Simplicial cohomology of l 1 (N) and other semi-group algebras

Transcription

1 Simplicial cohomology of l 1 (N) and other semi-group algebras Yasser Farhat, Département de mathématiques et statistique Université Laval Banach algebras 2011

2 Outline 1. Introduction 2. Simplicial cohomology of l 1 (semilattice) 3. Simplicial cohomology of l 1 (Z + ) 4. Simplicial cohomology of l 1 (S) for discrete semigroup of R +

3 Introduction Let A be a Banach algebra, A the dual of A. Let n 1 and C n (A, A) the Banach space generated by {a 1 a 2 a n+1, a k A and 1 k n + 1}. Let d n : C n+1 (A, A) C n (A, A) X = a 1 a 2 a n+2 k=n+1 k=0 dk n (X ), where d n 0 (X ) = a 2 a 3 a n+1 a n+2 a 1, d n k (X ) = ( 1)k a 1 a k a k+1 a n+2, if 1 k n + 1,

4 Introduction C n 1 (A, A) dn 1 C n (A, A) dn C n+1 (A, A). We have that d n 1 d n (X ) = 0, for all X C n (A, A). Let δ n = (d n ) and (C n (A, A)) = C n (A, A ), then δ n δ n 1 = 0. Thus we get the following chain C n 1 (A, A ) δn 1 C n (A, A ) δn C n+1 (A, A ).

5 Introduction Let Φ kerδ n. Does there exist ψ C n 1 (A, A ) : δ n 1 ψ = Φ? Can we find a bounded linear operator σ : C n (A, A ) C n 1 (A, A ) : Φ = δ n 1 σφ

6 Introduction As we have δ n = (d n ) and C n (A, A ) = (C n (A, A)),it suffices to show that there exist a bounded linear operator s p : C p (A, A) C p+1 (A, A), for p {n, n 1}, I (X ) = (s n 1 d n 1 + d n s n )(X ) for all X C n (A, A), where I is identity of the space. To reach the result, we take σ p = (s p ), for p {n 1, n}, then we obtain φ = δ n 1 (σ n 1 φ).

7 Cohomology of l 1 (semilattice) In the paper [C G W 2010] Y.Choi et al showed that simplicial cohomology groups H n (A, A ) vanish for all n 2, where A is a Band semigoup algebra, A = l 1 (S), (S is a Band semigroup if S is a semigroup and ss = s, for all s S). To do so, they determined the cyclic cohomology of A, HC n (A, A ), and used the Connes-Tzygan long exact sequence to deduce the result for H n (A, A ).

8 Cohomology of l 1 (semilattice) As part of my PhD work (supervisor : F. Gourdeau), I have succeeded in obtaining H n (A, A ) = 0, for all n 2, without using cyclic cohomology. To save time, I ll present the proof for particular case S that is a semilattice, (S is a commutative Band semigroup).

9 Cohomology of l 1 (semilattice) Definition s, t S : s t st = s Choi et al showed the result through two steps.

10 Cohomology of l 1 (semilattice), First step We denote δ s by s, where s S. Definition Let X = s 1 s 2 s n+1 C n (A, A). We say that X has a minimal element, say s k0 for 1 k 0 n + 1, if s k0 s k for all 1 k n + 1. In this step they showed that there exist s such that ( sd + ds ) (X ) = if X (x x x) and X + p X Y m m=1 else, where X has at least one minimal element, 0

11 Cohomology of l 1 (semilattice), First step And Y m has strictly more minimal element than X. As X has at most n + 1 minimal element, by repeating the algorithm (n + 1)-times, we get a finite combination of form (x x x). Later, they showed that we can cobound in this space. At this step, we can omit the word cyclic to get the same result in the simplicial homology.

12 Cohomology of l 1 (semilattice), Second step The purpose of this part is to find a suitable application Q such that Q(X ) is a finite linear combination of elements which admit at least one minimal element. In this section we will also need some definitions.

13 Cohomology of l 1 (semilattice), Second step Definition Let X = s 1 s n+1 be without minimal element. We say that a subtensor s k s l has a minimal left element if s k s i for all i in the cyclic interval [k, l]. A subtensor is a left-block if it has a minimal left element and is not strictly included in another subtensor which has a minimal left element.

14 For elementary tensor X without minimal element, it is easy to see that it has a unique decomposition into left-blocks, and therefore we can define I X = {i {1, 2,..., n+1} : x i is the first component of a left-block} If X has a minimal element, I X =.

15 Cohomology of l 1 (semilattice), Second step Definition If T is a finite semilattice, and α T, the height of α in T, denoted by h T (α), is the length of the longest descending chain in T which starts at α. If X = x 1 x n+1 is an elementary tensor, let L(X ) be the subsemilattice that is generated by {x 1,..., x n+1 } and we define the height of X to be n+1 h(x ) = h L(X ) (x i ). It s important to see that if X has no minimal element then i=1 n + 1 h(x ) n(n + 1).

16 Cohomology of l 1 (semilattice), Second step Now, we can give a bounded linear map s. Let X an elementary tensor, and s(x ) = i I X ( 1) i x 1 x i x i x n+1. Then p q { (s n 1 d n 1 +d n s n t(x ) if x1 < x )(X ) = kx + Y i + Z j + n+1 0 else i=1 j=1

17 where Y i has strictly less left blocks than X, h(z j ) < h(x ), 1 k n + 1, t(x ) = ( 1) n (x n+1 x 1 x n ).

18 Cohomology of l 1 (semilattice), Second step Let P r = I 1 r (sd + ds) for r > 0. Then P n 1 k (X ) = p q { t V i + W j + n 1 (X ) if x 1 < x n+1 < x n < < x 3 0 else i=1 j=1 where V i has strictly less left blocks than X and h(w j ) < h(x ). If in this case X has no minimal element, then t does not occur in the next iteration. Therefore, for P = P 1 P n+1, we have P n (X ) is a finite sum of elements which have strictly less left blocks than X or elements of height strictly smaller than the height of X

19 Cohomology of l 1 (semilattice), Second step Remember that if X has no minimal element then n + 1 h(x ) n(n + 1) and X has at most n + 1 left-block. Then for X an elementary tensor, we get P n3 (X ) is a finite sum of elements that have at least one minimal elelment. Which completes the work.

20 Cohomology of l 1 (Z + ) I managed to not use cyclic cohomology algebra in another algebra : l 1 (Z + ).

21 Cohomology of l 1 (Z + ) Let A = l 1 (Z + ) be the unital semigroup algebra. By [G. J. W, 2005] we know that H n (A, A ) vanish when n 2. To do so, Gourdeau et al determined the cyclic cohomology of A, HC n (A, A ) and used the Connes-Tzygan long exact sequence to deduce the result for H n (A, A ). In the fifth section of this paper the authors gave an explicit formula for the contracting homotopy in the cyclic case.

22 Cohomology of l 1 (Z + ) We give an explicit formula for the contracting homotopy in simplicial homology. ρ(m) = m 1 j=0 m j j Let s 1, s 2 : C n (A, A) C n+1 (A, A) where s 1 (m 1 m n+1 ) = n+1 k=1 if N 0, and if N = 0. ( 1) k+1 (m 1 ρ(m k ) m n+1 ) N ( )

23 Cohomology of l 1 (Z + ) s 2 (m 1 m n+1 ) = ( 1) n+1 m 1 1 (j m 2 m n m n+1 + m 1 j 0) N j=0 if N 0, and if N = 0, and (0 0 0) N = m m n+1

24 Cohomology of l 1 (Z + ) (s 1 d + ds 1 )(X ) = X + [ ( 1) n N m 2 m n ( m n+1 + ρ(m 1 ) )] + [ 1 N d 0(ρ(m 1 ) m 2 m n+1 ) ]

25 Cohomology of l 1 (Z + ) s 2 d + ds 2 = s 2 d 0 + d 0 s 2 + s 2 d 1 + d 1 s 2 + s 2 d 2 + d 2 s 2. s 2 d n + d n s 2 + d n+1 s 2 = d 0 s 2 + d n+1 s 2 Because s 2 d 0 + s 2 d 1 + d 1 s 2 = 0 and s 2 d k + d k s 2 = 0, for 2 k n.

26 Cohomology of l 1 (Z + ) As d 0 s 2 (X ) = ( 1)n+1 N m 1 1 j=0 m 2 m n+1 + m 1 j j = ( 1)n m 2 m n m n+1 + ρ(m 1 ), N d n+1 s 2 (X ) = 1 N m 1 1 j=0 j m 2 m n+1 + m 1 j = 1 N d 0(ρ(m 1 ) m 2 m n+1 ) We conclude that ( ) (s 1 + s 2 )d + d(s 1 + s 2 ) (X ) = X

27 Generalization to other ordered discrete semigroup Let G be a subgroup of R with the discrete topology, let S = G R +. Let B = l 1 (S). By [G J W 2005] we know that, for all n 2 : H n (B, B ) = 0 The proof uses Connes-Tzygan. The contracting homotopy, in cyclic cohomology is given by σ 1 T (m 1,..., m n )(m n+1 ) = n+1 ( 1) k+1 µ ( T [m 1,..., m k x, x,..., m n+1 ] a ) k k=1 0

28 Generalization to other ordered discrete semigroup where µ is an invariant mean on the amenable group S N = {s S : 0 s < N} with the addition is mod N, and N = m m n+1 T [m 1,..., m k x, x,..., m n+1 ] a k 0 l (S N ) defined by T [m 1,..., m k x, x,..., m n+1 ] a k 0 (s) = { T (m1,..., m k s, s,..., m n+1 ) if 0 s < a k 0 otherwise

29 Generalization to other ordered discrete semigroup The contracting homotopy, in the simplicial cohomology is given by σ = σ 1 + σ 2 where σ 2 T (m 1,..., m n )(m n+1 ) = ( 1) n+1 µ ( T [x, m 2,..., m n+1 + m 1 x, 0] m 1 0 )