Shifts and characterization of the elements of A

Matthieu Deneufchatel

Description

Shifts and characterization of the elements of A? Matthieu Deneufchatel 21/04/2010 We take the following notations : if A be an algebra and f ? A?, – fx : y 7? f(xy) (left shift) ; – xf : y 7? f(yx) (right shift) ; – xfy : z 7? f(yzx) ; – A? denotes the finite dual (also called Sweedler's dual). Theorem 0.1. Theorem by Abe (extended by Schutzenberger's condition (property (vi) be- low)) - Characterization of the elements of A? : Let A be an algebra and f ? A?. The following properties are equivalent : – (i) tµ(f) ? A? ?A? ; – (ii) The family (fx)x?A is of finite rank ; – (iii) The family (xf)x?A is of finite rank ; – (iv) The family (xfy)x,y?A is of finite rank ; – (v) f(xy) = n∑ i=1 fi(x)gi(y) ; – (vi) ? ? : A ? kn?n and (?, ?) ? k1?n ? kn?1 such that ?x ? A, f(x) = ??(x) ?; – (vii) Ker(f) contains an ideal of finite codimension (i.

• therefore tµ


• argument allows


• now


• argument also applies


• take ?


• finite rank


• among all


• ?fi ?


• therefore

Sujets

• Logical consequence

Informations

• Publié par: profil-urra-2012
• Publié le: 01 avril 2010
• Nombre de lectures: 30
• Langue: English

Extrait

◦ Shifts and characterization of the elements ofA

MatthieuDeneufchˆatel 21/04/2010

∗ We take the following notations : ifAbe an algebra andf∈ A, –f x :y7→f(xy) (left shift); –f:y7→f(yx;) (right shift) x –f y :z7→f(yzx) ; x ◦ –Adenotes the finite dual (also called Sweedler’s dual). Theorem 0.1.u¨hcneztgrebs’rebeyAxt(edeenySdbytcmobnrdeietoiTohn(proper(vi)be- ◦ low))-Characterization of the elements ofA: ∗ LetAbe an algebra andf∈ A. The following properties are equivalent : t∗ ∗ –(i)µ(f)∈ A⊗ A; –(ii)The family(f x )is of finite rank; x∈A –(iii)The family(f);is of finite rank x x ∈A f;is of finite rank –(iv)The family( x y ) y∈A x, n X –(v)f(xy) =f i (x)g i (y); i=1 n×n1×n n×1 –(vi)∃α:A →kand(λ, γ)∈k×ksuch that ∀x∈ A, f(x) =λ α(x)γ; –(vii) Ker(f)contains an ideal of finite codimension (i.e. there exists an idealIsuch that dim(Ker(f)/I)<∞).(1) Proof :We begin by two lemmas : n X Lemma 0.2.Lettbe an element ofV⊗W. If we decomposetasf i ⊗g i withnminimal, then the i=1 f i ’s and theg i ’s form free families. n−1 X Proof :Indeed, if that was not the case, one could write thaty n =α i y i and i=1 n n−1 X X t=x n ⊗y n +x i ⊗y i = (α i x n +x i )⊗y i . i=1i=1 But this equation shows thatnis not minimal and it is impossible by hypothesis.✷ Lemma 0.3.LetU, VandWbe vector spaces andφ:U×V→Wa bilinear map. Then ifz∈Im(φ) P n and ifz=φ(x i , y i )withnminimal among all decompositions of this form, the families(x i ) 1≤i≤n i=1 and(y i ) 1≤i≤n are free in their respective spaces. Note that the first lemma is a consequence of the second one : takeφ=⊗.

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