# Completely Dissipative Maps and Stinespring's Dilation-Type Theorem onσ-C*-Algebras

Authors: Tian Zhou Xu
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

The notion of the completely dissipative maps onσ-C*-algebras is introduced. We show that a completely dissipative map induces a representation of a σ-C*-algebra. The classical Stinespring's dilation-type theorem is extended to a more general setting.


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## Body

## 1. Introduction

There has been increased interest [1–12] in topological *-algebras that are inverse limits of C*-algebras, called Pro-C*-algebras. These algebras were introduced in [5] as a generalization of C*-algebras and were called locally C*-algebras. The same objects have been studied by other authors, for instance in [2, 13]. It is shown in [7, 14] that they arise naturally in the study of certain aspects of C*-algebras such as the tangent algebras of C*-algebras, multipliers of Pedersen's ideal, noncommutative analogues of classical Lie groups, and K-theory.Motivated mainly by the works [11, 12, 15–18], in this paper, we shall introduce the notion of the completely dissipative maps on σ-C*-algebras consider an abstract version of Stinespring's dilation-type theorem, and explore the natural relation completely dissipative maps to σ-C*-algebras. We shall show that a completely positive definite map induces a representation of a σ-C*-algebra. We also get a classification of completely positive definite maps on σ-C*-algebras. Our approach is motivated by the setting of Banach algebras and C*-algebras. This paper is organized as follows. In Section 1, introduction and some preliminaries are given. In Section 2, we present some definitions, results and notation which will be used throughout the paper. In Section 3, we shall prove the main result of this paper.All the vector spaces and algebras considered throughout this paper are over the fieldℂ of complexes, and the topological spaces are supposed to be Hausdorff.A locallym-convex (lmc-) algebra is a topological algebra A whose topology is defined by a family (pα), α∈I (I a directed index set), of submultiplicative seminorms. A is called an lmc*-algebra if, in addition, has an involution * such that pα(x*)=pα(x), for any α∈I, x∈A. If moreover, each pα, α∈I, is a C*-seminorm, that is, pα(x*x)=pα(x)2 for any α∈I, x∈A, A is called to be an lmc-C*-algebra. Given an lmc-algebra A denote by (Aα) the inverse system of Banach algebras corresponding to A, that is, Aα is the completion of the normed algebra A/Nα, Nα=Ker(pα), α∈I, under norm ∥·∥α, with ∥xα∥α=pα(x), xα=x+Nα, x∈A. A Pro-C*-algebra A is a complete lmc-C*-algebra. In this setting, (pα) can be replaced with the collection S(A), the set of all continuous C*-seminorms on A. For a Pro-C*-algebra A, and for a p∈S(A), the normed *-algebra (A/Nα, ∥·∥α) is automatically complete (see [13]), so that Aα=A/Nα is a C*-algebra. Thus, A is an inverse limit of C*-algebras, that is, A=lim←Aα(see [6]). A metrizable Pro-C*-algebra is called a σ-C*-algebra with its topology determined by a countable subfamily (pn) of S(A), n∈ℕ.


## 2. Preliminaries

In this section, we present some definitions and results used in this paper. Our first goal is to define completely dissipative maps onσ-C*-algebras. For this, we need to recall the following theorem in [15, Theorem 3.1.2 and Corollary 3.2.1].Theorem 2.1 . (see [15]).
LetA be a  C*-algebra with unit  e,  L:A→A a bounded *-map and  L(e)=0. Let  ϕt=exp(tL). Then  ϕt(t∈ℝ+) is completely positive on  A if and only if  L is completely dissipative, that is,  D(Ln;X,X)≥0 for all  X∈Mn(A), where  D(Ln;X,Y)=Ln(X*Y)-Ln(X*)Y-X*Ln(Y), for all  X,Y∈Mn(A), and  Ln=L⊗In.We can now introduce completely dissipative maps onσ-C*-algebras as follows.Definition 2.2.
LetA be a σ-C*-algebra, H a Hilbert space, ϕ:A→B(H) a continuous map. Set D:A×A→B(H) as D(a,b)=ϕ(ab)-ϕ(a)-ϕ(b). If the following conditions are satisfied:(i)
(2.1)[D(ai*,aj)]i,j=1n∈Mn+(B(H));(ii)
there exists a functionρ:A→ℝ+ such that
(2.2)∑i=1n∑j=1n〈[D(bi*ai*a*aaj,bj)-D(ai*a*aaj,bj)]xj,xi〉H≤ρ(a*a)∑i=1n∑j=1n〈[D(bi*ai*aj,bj)-D(ai*aj,bj)]xj,xi〉Hfor every a,a1,a2,…,an,b1,b2,…,bn∈A, x1,x2,…,xn∈H, and n∈ℕ, then ϕ is called a completely dissipative map.From the corresponding facts in the Aronszajn-Kolmogorov theorem of [17], we have the following result.Lemma 2.3 (see [17]).
LetA be a  σ-C*-algebra,  H a Hilbert space, and a map  T:A×A→B(H) be given such that, for all  a1,a2,…,an∈A,
(2.3)[T(ai*,aj)]i,j=1n∈Mn+(B(H)).
Then there exists a Hilbert space K and a map V:A→B(H,K) such that
(2.4)T(a,b)=V(a*)*V(b),∀a,b∈A.Definition 2.4.
LetA be a σ-C*-algebra, H a Hilbert space, and a map ϕ:A×A×A→B(H). We say that ϕ is completely positive definite if for any ai,bi∈A, i=1,2,…,n∈ℕ, we have
(2.5)[ϕ(bi*,ai*aj,bj)]i,j=1n∈Mn+(B(H)).
If there exists a function ρ:A→ℝ+ such that the map ϕ:A×A×A→B(H) satisfies the following additional condition:
(2.6)∑i=1n∑j=1n〈ϕ(bi*,ai*a*aaj,bj)xj,xi〉≤ρ(a*a)∑i=1n∑j=1n〈ϕ(bi*,ai*aj,bj)xj,xi〉,
for every a,a1,a2,…,an,b1,b2,…,bn∈A, x1,x2,…,xn∈H, and n∈ℕ, then we say, ϕ is relatively bounded.


## 3. The Stinespring's Dilation-Type Theorem

To prove the main theorem, the following things have to be elucidated. IfX and Y are vector spaces, we denote by X⊗Y their algebraic tensor product. This is linearly spanned by elements x⊗y(x∈X, y∈Y). If σ:X×Y→Z is a bilinear map, where X,Y, and Z are vector spaces, then there is a unique linear map σ′:X⊗Y→Z such that σ′(x⊗y)=σ(x,y) for all x∈X and y∈Y.Ifτ,μ are linear functionals on the vector spaces X,Y, respectively, then there is a unique linear functional τ⊗μ on X⊗Y such that (3.1)(τ⊗μ)(x⊗y)=τ(x)μ(y),∀x∈X,y∈Y,
since the function (3.2)X×Y⟶C,(x,y)⟶τ(x)μ(y)
is bilinear.Suppose that∑j=1nxj⊗yj=0, where xj∈X and yj∈Y. If y1,…,yn are linearly independent, then x1=⋯=xn=0. For, in this case, there exist linear functionals μj:Y→ℂ such that μj(yi)=δij(Kronecker delta). If τ:X→ℂ is linear, we have (3.3)0=(τ⊗μj)(∑i=1nxi⊗yi)=∑i=1nτ(xi)ρj(yi)=∑i=1nτ(xi)δij=τ(xj).
Thus, τ(xj)=0 for arbitrary τ, and this shows that x1=⋯=xn=0.Similarly, if∑j=1nxj⊗yj=0, and x1,…,xn are linearly independent, then y1=⋯=yn=0.Theorem 3.1.
LetA be a  σ-C*-algebra with unit  e,  H a Hilbert space,  ϕ:A→B(H) be a completely dissipative map, and  D̂:A×A×A→B(H) defined by
(3.4)D̂(a,b,c)=D(ab,c)-D(b,c),
where D(a,b)=ϕ(ab)-ϕ(a)-ϕ(b), then(i)
D̂ is a relatively bounded completely positive definite map,(ii)
there exists a Hilbert spaceX, a representation φ of A on X, and a map V:A→B(H,X) such that
(3.5)D̂(a,b,c)=V(a*)*φ(b)V(c),∀a,b,c∈A,(iii)
{φ(a)V(b)x:a,b∈A,x∈H} spans a dense subspace of X,(iv)
the representation(φ,X,V) associated with ϕ is unique up to a unitary equivalence,(v)
the mapV:A→B(H,X) is a 1-cocycle with respect to the representation φ, that is, Δφ1V=φ(a)V(b)-V(ab)+V(a)=0.Proof.
(i) By the definition, for anyai,bi∈A,i=1,2,…,n,n∈ℕ, we have
(3.6)D̂(bi*,ai*aj,bj)=D(bi*ai*aj,bj)-D(ai*aj,bj)=ϕ(bi*ai*ajbj)-ϕ(bi*ai*aj)-ϕ(bj)-[ϕ(ai*ajbj)-ϕ(ai*aj)-ϕ(bj)]=ϕ(bi*ai*ajbj)-ϕ(bi*ai*aj)-ϕ(ai*ajbj)+ϕ(ai*aj),D(bi*ai*,ajbj)-D(bi*ai*,aj)+D(ai*,aj)-D(ai*,ajbj)=ϕ(bi*ai*ajbj)-ϕ(bi*ai*)-ϕ(ajbj)-[ϕ(bi*ai*aj)-ϕ(bi*ai*)-ϕ(aj)]+[ϕ(ai*aj)-ϕ(ai*)-ϕ(aj)]-[ϕ(ai*ajbj)-ϕ(ai*)-ϕ(ajbj)]=ϕ(bi*ai*ajbj)-ϕ(bi*ai*aj)+ϕ(ai*aj)-ϕ(ai*ajbj).Thus,
(3.7)D̂(bi*,ai*aj,bj)=D(bi*ai*,ajbj)-D(bi*ai*,aj)+D(ai*,aj)-D(ai*,ajbj).
From Lemma 2.3, there exists a Hilbert space K, and a map V:A→B(H,K) such that
(3.8)D(a,b)=V(a*)*V(b),∀a,b∈A.
It follows that
(3.9)D̂(bi*,ai*aj,bj)=V(aibi)*V(ajbj)-V(aibi)*V(aj)+V(ai)*V(aj)-V(ai)*V(ajbj)=V(aibi)*[V(ajbj)-V(aj)]+V(ai)*[V(aj)-V(ajbj)]=[V(aibi)*-V(ai)*][V(ajbj)-V(aj)]=[V(aibi)-V(ai)]*[V(ajbj)-V(aj)].
Therefore, we obtain
(3.10)[D̂(bi*,ai*aj,bj)]i,j=1n∈Mn+(B(H)).
Sinceϕ:A→B(H) be a completely dissipative map, then by Definition 2.2(ii), there exists a function ρ:A→ℝ+ such that
(3.11)∑i=1n∑j=1n〈[D(bi*ai*a*aaj,bj)-D(ai*a*aaj,bj)]xj,xi〉H≤ρ(a*a)∑i=1n∑j=1n〈[D(bi*ai*aj,bj)-D(ai*aj,bj)]xj,xi〉H,
where D(a,b)=ϕ(ab)-ϕ(a)-ϕ(b), and a,a1,a2,…,an,b1,b2,…,bn∈A, x1,x2,…,xn∈H. Thus, we have
(3.12)∑i=1n∑j=1n〈D̂(bi*,ai*a*aaj,bj)xj,xi〉H=∑i=1n∑j=1n〈[D(bi*ai*a*aaj,bj)-D(ai*a*aaj,bj)]xj,xi〉H≤ρ(a*a)∑i=1n∑j=1n〈[D(bi*ai*aj,bj)-D(ai*aj,bj)]xj,xi〉H=ρ(a*a)∑i=1n∑j=1n〈D̂(bi*,ai*aj,bj)xj,xi〉H.
It follows that D̂ is relatively bounded.
(ii) LetA⊗A⊗H be the algebraic tensor product of A,A, and H. The algebraic tensor product of A,A, and H consists of elements of the form ∑i=1nai⊗bi⊗xi for all ai,bi in A, xi in H,i=1,2,…,n and n in ℕ. A map 〈·,·〉:(A⊗A⊗H)×(A⊗A⊗H)→ℂ is defined on elementary terms by
(3.13)〈a⊗b⊗x,c⊗d⊗y〉=〈D̂(d*,c*a,b)x,y〉H,
where 〈·,·〉H refers to the Hilbert inner product on H, and extending by linearity. Hence, we have
(3.14)〈ξ,η〉=∑i=1n∑j=1m〈D̂(dj*,cj*ai,bi)xi,yj〉H,
for ξ=∑i=1nai⊗bi⊗xi, and η=∑j=1mcj⊗dj⊗yj in A⊗A⊗H.
It is clear that〈·,·〉 is linear in the first variable and conjugate linear in the second. For such a form, the polarisation identity
(3.15)〈ξ,η〉=14∑k=03ik〈ξ+ikη,ξ+ikη〉
holds. The existence of such sesquilinear form is easily seen if we show that
(3.16)∑j=1naj⊗bj⊗xj=0⟹〈∑j=1naj⊗bj⊗xj,∑j=1naj⊗bj⊗xj〉=0.
Choose linearly independent elements e1,…,em in H having the same linear span as x1,…,xn. Then xj=∑i=1mλijei for unique scalars λij. Since ∑j=1naj⊗bj⊗xj=0, we have
(3.17)∑j=1naj⊗bj⊗(∑i=1mλijei)=∑i,jλijaj⊗bj⊗ei=0,
and, therefore, ∑j=1nλijaj⊗bj=0, for i=1,2,…,m, because e1,…,em are linearly independent.
Similarly, choose linearly independent elementsε1,…,εp in A having the same linear span as b1,…,bn. Then bj=∑k=1pγkjεk for unique scalars γkj. Since ∑j=1nλijaj⊗bj=0, for i=1,2,…,m, we have
(3.18)∑j=1nλijaj⊗bj=∑j=1nλijaj⊗(∑k=1pγkjεk)=∑j,kλijγkjaj⊗εk=0,
and hence ∑j=1nλijγkjaj=0, for i=1,…,m, k=1,…,p, because ε1,…,εp are linearly independent. Hence,
(3.19)〈∑j=1naj⊗bj⊗xj,∑j=1naj⊗bj⊗xj〉=〈∑i,j,kλijγkjaj⊗εk⊗ei,∑i,j,kλijγkjaj⊗εk⊗ei〉=〈∑i,k0⊗εk⊗ei,∑i,k0⊗εk⊗ei〉=∑i,k,s,l〈D̂(εs*,0,εk*)ei,el〉=∑i,k,s,l〈0ei,el〉=0.
It follows from the polarisation identity that a sesquilinear form〈·,·〉 is hermitian if and only if 〈ξ,ξ〉∈ℝ(ξ∈A⊗A⊗H). Thus, positive sesquilinear forms are hermitian.
Letξ∈A⊗A⊗H, and suppose that ξ=∑i=1nai⊗bi⊗xi, then we have the formula
(3.20)〈ξ,ξ〉=∑i,j=1n〈D̂(bj*,aj*ai,bi)xi,yj〉H.
Since D̂:A⊗A⊗A→B(H) is relatively bounded completely positive definite map from (i), it follows that
(3.21)〈ξ,ξ〉≥0,∀ξ∈A⊗A⊗H.
Then, the sesquilinear form 〈·,·〉 is positive semidefinite. Setting
(3.22)N={ξ∈A⊗A⊗H:〈ξ,ξ〉=0},
it follows from the Cauchy-Schwartz inequality that the set N is a subspace of A⊗A⊗H. The induced form
(3.23)〈ξ+N,η+N〉=〈ξ,η〉,∀ξ+N,η+N∈(A⊗A⊗H)N
on the quotient space (A⊗A⊗H)/N is thus positive definite, and it is an inner product on (A⊗A⊗H)/N. Thus, the quotient space (A⊗A⊗H)/N is a pre-Hilbert space under this inner product. By completing it with respect to the induced inner product, we get a Hilbert space X.
Fora∈A, define a linear map φ(a) on A⊗A⊗H by φ(a)(c⊗d⊗x)=(ac)⊗d⊗x, where c,d∈A, x∈H, and extending by linearity. Hence for each ξ=∑i=1nai⊗bi⊗xi in A⊗A⊗H, we have
(3.24)φ(a)(∑i=1nai⊗bi⊗xi)=∑i=1n(aai)⊗bi⊗xi.
SinceD̂:A⊗A⊗A→B(H) is relatively bounded completely positive definite map from (i); hence, there exists a function ρ:A→ℝ+ such that
(3.25)∑i,j=1n〈D̂(bi*,ai*a*aaj,bj)xj,xi〉H≤ρ(a*a)∑i,j=1n〈D̂(bi*,ai*aj,bj)xj,xi〉H,
for every a,a1,a2,…,an∈A, x1,x2,…,xn∈H, and n∈ℕ. Thus, for any ξ=∑i=1nai⊗bi⊗xi∈A⊗A⊗H, by an easy calculation, we have
(3.26)〈φ(a)ξ,φ(a)ξ〉=∑i,j=1n〈D̂(bj*,aj*a*aai,bi)xi,xj〉H≤ρ(a*a)∑i,j=1n〈D̂(bj*,aj*ai,bi)xi,xj〉H=ρ(a*a)〈∑i=1nai⊗bi⊗xi,∑j=1naj⊗bj⊗xj〉=ρ(a*a)〈ξ,ξ〉.
So φ(A) leaves N invariant, and the induced map
(3.27)φ(a)(ξ+N)=φ(a)(∑i=1nai⊗bi⊗xi+N)=φ(a)(ξ)+N=∑i=1n(aai)⊗bi⊗xi+N
defines a linear operator on the quotient space (A⊗A⊗H)/N. Since ∥φ(a)∥≤ρ(a*a)1/2 for all a∈A, therefore, φ(a) extends to a bounded linear operator from X to X by continuity, which will be also denoted by φ(a), thus φ(a) in B(X). It is easily to see that φ is representation, and that φ is a *-representation, that is, φ(a)*=φ(a*),foralla∈A. Indeed, for ξ=∑i=1nai⊗bi⊗xi and η=∑j=1mcj⊗dj⊗yj in A⊗A⊗H, it follows that
(3.28)〈ξ,φ(a)*η〉=〈φ(a)ξ,η〉=〈φ(a)(∑i=1nai⊗b1⊗xi),∑j=1mcj⊗dj⊗yj〉=〈∑i=1n(aai)⊗bi⊗xi,∑j=1mcj⊗dj⊗yj〉=∑i,j〈D̂(dj*,cj*aai,bi)xi,yj〉H=∑i,j〈D̂(dj*,(a*cj)*ai,bi)xi,yj〉H=〈∑i=1nai⊗bi⊗xi,∑j=1ma*cj⊗dj⊗yj〉=〈∑i=1nai⊗bi⊗xi,φ(a*)(∑j=1mcj⊗dj⊗yj)〉=〈ξ,φ(a*)η〉.
Fora∈A and x∈H, we define V:A→B(H,X) as follows:
(3.29)V(a)x=e⊗a⊗x+N.
It follows that
(3.30)〈V(a)x,V(a)x〉=〈D̂(a*,e,a)x,x〉H≤‖D̂(a*,e,a)‖⋅‖x‖2.
So
(3.31)‖V(a)‖≤‖D̂(a*,e,a)‖1/2,
and for any a,b,c∈A and x,y∈H, we obtain
(3.32)〈V(a*)*φ(b)V(c)x,y〉H=〈φ(b)V(c)x,V(a*)y〉H=〈D̂(a,b,c)x,y〉H.
Hence,
(3.33)D̂(a,b,c)=V(a*)*φ(b)V(c),∀a,b,c∈A.
(iii) For everya,b in A and x in H, we have
(3.34)φ(a)V(b)(x)=φ(a)(e⊗b⊗x+N)=a⊗b⊗x+N.
So, the linear span of the set {φ(a)V(b)x:a,b∈A,x∈H} is precisely (A⊗A⊗H)/N since every element of (A⊗A⊗H)/N is the finite sum of a⊗b⊗x+N with a,b in A and x∈H. Thus, it follows that {φ(a)V(b)x:a,b∈A,x∈H} spans a dense subspace of X.
(iv) Let(φ,X,V) and (ψ,Y,W) be the representations associated with ϕ. For each a1,a2,…,an, b1,b2,…,bn∈A, and x1,x2,…,xn∈H, we have
(3.35)‖∑i=1nφ(ai)V(bi)xi‖X2=〈∑i=1nφ(ai)V(bi)xi,∑i=1nφ(ai)V(bi)xi〉=∑i=1n∑j=1n〈V(bj)*φ(aj*ai)V(bi)xi,xj〉H=∑i=1n∑j=1n〈D̂(bj*,aj*ai,bi)xi,xj〉H=∑i=1n∑j=1n〈W(bj)*ψ(aj*ai)W(bi)xi,xj〉H=〈∑i=1nψ(ai)W(bi)xi,∑i=1nψ(ai)W(bi)xi〉=‖∑i=1nψ(ai)Wxi‖Y2
and so the linear map defined by
(3.36)U0(∑i=1nφ(ai)V(bi)xi)=∑i=1nψ(ai)W(bi)xi
extends to an isometry from X to Y, which will be denoted by U. Since the range of U0 is dense in Y; thus, U is a unitary operator of X onto Y.
From the relationD̂(a,b,c)=V(a*)*φ(b)V(c)=W(a*)*ψ(b)W(c),a,b,c∈A, for all ai,bi, cj,dj∈A,i=1,2,…,n, j=1,2,…,m and x1,…,xn,y1,…,ym∈H, we get
(3.37)〈U(∑i=1nφ(ai)V(bi)xi),∑j=1mψ(cj)W(dj)yj〉=〈∑i=1nψ(ai)W(bi)xi,∑j=1mψ(cj)W(dj)yj〉=∑i=1n∑j=1m〈W(dj)*ψ(cj*ai)W(bi)xi,yj〉H=∑i=1n∑j=1m〈D̂(dj*,cj*ai,bi)xi,yj〉H=〈∑i=1nφ(ai)V(bi)xi,∑j=1mφ(cj)V(dj)yj〉.
Therefore, U*(∑j=1mψ(cj)W(dj)yj)=∑j=1mφ(cj)V(dj)yj. For each a,ai,bi∈A,i=1,2,…,n and x1,…,xn∈H, we have
(3.38)Uφ(a)U*(∑i=1nψ(ai)W(bi)xi)=U(∑i=1nφ(aai)V(bi)xi)=ψ(a)(∑i=1nψ(ai)W(bi)xi).
Since Uφ(a)U* and ψ(a) are bounded, and {ψ(a)W(b)x:a,b∈A,x∈H} spans a dense subspace of Y, we then have Uφ(a)U*=ψ(a) for each a∈A.
(v) For alla,b∈A, we let T=φ(a)V(b)-V(ab)+V(a). Then, T∈B(H,X). Thus,
(3.39)T*T=[φ(a)V(b)-V(ab)+V(a)]*[φ(a)V(b)-V(ab)+V(a)]=[V(b)*φ(a)*-V(ab)*+V(a)*][φ(a)V(b)-V(ab)+V(a)]=V(b)*φ(a*a)V(b)-V(b)*φ(a)*V(ab)+V(b)*φ(a*a)-V(ab)*φ(a)V(b)+V(ab)*V(ab)-V(ab)*V(a)+V(a)*φ(a)V(b)-V(a)*V(ab)+V(a)*V(a)=D̂(b*,a*a,b)-D̂(b*,a*,ab)+D̂(b*,a*,a)-D̂(b*a*,a,b)+D̂(a*,a,b)+D̂(b*a*,e,ab)-D̂(b*a*,e,a)-D̂(a*,e,ab)+D̂(a*,e,a)=D(b*a*a,b)-D(a*a,b)-D(b*a*,ab)+D(a*,ab)+D(b*a*,a)-D(a*,a)-D(b*a*a,b)+D(a,b)+D(a*a,b)-D(a,b)+D(b*a*,ab)-D(e,ab)-D(b*a*,a)+D(e,a)-D(a*,ab)+D(e,ab)+D(a*,a)-D(e,a)=0.
We, therefore, get T=0, that is, Δφ1V=φ(a)V(b)-V(ab)+V(a)=0. This completes the proof.Finally, as in cohomology theory of Banach algebras [16], we introduce the notion of n-cocycles.Definition 3.2.
LetA be a σ-C*-algebra, and φ a representation of A in a Hilbert space H. Let K be another Hilbert space. For n≥1, the group (under pointwise addition) of all n-cochains on A with values in B(K,H) is denoted by Cn(A;B(K,H)) consisting of all continuous n-linear maps of An into B(K,H). We set Cn(A;B(K,H))=0 for n≤0. We define a group homomorphism (coboundary operator) Δφn:Cn(A;B(K,H))→Cn+1(A;B(K,H)) with respect to φ by Δφn=0 for n≤0 and, n≥1,
(3.40)Δφnf(a1,a2,…,an+1)=φ(a1)f(a2,a3,…,an+1)+∑i=1n(-1)if(a1,a2,ai-1,aiai+1,ai+2,…,an+1)+(-1)n+1f(a1,a2,…,an),
where a1,a2,…,an+1∈A.
We set(3.41)Zφn(A;B(K,H))={f∈Cn(A;B(K,H)):Δφnf=0}.
Elements of Zφn(A;B(K,H)) are called n-cocycles with respect to the representation φ of A. We see that a continuous linear map f:A→B(K,H) is called 1-cocycle with respect to the representation φ of A if it satisfied Δφ1f=φ(a)f(b)-f(ab)+f(a)=0. The 1-cocycle appears at (v) in Theorem 3.1. For further, results will report in another paper.



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*Source: 702989-2011-11-13.xml*