products of generators h _s containing at least k + 1 pairwise different generators. We also denote by $I_k^{\mathit{\text{ab}}}$ the image of I _k in ${\mathcal{C}}_{\mathrm{\Delta }}^{\mathit{\text{ab}}}$. The algebra ${\mathcal{C}}_{\mathrm{\Delta }}$ and their K-Theory was studied in details in (Omran and Gouda 2012). For any vertex t in Δ there is a natural evaluation map ${\mathcal{C}}_{△}\to \mathbb{C}$ mapping the generators h _t to 1 and all the other generators to 0. The following propositions are due to Cuntz (2002).

Proposition 1

(i) The evaluation map${\mathcal{C}}_{△}\to \mathbb{C}$defined above induces an isomorphism in K-theory. (ii) The surjective map${\mathcal{I}}_{△}\to {\mathcal{I}}_{△}^{\mathit{\text{ab}}}$induces an isomorphism in K-theory, where${\mathcal{I}}_{△}^{\mathit{\text{ab}}}$is the abelianization of${\mathcal{I}}_{△}$.

We can observe that I _k is the kernel of the evaluation map which define above so we can conclude that I _k is closed.

Remark 1

Let Δ and${\mathcal{I}}_{△}\subset {\mathcal{C}}_{△}$as above. Then$K_{\ast }\left({\mathcal{I}}_{△}\right)\cong K_{\ast }\left(\mathbb{C}\right),\ast =0,1$, if the dimension n of △ is even and$K_{\ast }\left({\mathcal{I}}_{△}\right)\cong K_{\ast }\left(C_0\right(0,1\left)\right),\ast =0,1$, if the dimension n of △ is odd.

Proposition 2

Let Σ be a locally finite simplicial complex. Then${\mathcal{C}}_{\mathrm{\Sigma }}^{\mathit{\text{ab}}}$is isomorphic to C₀(|Σ|), the algebra of continuous functions vanishing at infinity on the geometric realization |Σ| of Σ.

Universal C^∗-algebras associated to certain complexes

Universal C^∗-algebras is a C^∗-algebras generated by generators and relations. Many C^∗-algebras can be constructed in the form of universal C^∗-algebras an important example for universal C^∗-algebras is Cuntz algebras O _n the existence of this algebras and their K-theory was introduced by Cuntz (1981, 1984) more other examples of universal C^∗-algebras can be found in (Cuntz 1993; Davidson 1996). In the following, we introduce a general technical lemma to compute the quotient of the skeleton filtration for a general algebra associated to simplicial complex.

For a subset W ⊂ V _Σ , let Γ ⊂ Σ be the subcomplex generated by W and let ${\mathcal{I}}_{\mathrm{\Gamma }}$ be the ideal in ${\mathcal{C}}_{\mathrm{\Gamma }}$ generated by products containing all generators of ${\mathcal{C}}_{\mathrm{\Gamma }}$.

Lemma 1

Let ${\mathcal{C}}_{\mathrm{\Sigma }}$ and ${\mathcal{C}}_{\mathrm{\Gamma }}$ as above, then we have

$$I_k/I_{k+1}\cong {\oplus }_{W\subset V_{\mathrm{\Sigma }},\left|W\right|=k+1}{\mathcal{I}}_{\mathrm{\Gamma }}$$

Proof

${\mathcal{C}}_{\mathrm{\Sigma }}/I_{k+1}$ is generated by the images $\dot{h_i}$, i ∈ V _Σ of the generators in the quotient.

Given a subset W ⊂ V _Σ with |W| = k + 1, let

$${\mathcal{C}}_{{\mathrm{\Gamma }}^{\prime }}=C^{\ast }\left(\right\{\dot{h_i}|i\in W\})\subset {\mathcal{C}}_{\mathrm{\Sigma }}/I_{k+1}.$$

Let ${\mathcal{I}}_{{\mathrm{\Gamma }}^{{}^{\prime }}}$ denote the ideal in ${\mathcal{C}}_{{\mathrm{\Gamma }}^{{}^{\prime }}}$ generated by products containing all generators $\dot{h_i}$, i ∈ Γ^′, and let ${\mathcal{B}}_{\mathrm{\Gamma }}$ denote its closure. If $W\ne W^{{}^{\prime }}$, then ${\mathcal{B}}_{\mathrm{\Gamma }}{\mathcal{B}}_{{\mathrm{\Gamma }}^{{}^{\prime }}}=0$, because the product of any two elements in ${\mathcal{B}}_{\mathrm{\Gamma }}$ and ${\mathcal{B}}_{{\mathrm{\Gamma }}^{{}^{\prime }}}$ contains products of more than k + 1-different generators, which are equal to zero in the algebra ${\mathcal{C}}_{\mathrm{\Sigma }}/I_{k+1}$

It is clear that ${\mathcal{B}}_{\mathrm{\Gamma }}\subset I_k/I_{k+1}$ so that

$${\oplus }_{W\subset V_{\mathrm{\Sigma }},\left|W\right|=k+1}{\mathcal{B}}_{\mathrm{\Gamma }}\subset I_k/I_{k+1}.$$

Conversely, let x ∈ I _k  / I_k+1. Then there is a sequence (x _n ) converging to x, such that each x _n is a sum of monomials m _s in $\dot{h_i}$ containing at least k + 1-different generators. Then $m_s\in {\mathcal{B}}_{\mathrm{\Gamma }}$ for some W and

$$x_n=\sum m_s\in {\oplus }_{W\subset V_{\mathrm{\Sigma }},\left|W\right|=k+1}{\mathcal{B}}_{\mathrm{\Gamma }}.$$

The space ${\oplus }_{W\subset V,\left|W\right|=k+1}{\mathcal{B}}_{\mathrm{\Gamma }}$ is closed, because it is a direct sum of closed ideals. It follows that

$$I_k/I_{k+1}={\oplus }_{W\subset V_{\mathrm{\Sigma }},\left|W\right|=k+1}{\mathcal{B}}_{\mathrm{\Gamma }}$$

Let now

$${\pi }_W:{\mathcal{C}}_{\mathrm{\Sigma }}\to {\mathcal{C}}_{\mathrm{\Gamma }}.$$

be the canonical evaluation map defined by

$${\pi }_W\left(h_i\right)=\left\{\begin{array}{ll}{h_i}^{{}^{\prime }}& \forall i\in W\\ 0& \text{if}i\notin W,\end{array}\right.$$

where ${h_i}^{{}^{\prime }}$ denotes the generator in ${\mathcal{C}}_{\mathrm{\Gamma }}$ corresponding to the index i in W, in other words

$${\mathcal{C}}_{\mathrm{\Gamma }}=C^{\ast }\left(h_i^{{}^{\prime }}\right|i\in W).$$

We prove that π _W (I_k+1) = 0. Since polynomials of the form

$$\sum \dots h_{i_0}\dots h_{i_j}\dots h_{i_{k+1}}\dots ,i_0,\dots ,i_j,\dots ,i_{k+1},\dots \in V_{\mathrm{\Sigma }}$$

are dense in I_k+1, it is enough to show that π _W (x) = 0 for each such polynomial x. We have

$${\pi }_W\left(x\right)=\sum \dots \underset{i_0}{\overset{{}^{\prime }}{h}}\dots h_{i_j}^{{}^{\prime }}\dots h_{i_{k+1}}^{{}^{\prime }}\dots =0,$$

since there is at least one i _l which is not in W. For this index ${\pi }_W\left(h_{i_l}\right)=0$. Thus π _W (x) = 0. Therefore π _W descends to a homomorphism

$$\dot{{\pi }_W}:{\mathcal{C}}_{\mathrm{\Sigma }}/I_{k+1}\to {\mathcal{C}}_{\mathrm{\Gamma }}$$

Now we show that π _W is surjective as follows: Since π _W (I_k+1) = 0, we have Ker π _W  ⊃ I_k+1. It follows that the following diagram

$$\begin{array}{lll}{\mathcal{C}}_{\mathrm{\Sigma }}& \to & \hfill {\mathcal{C}}_{\mathrm{\Gamma }}\hfill \\ \searrow & \hfill \uparrow \hfill \\ \hfill {\mathcal{C}}_{\mathrm{\Sigma }}/I_{k+1}\hfill \end{array}$$

commutes and $\dot{{\pi }_W}\left(\dot{h_i}\right):={\pi }_W\left(h_i\right)={h_i}^{{}^{\prime }},i\in W$ is well defined. This shows that ${\pi }_W\left({\mathcal{C}}_{\mathrm{\Sigma }}\right)$ is a closed subalgebra in ${\mathcal{C}}_{\mathrm{\Gamma }}$ and isomorphic to $\dot{{\pi }_W}({\mathcal{C}}_{\mathrm{\Sigma }}/I_{k+1})$. We have $\dot{{\pi }_W}\left({\mathcal{B}}_{\mathrm{\Gamma }}\right)={\mathcal{I}}_{\mathrm{\Gamma }}$. It is clear that Ker π _W is the ideal generated by h _i for i not in W and therefore $\text{Ker}\dot{{\pi }_W}$ is generated by $\dot{h_i}$ for i not in W. This comes at once from the definitions of $\dot{{\pi }_W}\left(\dot{h_i}\right)$ and π _W (h _i ) above and the fact that both are equal. We conclude that ${\mathcal{B}}_{\mathrm{\Gamma }}\text{Ker}\dot{{\pi }_W}=0$. This again implies that ${\mathcal{B}}_{\mathrm{\Gamma }}\cap \text{Ker}\dot{{\pi }_W}=0$. Moreover the following diagram is commutative:

$$\begin{array}{lll}{\mathcal{C}}_{\mathrm{\Sigma }}& \to & \hfill {\mathcal{C}}_{\mathrm{\Gamma }}\hfill \\ \bigcup & \hfill \bigcup \hfill \\ {\mathcal{B}}_{\mathrm{\Gamma }}& \to & \hfill {\mathcal{I}}_{\mathrm{\Gamma }}\hfill \\ \searrow & \hfill \uparrow \hfill \\ \hfill {\mathcal{B}}_{\mathrm{\Gamma }}/\text{Ker}\dot{{\pi }_W}.\hfill \end{array}$$

So, $\dot{{\pi }_W}\left({\mathcal{B}}_{\mathrm{\Gamma }}\right)$ is dense and closed in ${\mathcal{I}}_{\mathrm{\Gamma }}$. Therefore $\dot{{\pi }_W}:{\mathcal{B}}_{\mathrm{\Gamma }}\to {\mathcal{I}}_{\mathrm{\Gamma }}$ is injective and surjective.

As a consequence of the above lemma we have the following.

Proposition 3

Let ${\mathcal{C}}_{\mathrm{\Delta }}$ and I _k defined as above. Then we have an isomorphism

$$I_k/I_{k+1}\cong \underset{△}{\oplus }{\mathcal{I}}_{△},$$

where the sum is taken over all k-simplexes △ in Σ.

Proof

As in the proof of lemma 1 above with Σ = Δ, we find that:

$$I_k/I_{k+1}=\underset{△}{\oplus }{\mathcal{I}}_{△}.$$

In the following we study the C^∗-algebras ${\mathcal{C}}_{{\mathrm{\Gamma }}^n}$ associated to simplicial flag complexes Γ of a specific simple type. These simplicial complexes is a subcomplex of the “non-commutative spheres” in the sense of Cuntz work (Cuntz 2002). We determine the K-theory of ${\mathcal{C}}_{{\mathrm{\Gamma }}^n}$ and also the K-theory of its skeleton filtration. The K-theory of C^∗-algebras is a powerful tool for classifying C^∗-algebras up to their Projections and unitaries, more details about K-theory of C^∗-algebras found in the references (Blackadar 1986; Murphy 1990; Rørdam et al. 2000; Wegge-Olsen 1993).

We denote by Γⁿ the simplicial complex with n + 2 vertices, given in the form

$$V_{{\mathrm{\Gamma }}^n}=\{0^{+},0^{-},1,\dots ,n\},$$

and

$${\mathrm{\Gamma }}^n=\{\gamma \subset V_{{\mathrm{\Gamma }}^n}|\{0^{+},0^{-}\}⊈\gamma \}.$$

Let

$$\begin{array}{ll}{\mathcal{C}}_{{\mathrm{\Gamma }}^n}& =C^{\ast }(h_{0^{-}},h_{0^{+}},h_1,h_2,\dots ,h_n|h_{0^{-}}{h}_{0^{+}}\\ =0,h_i\ge 0,\sum _i{h}_i=1,\forall i)\end{array}$$

be the universal C^∗- algebra associated to Γⁿ. The existence of such algebras is due to Cuntz (2002). It is clear that for any element $h_i\in {\mathcal{C}}_{{\mathrm{\Gamma }}^n}$, we have ∥h _i ∥ ≤ 1.

Denote by the natural ideal in ${\mathcal{C}}_{{\mathrm{\Gamma }}^n}$ generated by products of generators containing all h _i , $i\in V_{{\mathrm{\Gamma }}^n}$. Then we have the skeleton filtration

$${\mathcal{C}}_{{\mathrm{\Gamma }}^n}=I_0\supset I_1\supset I_2\supset \mathrm{.....}\supset I_{n+1}:=\mathcal{I}$$

The aim of this section is to prove that the K-theory of the ideals in the algebras ${\mathcal{C}}_{{\mathrm{\Gamma }}^n}$ is equal to zero. We have the following

Lemma 2

Let${\mathcal{C}}_{{\mathrm{\Gamma }}^n}$be as above. Then${\mathcal{C}}_{{\mathrm{\Gamma }}^n}$is homotopy equivalent to.

Proof

Let $\beta :\mathbb{C}\to {\mathcal{C}}_{{\mathrm{\Gamma }}^n}$ be the natural homomorphism which sends 1 to $1_{{\mathcal{C}}_{{\mathrm{\Gamma }}^n}}$. For a fixed $i\in V_{{\mathrm{\Gamma }}^n}$ such that i ≠ 0^-,0⁺, define the homomorphism

$$\alpha :{\mathcal{C}}_{{\mathrm{\Gamma }}^n}\to \mathbb{C}$$

by α(h _i ) = 1 and α(h _j ) = 0 for any j ≠ i. Notice that $\alpha \circ \beta ={\mathit{\text{id}}}_{\mathbb{C}}$. Now define ${\phi }_t:{\mathcal{C}}_{{\mathrm{\Gamma }}^n}\to {\mathcal{C}}_{{\mathrm{\Gamma }}^n}$, $h_i\mapsto h_i+(1-t)\left(\sum _{j\ne i}{h}_j\right)$, $h_j\mapsto t\left(h_j\right)j\in V_{{\mathrm{\Gamma }}^n}\setminus \left\{i\right\}.$ The elements φ _t (h _j ), $j\in V_{{\mathrm{\Gamma }}^n}$, satisfy the same relations as the elements h _j in ${\mathcal{C}}_{{\mathrm{\Gamma }}^n}$:

• (i) φ _t (h _j ) ≥ 0

• (ii)
  $$\begin{array}{ll}{\phi }_t\left({\sum }_j{h}_j\right)& ={\phi }_t\left(h_i\right)+{\sum }_{j\ne i}{\phi }_t\left(h_j\right)\\ =h_i+(1-t)\left({\sum }_{j\ne i}{h}_j\right)+t\left({\sum }_{j\ne i}{h}_j\right)\\ =h_i+\sum _{j\ne i}{h}_j\text{for fixed}i\\ ={\sum }_j{h}_j=1\text{for all}j,\end{array}$$

• (iii) ${\phi }_t\left(h_{0^{-}}\right){\phi }_t\left(h_{0^{+}}\right)=t^2\left(h_{0^{-}}{h}_{0^{+}}\right)=0.$

We note that ${\phi }_1={\mathit{\text{id}}}_{{\mathcal{C}}_{{\mathrm{\Gamma }}^n}}$ and φ₀ = β ∘ α.

This implies that

$${\phi }_0=\beta \circ \alpha \sim {\mathit{\text{Id}}}_{{\mathcal{C}}_{{\mathrm{\Gamma }}^n}}.$$

This means that ${\mathcal{C}}_{{\mathrm{\Gamma }}^n}$ is homotopy equivalent to .

From the above lemma, we have $K_{\ast }\left({\mathcal{C}}_{{\mathrm{\Gamma }}^n}\right)=K_{\ast }\left(\mathbb{C}\right)$, for ∗ = 0,1.

Now we describe the subquotients of the skeleton filtration in ${\mathcal{C}}_{{\mathrm{\Gamma }}^n}$.

Proposition 4

In the C^∗-algebra${\mathcal{C}}_{{\mathrm{\Gamma }}^n}$one has

$$I_k/I_{k+1}\cong {\oplus }_{△}{\mathcal{I}}_{△}\oplus {\oplus }_{\gamma }{\mathcal{I}}_{\gamma },$$

where the sum is taken over all subcomplexes △ of Γⁿwhich are isomorphic to the standard k-simplex △ and over all subcomplexes γ of Γⁿwhich contain both vertices 0⁺, 0^-and the second sum is taken over every subcomplex γ which contains both vertices 0⁺,0^-and whose number of vertices is k + 1.

Proof

We use Lemma 1 above. For every $W\subset V_{{\mathrm{\Gamma }}^n}$ with |W| = k + 1, we have two cases. Either {0⁺,0^-} is not a subset of W, then Γ is a k- simplex, or {0⁺,0^-} is a subset of W, then Γ is a subcomplex in Γⁿ isomorphic to γ. This proves our proposition.

Lemma 3

For the complex Γⁿwith n + 2 vertices, ${\mathcal{C}}_{{\mathrm{\Gamma }}^n}/I_1$is commutative and isomorphic to${\mathbb{C}}^{n+2}$.

Proof

Let $\dot{h_i}$ denote the image of a generator h _i for ${\mathcal{C}}_{{\mathrm{\Gamma }}^n}$. One has the following relations:

$$\sum _i\dot{h_i}=1,\dot{h_i}\dot{h_j}=0,i\ne j.$$

For every $\dot{h_i}$ in ${\mathcal{C}}_{{\mathrm{\Gamma }}^n}/I_1$ we have

$$\dot{h_i}=\dot{h_i}\left(\sum _i\dot{h_i}\right)={\dot{h}}_i^2.$$

Hence ${\mathcal{C}}_{{\mathrm{\Gamma }}^n}/I_1$ is generated by n + 2 different orthogonal projections and therefore ${\mathcal{C}}_{{\mathrm{\Gamma }}^n}/I_1\cong {\mathbb{C}}^{n+2}$.

Lemma 4

I₁ / I₂in${\mathcal{C}}_{{\mathrm{\Gamma }}^n}$is isomorphic to$I_1^{\mathit{\text{ab}}}/I_2^{\mathit{\text{ab}}}$in${\mathcal{C}}_{{\mathrm{\Gamma }}^n}^{\mathit{\text{ab}}}$.

Proof

From the proposition 4 above, one has

$$I_1/I_2\cong {\oplus }_{{△}^1}{\mathcal{I}}_{{△}^1}$$

where △¹ is 1-simplex, and

$$I_1^{\mathit{\text{ab}}}/I_2^{\mathit{\text{ab}}}\cong {\oplus }_{{△}^1}{\mathcal{I}}_{{△}^1}^{\mathit{\text{ab}}}.$$

Since ${\mathcal{I}}_{{△}^1}\subset {\mathcal{C}}_{{△}^1}$ is commutative because the generators of ${\mathcal{C}}_{{△}^1}$ commute (since $h_{s_1}=1-h_{s_0}$). We get

$${\mathcal{I}}_{{△}^1}\cong {\mathcal{I}}_{{△}^1}^{\mathit{\text{ab}}}\cong C_0(0,1).$$

Lemma 5

In ${\mathcal{C}}_{{\mathrm{\Gamma }}^n}$, we have K₀(I₁ / I₂) = 0 and $K_1(I_1/I_2)={\mathbb{Z}}^{\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)+2n}.$

Proof

By applying above lemma, and proposition 4, we have

$$I_1/I_2\cong {\oplus }_{{△}^1}{\mathcal{I}}_{{△}^1}$$

The sum contain $\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)+2n$ 1-simplex, △¹ ≅ C₀(0, 1). where K₀(C₀(0, 1)) = 0 and $K_1\left(C_0\right(0,1\left)\right)=\mathbb{Z}$.

Lemma 6

${\mathcal{C}}_{{\mathrm{\Gamma }}^n}/I_2$is a commutative C^∗-algebra.

Proof

Consider the extension

$$\begin{array}{lllllll}0\to & I_1/I_2& \to & {\mathcal{C}}_{{\mathrm{\Gamma }}^n}/I_2& \to & {\mathcal{C}}_{{\mathrm{\Gamma }}^n}/I_1& \to 0\end{array}$$

and the analogous extension for the abelianized algebras.

The extensions above induce the following commutative diagram:

$$\begin{array}{ccccccc}0\to & I_1/I_2& \to & {\mathcal{C}}_{{\mathrm{\Gamma }}^n}/I_2& \to & {\mathcal{C}}_{{\mathrm{\Gamma }}^n}/I_1& \to 0\\ \downarrow & \downarrow & \downarrow \\ 0\to & I_1^{\mathit{\text{ab}}}/I_2^{\mathit{\text{ab}}}& \to & {\mathcal{C}}_{{\mathrm{\Gamma }}^n}^{\mathit{\text{ab}}}/I_2^{\mathit{\text{ab}}}& \to & {\mathcal{C}}_{{\mathrm{\Gamma }}^n}^{\mathit{\text{ab}}}/I_1^{\mathit{\text{ab}}}& \to 0\end{array}$$

We have from 3 isomorphisms ${\mathcal{C}}_{{\mathrm{\Gamma }}^n}/I_1\cong {\mathcal{C}}_{{\mathrm{\Gamma }}^n}^{\mathit{\text{ab}}}/I_1^{\mathit{\text{ab}}}\cong {\mathbb{C}}^{n+2}$ and from 4 that $I_1/I_2\cong I_1^{\mathit{\text{ab}}}/I_2^{\mathit{\text{ab}}}$, so

$${\mathcal{C}}_{{\mathrm{\Gamma }}^n}/I_2\cong {\mathcal{C}}_{{\mathrm{\Gamma }}^n}^{\mathit{\text{ab}}}/I_2^{\mathit{\text{ab}}}.$$

Lemma 7

C^∗-algebra${\mathcal{C}}_{{\mathrm{\Gamma }}^1}$is commutative and K_∗(I₂) = 0, ∗ = 0, 1 where I₂is an ideal in${\mathcal{C}}_{{\mathrm{\Gamma }}^1}$defined as in the above.

Proof

${\mathcal{C}}_{{\mathrm{\Gamma }}^1}$ is generated by three positive generators, $h_{0^{-}},h_{0^{+}},h_1$. Consider the product of two generators, say $h_1{h}_{0^{-}}$. We have that $1,h_{0^{-}}$ and $h_{0^{+}}$ commute with $h_{0^{-}}$, therefore also $h_1=1-h_{0^{-}}-h_{0^{+}}$.

By a similar computation we can show that $h_{0^{+}}$ and h₁ commute. This implies that ${\mathcal{C}}_{{\mathrm{\Gamma }}^1}$ is commutative. Therefore I₂ = 0 in ${\mathcal{C}}_{{\mathrm{\Sigma }}^1}$ Then, at once K_∗(I₂) = 0.