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 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 (2005Omran ( , 2013.

Definition 1. A simplicial complex
consists of a set of vertices V and a set of non-empty subsets of V , the simplexes in , such that: • If s ∈ V , then {s} ∈ .
• If F ∈ and ∅ = E ⊂ F then E ∈ .
A simplicial complex is called flag or full, if it is determined by its 1-simplexes in the sense that {s 0 , . . . , s n } ∈ ⇐⇒ {s i , s j } ∈ for all 0 ≤ i < j ≤ n.
is called locally finite if every vertex of is contained in only finitely many simplexes of , and finitedimensional (of dimension n) if it contains no simplexes with more than n + 1-vertices. For a simplicial complex one can define the topological space | | associated to this complex. It is called the "geometric realization" of the complex and can be defined as the space of maps ∈ . If is locally finite, then | | is locally compact.
Let be a locally finite flag simplicial complex. Denote by V the set of its vertices. Define C as the universal C *algebra with positive generators h s , s ∈ V , satisfying the relations Here the sum is finite, because is locally finite. C 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 . Denote by I k the ideal in C generated by products containing at least n+1 different generators. The filtration (I k ) of C is called the skeleton filtration. Let be the standard n-simplex. Denote by C the associated universal C * -algebra with generators h s , s ∈ {s 0 , . . . , s n }, such that h s ≥ 0 and s h s = 1. Denote by I the ideal in C generated by products of generators containing all the h s i , i = 0, . . . , n. For each k, denote by I k the ideal in C generated by all products of generators h s containing at least k +1 pairwise different generators. We also denote by I ab k the image of I k in C ab . The algebra C and their http://www.springerplus.com/content/3/1/258 K-Theory was studied in details in (Omran and Gouda 2012). For any vertex t in there is a natural evaluation map C −→ 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 C −→ C defined above induces an isomorphism in K-theory. (ii) The surjective map I −→ I ab induces an isomorphism in K-theory, where I ab is the abelianization of 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.
Proposition 2. Let be a locally finite simplicial complex. Then C ab is isomorphic to C 0 (| |), the algebra of continuous functions vanishing at infinity on the geometric realization | | of .

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 (1981Cuntz ( , 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 , let ⊂ be the subcomplex generated by W and let I be the ideal in C generated by products containing all generators of C . Lemma 1. Let C and C as above, then we have Given a subset W ⊂ V with |W | = k + 1, let Let I denote the ideal in C generated by products containing all generatorsḣ i , i ∈ , and let B denote its closure. If W = W , then B B = 0, because the product of any two elements in B and B contains products of more than k + 1-different generators, which are equal to zero in the algebra C /I k+1 It is clear that 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 inḣ i containing at least k+1-different generators. Then m s ∈ B for some W and The space W ⊂V ,|W |=k+1 B is closed, because it is a direct sum of closed ideals. It follows that be the canonical evaluation map defined by where h i denotes the generator in C corresponding to the index i in W , in other words We prove that π W (I k+1 ) = 0. Since polynomials of the form since there is at least one i l which is not in W . For this index π W (h i l ) = 0. Thus π W (x) = 0. Therefore π W descends to a homomorphisṁ 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 defined. This shows that π W (C ) is a closed subalgebra in http://www.springerplus.com/content/3/1/258 C and isomorphic toπ W (C /I k+1 ). We haveπ W (B ) = I . It is clear that Ker π W is the ideal generated by h i for i not in W and therefore Kerπ W is generated byḣ i for i not in W . This comes at once from the definitions oḟ π W (ḣ i ) and π W (h i ) above and the fact that both are equal. We conclude that B Kerπ W = 0. This again implies that B ∩ Kerπ W = 0. Moreover the following diagram is commutative: So,π W (B ) is dense and closed in I . Thereforeπ W : B −→ I is injective and surjective.
As a consequence of the above lemma we have the following.

Proposition 3. Let C and I k defined as above. Then we have an isomorphism
where the sum is taken over all k-simplexes in .
Proof. As in the proof of lemma 1 above with = , we find that: In the following we study the C * -algebras C n associated to simplicial flag complexes of a specific simple type. These simplicial complexes is a subcomplex of the "noncommutative spheres" in the sense of Cuntz work (Cuntz 2002). We determine the K-theory of C 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 Ktheory of C * -algebras found in the references (Blackadar 1986;Murphy 1990;Rørdam et al. 2000;Wegge-Olsen 1993).
We denote by n the simplicial complex with n + 2 vertices, given in the form V n = {0 + , 0 − , 1, . . . , n}, be the universal C * -algebra associated to n . The existence of such algebras is due to Cuntz (2002). It is clear that for any element h i ∈ C n , we have h i ≤ 1.
Denote by I the natural ideal in C n generated by products of generators containing all h i , i ∈ V n . Then we have the skeleton filtration The aim of this section is to prove that the K-theory of the ideals I in the algebras C n is equal to zero. We have the following Lemma 2. Let C n be as above. Then C n is homotopy equivalent to C.
Proof. Let β : C−→C n be the natural homomorphism which sends 1 to 1 C n . For a fixed i ∈ V n such that i = 0 − , 0 + , define the homomorphism α : C n −→C by α(h i ) = 1 and α(h j ) = 0 for any j = i. Notice that The elements ϕ t (h j ), j ∈ V n , satisfy the same relations as the elements h j in C n : We note that ϕ 1 = id C n and ϕ 0 = β • α.
This implies that This means that C n is homotopy equivalent to C.
From the above lemma , we have K * (C n ) = K * (C), for * = 0, 1. http://www.springerplus.com/content/3/1/258 Now we describe the subquotients of the skeleton filtration in C n . Proposition 4. In the C * -algebra C n one has where the sum is taken over all subcomplexes of n which are isomorphic to the standard k-simplex and over all subcomplexes γ of n which 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 ⊂ V n with |W | = k + 1, we have two cases. Either {0 + , 0 − } is not a subset of W , then is a ksimplex, or {0 + , 0 − } is a subset of W , then is a subcomplex in n isomorphic to γ . This proves our proposition.
Lemma 3. For the complex n with n+2 vertices, C n /I 1 is commutative and isomorphic to C n+2 .
Proof. Letḣ i denote the image of a generator h i for C n . One has the following relations: For everyḣ i in C n /I 1 we havė Hence C n /I 1 is generated by n + 2 different orthogonal projections and therefore C n /I 1 ∼ = C n+2 . Lemma 4. I 1 /I 2 in C n is isomorphic to I ab 1 /I ab 2 in C ab n .
Proof. From the proposition 4 above, one has where 1 is 1-simplex, and I ab 1 /I ab 2 ∼ = 1 I ab 1 .
Lemma 6. C n /I 2 is a commutative C * -algebra.
Lemma 7. C * -algebra C 1 is commutative and K * (I 2 ) = 0, * = 0, 1 where I 2 is an ideal in C 1 defined as in the above.
By a similar computation we can show that h 0 + and h 1 commute. This implies that C 1 is commutative. Therefore I 2 = 0 in C 1 Then, at once K * (I 2 ) = 0.