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

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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

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 Cdefined 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 ,iW 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 Γn 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 Γn. 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 αβ= 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 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 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 Γn isomorphic to γ. This proves our proposition.

Lemma 3

For the complex Γnwith 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 +2n 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.

References

  1. Blackadar B: K-theory for operator algebras. MSRI Publ. 5, Cambridge University Press; 1986.

  2. Cuntz J: K-theory for certain C-algebras. Ann Math 1981, 113: 181-197. 10.2307/1971137

  3. Cuntz J: K-theory and C-algebras. Proc. Conf. on K-theory (Bielefeld,1982). Springer Lecture Notes in Math. 1046 1984, 55-79.

  4. Cuntz J: A survey of some aspects of noncommutative geometry. Jahresber D Dt Math-Verein 1993, 95: 60-84.

  5. Cuntz J: Non-commutative simplicial complexes and Baum-Connes-conjecture. GAFA, Geom Func Anal 2002, 12: 307-329. 10.1007/s00039-002-8248-6

  6. Davidson KR: C-algebras by example. Providence: Fields Institute monographs, Amer. Math. Soc; 1996.

  7. Murphy GJ: C-algebras and operator theory. Academic Press; 1990.

  8. Omran S: C-algebras associated with higher-dimensional noncommutative simplicial complexes and their K-theory. Dissertation, Münster: Univ. Münster-Germany; 2005.

  9. Omran S: C-algebras associated noncommutative circle and their K -theory. Aust J Math Anal Appl 2013, 10(1):1-8. Article 8

  10. Omran S, Gouda GhY: On the K-theory of C-algebras associated with n -Simplexes. Int J Math Anal 2012, 6(17):847-855.

  11. Rørdam M, Larsen F, Laustsen N: An introduction to K-theory for C-algebras. London Mathematical Society Student Text 49, Cambridge University Press; 2000.

  12. Wegge-Olsen NE: K-theory and C-algebras. New York: Oxford University Press; 1993.

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Correspondence to Saleh Omran.

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Keywords

  • Simplicial complexes
  • K- theory of C-algebras; Universal C-algebras