Open Access

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

SpringerPlus20143:258

https://doi.org/10.1186/2193-1801-3-258

Received: 20 December 2013

Accepted: 17 March 2014

Published: 21 May 2014

Abstract

In this article we introduce a universal C-algebras associated to certain simplicial flag complexes. We denote it by C Γ 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

Keywords

Simplicial complexesK- theory of C-algebras; Universal C-algebras

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  ≠ EF then EΣ.

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 s V Σ f ( s ) = 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 C Σ as the universal C-algebra with positive generators h s , sV, satisfying the relations
h s 0 h s 1 h s n = 0 whenever { s 0 , s 1 , , s n } V Σ ,
s V Σ h s h t = h t t V Σ .

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

C Σ 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, tV Σ . Denote by I k the ideal in C Σ generated by products containing at least n + 1 different generators. The filtration (I k ) of C Σ is called the skeleton filtration.

Let
Δ : = ( s 0 , , s n ) R n + 1 | 0 s i 1 , i = 1 n s i = 1

be the standard n-simplex. Denote by C Δ the associated universal C-algebra with generators h s , s {s0, …, s n }, such that h s  ≥ 0 and s h s = 1 . Denote by I Δ the ideal in C Δ 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 Δ generated by all products of generators h s containing at least k + 1 pairwise different generators. We also denote by I k ab the image of I k in C Δ ab . The algebra C Δ 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 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.

Remark 1

Let Δ and I C as above. Then K ( I ) K ( C ) , = 0 , 1 , if the dimension n of is even and K ( I ) K ( C 0 ( 0 , 1 ) ) , = 0 , 1 , if the dimension n of is odd.

Proposition 2

Let Σ be a locally finite simplicial complex. Then C Σ 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 WV Σ , let ΓΣ be the subcomplex generated by W and let I Γ be the ideal in C Γ generated by products containing all generators of C Γ .

Lemma 1

Let C Σ and C Γ as above, then we have
I k / I k + 1 W V Σ , | W | = k + 1 I Γ

Proof

C Σ / I k + 1 is generated by the images h i ̇ , iV Σ of the generators in the quotient.

Given a subset WV Σ with |W| = k + 1, let
C Γ = C ( { h i ̇ | i W } ) C Σ / I k + 1 .

Let I Γ denote the ideal in C Γ generated by products containing all generators h i ̇ , iΓ, and let B Γ denote its closure. If W W , then B Γ B Γ = 0 , because the product of any two elements in B Γ and B Γ contains products of more than k + 1-different generators, which are equal to zero in the algebra C Σ / I k + 1

It is clear that B Γ I k / I k + 1 so that
W V Σ , | W | = k + 1 B Γ I k / I k + 1 .
Conversely, let xI k  / Ik+1. Then there is a sequence (x n ) converging to x, such that each x n is a sum of monomials m s in h i ̇ containing at least k + 1-different generators. Then m s B Γ for some W and
x n = m s W V Σ , | W | = k + 1 B Γ .
The space W V , | W | = k + 1 B Γ is closed, because it is a direct sum of closed ideals. It follows that
I k / I k + 1 = W V Σ , | W | = k + 1 B Γ
Let now
π W : C Σ C Γ .
be the canonical evaluation map defined by
π W ( h i ) = h i i W 0 if i W ,
where h i denotes the generator in C Γ corresponding to the index i in W, in other words
C Γ = C ( h i | i W ) .
We prove that π W (Ik+1) = 0. Since polynomials of the form
h i 0 h i j h i k + 1 , i 0 , , i j , , i k + 1 , V Σ
are dense in Ik+1, it is enough to show that π W (x) = 0 for each such polynomial x. We have
π W ( x ) = h i 0 h i j h i k + 1 = 0 ,
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 homomorphism
π W ̇ : C Σ / I k + 1 C Γ
Now we show that π W is surjective as follows: Since π W (Ik+1) = 0, we have Ker π W Ik+1. It follows that the following diagram
C Σ C Γ C Σ / I k + 1
commutes and π W ̇ ( h i ̇ ) : = π W ( h i ) = h i , i W is well defined. This shows that π W ( C Σ ) is a closed subalgebra in C Γ and isomorphic to π W ̇ ( C Σ / I k + 1 ) . We have π W ̇ ( B Γ ) = I Γ . It is clear that Ker π W is the ideal generated by h i for i not in W and therefore Ker π W ̇ is generated by h i ̇ for i not in W. This comes at once from the definitions of π W ̇ ( h i ̇ ) and π W (h i ) above and the fact that both are equal. We conclude that B Γ Ker π W ̇ = 0 . This again implies that B Γ Ker π W ̇ = 0 . Moreover the following diagram is commutative:
C Σ C Γ B Γ I Γ B Γ / Ker π W ̇ .

So, π W ̇ ( B Γ ) is dense and closed in I Γ . Therefore π W ̇ : B Γ I Γ is injective and surjective.

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

Proposition 3

Let C Δ and I k defined as above. Then we have an isomorphism
I k / I k + 1 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 = I .

In the following we study the C-algebras C Γ 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 C Γ 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 Γ n the simplicial complex with n + 2 vertices, given in the form
V Γ n = { 0 + , 0 - , 1 , , n } ,
and
Γ n = { γ V Γ n | { 0 + , 0 - } γ } .
Let
C Γ n = C ( h 0 - , h 0 + , h 1 , h 2 , , h n | h 0 - h 0 + = 0 , h i 0 , i h i = 1 , i )

be the universal C- algebra associated to Γ n . The existence of such algebras is due to Cuntz (2002). It is clear that for any element h i C Γ n , we have h i  ≤ 1.

Denote by the natural ideal in C Γ n generated by products of generators containing all h i , i V Γ n . Then we have the skeleton filtration
C Γ n = I 0 I 1 I 2 ..... I n + 1 : = I

The aim of this section is to prove that the K-theory of the ideals in the algebras C Γ n is equal to zero. We have the following

Lemma 2

Let C Γ n be as above. Then C Γ n is homotopy equivalent to .

Proof

Let β : C C Γ n be the natural homomorphism which sends 1 to 1 C Γ n . For a fixed i V Γ n such that i ≠ 0-,0+, define the homomorphism
α : C Γ n C
by α(h i ) = 1 and α(h j ) = 0 for any j ≠ i. Notice that α β = id C . Now define φ t : C Γ n C Γ n , h i h i + ( 1 - t ) ( j i h j ) , h j t ( h j ) j V Γ n { i } . The elements φ t (h j ), j V Γ n , satisfy the same relations as the elements h j in C Γ n :
  1. (i)

    φ t (h j ) ≥ 0

     
  2. (ii)
    φ t j h j = φ t ( h i ) + j i φ t ( h j ) = h i + ( 1 - t ) j i h j + t j i h j = h i + j i h j for fixed i = j h j = 1 for all j ,
     
  3. (iii)

    φ t ( h 0 - ) φ t ( h 0 + ) = t 2 ( h 0 - h 0 + ) = 0 .

     

We note that φ 1 = id C Γ n and φ0 = βα.

This implies that
φ 0 = β α Id C Γ n .

This means that C Γ n is homotopy equivalent to .

From the above lemma, we have K ( C Γ n ) = K ( C ) , for  = 0,1.

Now we describe the subquotients of the skeleton filtration in C Γ n .

Proposition 4

In the C-algebra C Γ n one has
I k / I k + 1 I γ I γ ,

where the sum is taken over all subcomplexes of Γ n which are isomorphic to the standard k-simplex and over all subcomplexes γ of Γ 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 Γ 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 Γ n isomorphic to γ. This proves our proposition.

Lemma 3

For the complex Γ n with n + 2 vertices, C Γ n / I 1 is commutative and isomorphic to C n + 2 .

Proof

Let h i ̇ denote the image of a generator h i for C Γ n . One has the following relations:
i h i ̇ = 1 , h i ̇ h j ̇ = 0 , i j .
For every h i ̇ in C Γ n / I 1 we have
h i ̇ = h i ̇ i h i ̇ = h ̇ i 2 .

Hence C Γ n / I 1 is generated by n + 2 different orthogonal projections and therefore C Γ n / I 1 C n + 2 .

Lemma 4

I1 / I2in C Γ n is isomorphic to I 1 ab / I 2 ab in C Γ n ab .

Proof

From the proposition 4 above, one has
I 1 / I 2 1 I 1
where 1 is 1-simplex, and
I 1 ab / I 2 ab 1 I 1 ab .
Since I 1 C 1 is commutative because the generators of C 1 commute (since h s 1 = 1 - h s 0 ). We get
I 1 I 1 ab C 0 ( 0 , 1 ) .

Lemma 5

In C Γ n , we have K0(I1 / I2) = 0 and K 1 ( I 1 / I 2 ) = Z n 2 + 2 n .

Proof

By applying above lemma, and proposition 4, we have
I 1 / I 2 1 I 1

The sum contain n 2 + 2 n 1-simplex, 1C0(0, 1). where K0(C0(0, 1)) = 0 and K 1 ( C 0 ( 0 , 1 ) ) = Z .

Lemma 6

C Γ n / I 2 is a commutative C-algebra.

Proof

Consider the extension
0 I 1 / I 2 C Γ n / I 2 C Γ n / I 1 0

and the analogous extension for the abelianized algebras.

The extensions above induce the following commutative diagram:
0 I 1 / I 2 C Γ n / I 2 C Γ n / I 1 0 0 I 1 ab / I 2 ab C Γ n ab / I 2 ab C Γ n ab / I 1 ab 0
We have from 3 isomorphisms C Γ n / I 1 C Γ n ab / I 1 ab C n + 2 and from 4 that I 1 / I 2 I 1 ab / I 2 ab , so
C Γ n / I 2 C Γ n ab / I 2 ab .

Lemma 7

C-algebra C Γ 1 is commutative and K(I2) = 0,  = 0, 1 where I2is an ideal in C Γ 1 defined as in the above.

Proof

C Γ 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 C Γ 1 is commutative. Therefore I2 = 0 in C Σ 1 Then, at once K(I2) = 0.

Declarations

Authors’ Affiliations

(1)
Faculty of Science, Taif University
(2)
Faculty of science, Math. Department, KSA South valley university

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© Omran; licensee Springer. 2014

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