Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 702989, 13 pages
doi:10.1155/2011/702989
Research Article
Completely Dissipative Maps and Stinespring’s
Dilation-Type Theorem on σ-C∗ -Algebras
Tian Zhou Xu
School of Mathematics, Beijing Institute of Technology, Beijing 100081, China
Correspondence should be addressed to Tian Zhou Xu, xutianzhou@bit.edu.cn
Received 18 July 2011; Accepted 6 September 2011
Academic Editor: Jean Michel Combes
Copyright q 2011 Tian Zhou Xu. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
The notion of the completely dissipative maps on σ-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.
1. Introduction
There has been increased interest 1–12 in topological ∗ -algebras that are inverse limits of C∗ algebras, called P ro-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 field
C of complexes, and the topological spaces are supposed to be Hausdorff.
2
Abstract and Applied Analysis
A locally m-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 x2 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 Kerpα , α ∈ I, under norm · α , with xα α pα x, xα x Nα , x ∈ A. A
P ro-C∗ -algebra A is a complete lmc-C∗ -algebra. In this setting, pα can be replaced with the
collection SA, the set of all continuous C∗ -seminorms on A. For a P ro-C∗ -algebra A, and for
a p ∈ SA, 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 limAα see
←
6. A metrizable P ro-C∗ -algebra is called a σ-C∗ -algebra with its topology determined by a
countable subfamily pn of SA, 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 on σ-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. Let A be a C∗ -algebra with unit e, L : A → A a bounded ∗ -map and
Le 0. Let φt exptL. Then φt t ∈ R is completely positive on A if and only if L is completely
dissipative, that is, DLn ; X, X ≥ 0 for all X ∈ Mn A, where DLn ; 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 on σ-C∗ -algebras as follows.
Definition 2.2. Let A be a σ-C∗ -algebra, H a Hilbert space, φ : A → BH a continuous map.
Set D : A × A → BH as Da, b φab − φa − φb. If the following conditions are
satisfied:
i
∗ n
D ai , aj i,j1 ∈ Mn BH;
2.1
ii there exists a function ρ : A → R such that
n n
∗ ∗ ∗
D bi ai a aaj , bj − D a∗i a∗ aaj , bj xj , xi H
i1 j1
n n
∗ ∗
≤ ρa a
D bi ai aj , bj − D a∗i aj , bj xj , xi H
2.2
∗
i1 j1
for every a, a1 , a2 , . . . , an , b1 , b2 , . . . , bn ∈ A, x1 , x2 , . . . , xn ∈ H, and n ∈ N, then φ is
called a completely dissipative map.
Abstract and Applied Analysis
3
From the corresponding facts in the Aronszajn-Kolmogorov theorem of 17, we have
the following result.
Lemma 2.3 see 17. Let A be a σ-C∗ -algebra, H a Hilbert space, and a map T : A × A → BH
be given such that, for all a1 , a2 , . . . , an ∈ A,
∗ n
T ai , aj i,j1 ∈ Mn BH.
2.3
Then there exists a Hilbert space K and a map V : A → BH, K such that
T a, b V a∗ ∗ V b,
∀a, b ∈ A.
2.4
Definition 2.4. Let A be a σ-C∗ -algebra, H a Hilbert space, and a map φ : A × A × A → BH.
We say that φ is completely positive definite if for any ai , bi ∈ A, i 1, 2, . . . , n ∈ N, we have
n
∗ ∗
φ bi , ai aj , bj i,j1 ∈ Mn BH.
2.5
If there exists a function ρ : A → R such that the map φ : A × A × A → BH satisfies the
following additional condition:
n
n
n n ∗ ∗ ∗
∗ ∗
φ bi , ai a aaj , bj xj , xi ≤ ρa∗ a
φ bi , ai aj , bj xj , xi ,
i1 j1
2.6
i1 j1
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. If X 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 σx, y
for all x ∈ X and y ∈ Y .
If τ, 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
τ ⊗ μ x ⊗ y τxμ y ,
∀x ∈ X, y ∈ Y,
3.1
since the function
X × Y −→ C,
is bilinear.
x, y −→ τxμ y
3.2
4
Abstract and Applied Analysis
Suppose that nj1 xj ⊗ 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
0 τ ⊗ μj
n
xi ⊗ yi
i1
n
n
τxi ρj yi τxi δij τ xj .
i1
3.3
i1
Thus, τxj 0 for arbitrary τ, and this shows that x1 · · · xn 0.
Similarly, if nj1 xj ⊗ yj 0, and x1 , . . . , xn are linearly independent, then y1 · · · yn 0.
Theorem 3.1. Let A be a σ-C∗ -algebra with unit e, H a Hilbert space, φ : A → BH be a
: A × A × A → BH defined by
completely dissipative map, and D
Da,
b, c Dab, c − Db, c,
3.4
where Da, b φab − φa − φb, then
is a relatively bounded completely positive definite map,
i D
ii there exists a Hilbert space X, a representation ϕ of A on X, and a map V : A → BH, X
such that
Da,
b, c V a∗ ∗ ϕbV c,
∀a, b, c ∈ A,
3.5
iii {ϕaV bx : 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 map V : A → BH, X is a 1-cocycle with respect to the representation ϕ, that is,
Δ1ϕ V ϕaV b − V ab V a 0.
Proof. i By the definition, for any ai , bi ∈ A, i 1, 2, . . . , n, n ∈ N, we have
b∗ , a∗ aj , bj
D
i
i
D bi∗ a∗i aj , bj − D a∗i aj , bj
φ bi∗ a∗i aj bj − φ bi∗ a∗i aj − φ bj − φ a∗i aj bj − φ a∗i aj − φ bj
φ bi∗ a∗i aj bj − φ bi∗ a∗i aj − φ a∗i aj bj φ a∗i aj ,
D bi∗ a∗i , aj bj − D bi∗ a∗i , aj D a∗i , aj − D a∗i , aj bj
φ bi∗ a∗i aj bj − φ bi∗ a∗i − φ aj bj − φ bi∗ a∗i aj − φ bi∗ a∗i − φ aj
φ a∗i aj − φ a∗i − φ aj − φ a∗i aj bj − φ a∗i − φ aj bj
φ bi∗ a∗i aj bj − φ bi∗ a∗i aj φ a∗i aj − φ a∗i aj bj .
3.6
Abstract and Applied Analysis
5
Thus,
b∗ , a∗ aj , bj D b∗ a∗ , aj bj − D b∗ a∗ , aj D a∗ , aj − D a∗ , aj bj .
D
i
i
i i
i i
i
i
3.7
From Lemma 2.3, there exists a Hilbert space K, and a map V : A → BH, K such that
Da, b V a∗ ∗ V b,
∀a, b ∈ A.
It follows that
b∗ , a∗ aj , bj V ai bi ∗ V aj bj − V ai bi ∗ V aj V ai ∗ V aj − V ai ∗ V aj bj
D
i
i
V ai bi ∗ V aj bj − V aj V ai ∗ V aj − V aj bj
V ai bi ∗ − V ai ∗ V aj bj − V aj
V ai bi − V ai ∗ V aj bj − V aj .
3.8
3.9
Therefore, we obtain
n
b∗ , a∗ aj , bj
D
i
i
i,j1
∈ Mn BH.
3.10
Since φ : A → BH be a completely dissipative map, then by Definition 2.2ii, there
exists a function ρ : A → R such that
n
n ∗ ∗ ∗
D bi ai a aaj , bj − D a∗i a∗ aaj , bj xj , xi H
i1 j1
n n
∗ ∗
≤ ρa a
D bi ai aj , bj − D a∗i aj , bj xj , xi H ,
3.11
∗
i1 j1
where Da, b φab − φa − φb, and a, a1 , a2 , . . . , an , b1 , b2 , . . . , bn ∈ A, x1 , x2 , . . . , xn ∈ H.
Thus, we have
n n b∗ , a∗ a∗ aaj , bj xj , xi
D
i
i
i1 j1
H
n
n ∗ ∗ ∗
D bi ai a aaj , bj − D a∗i a∗ aaj , bj xj , xi H
i1 j1
n
n ∗ ∗
≤ ρa a
D bi ai aj , bj − D a∗i aj , bj xj , xi H
∗
i1 j1
ρa∗ a
n n b∗ , a∗ aj , bj xj , xi .
D
i
i
i1 j1
is relatively bounded.
It follows that D
H
3.12
6
Abstract and Applied Analysis
ii Let A ⊗ 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 ni1 ai ⊗ 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
∗ , c∗ a, bx, y ,
a ⊗ b ⊗ x, c ⊗ d ⊗ y Dd
3.13
H
where ·, ·H refers to the Hilbert inner product on H, and extending by linearity. Hence, we
have
m n d∗ , c∗ ai , bi xi , yj ,
ξ, η D
j j
3.14
H
i1 j1
n
⊗ bi ⊗ xi , and η m
j1 cj ⊗ 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
for ξ i1 ai
3
1
ik ξ ik η, ξ ik η
ξ, η 4 k0
3.15
holds. The existence of such sesquilinear form is easily seen if we show that
n
aj ⊗ bj ⊗ xj 0 ⇒
j1
n
n
aj ⊗ bj ⊗ xj , aj ⊗ bj ⊗ xj
j1
0.
3.16
j1
Choose linearly independent elements e1 , . . . , em in H having the same linear span as
n
x1 , . . . , xn . Then xj m
i1 λij ei for unique scalars λij . Since
j1 aj ⊗ bj ⊗ xj 0, we have
n
aj ⊗ bj ⊗
j1
m
λij ei
λij aj ⊗ bj ⊗ ei 0,
3.17
i,j
i1
n
and, therefore,
j1 λij aj ⊗ bj 0, for i 1, 2, . . . , m, because e1 , . . . , em are linearly
independent.
Similarly, choose linearly independent elements ε1 , . . . , εp in A having the same linear
p
span as b1 , . . . , bn . Then bj k1 γkj εk for unique scalars γkj . Since nj1 λij aj ⊗ bj 0, for
i 1, 2, . . . , m, we have
n
n
λij aj ⊗ bj λij aj ⊗
j1
j1
p
k1
γkj εk
λij γkj aj ⊗ εk 0,
j,k
3.18
Abstract and Applied Analysis
7
and hence nj1 λij γkj aj 0, for i 1, . . . , m, k 1, . . . , p, because ε1 , . . . , εp are linearly
independent. Hence,
n
n
aj ⊗ bj ⊗ xj , aj ⊗ bj ⊗ xj
j1
j1
λij γkj aj ⊗ εk ⊗ ei , λij γkj aj ⊗ εk ⊗ ei
i,j,k
3.19
i,j,k
0 ⊗ εk ⊗ ei , 0 ⊗ εk ⊗ ei
i,k
i,k
εs∗ , 0, ε∗ ei , el D
0 ei , el 0.
k
i,k,s,l
i,k,s,l
It follows from the polarisation identity that a sesquilinear form ·, · is hermitian if
and only if ξ, ξ ∈ Rξ ∈ A ⊗ A ⊗ H. Thus, positive sesquilinear forms are hermitian.
Let ξ ∈ A ⊗ A ⊗ H, and suppose that ξ ni1 ai ⊗ bi ⊗ xi , then we have the formula
ξ, ξ n b∗ , a∗ ai , bi xi , yj .
D
j
j
H
i,j1
3.20
: A ⊗ A ⊗ A → BH is relatively bounded completely positive definite map from
Since D
i, it follows that
ξ, ξ ≥ 0,
∀ξ ∈ A ⊗ A ⊗ H.
3.21
Then, the sesquilinear form ·, · is positive semidefinite. Setting
N {ξ ∈ A ⊗ A ⊗ H : ξ, ξ 0},
3.22
it follows from the Cauchy-Schwartz inequality that the set N is a subspace of A ⊗ A ⊗ H. The
induced form
ξ N, η N ξ, η ,
∀ξ N, η N ∈
A ⊗ A ⊗ H
N
3.23
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.
For a ∈ A, define a linear map ϕa on A ⊗ A ⊗ H by ϕac ⊗ d ⊗ x ac ⊗ d ⊗ x, where
c, d ∈ A, x ∈ H, and extending by linearity. Hence for each ξ ni1 ai ⊗ bi ⊗ xi in A ⊗ A ⊗ H,
we have
n
ai ⊗ bi ⊗ xi
ϕa
i1
n
aai ⊗ bi ⊗ xi .
i1
3.24
8
Abstract and Applied Analysis
: A ⊗ A ⊗ A → BH is relatively bounded completely positive definite map
Since D
from i; hence, there exists a function ρ : A → R such that
n b∗ , a∗ a∗ aaj , bj xj , xi
D
i
i
i,j1
H
≤ ρa∗ a
n b∗ , a∗ aj , bj xj , xi ,
D
i
i
H
i,j1
for every a, a1 , a2 , . . . , an ∈ A, x1 , x2 , . . . , xn ∈ H, and n ∈ N. Thus, for any ξ A ⊗ A ⊗ H, by an easy calculation, we have
ϕaξ, ϕa ξ n
3.25
i1 ai ⊗bi ⊗xi
∈
n b∗ , a∗ a∗ aai , bi xi , xj
D
j
j
H
i,j1
≤ ρa∗ a
n b∗ , a∗ ai , bi xi , xj
D
j
j
i,j1
3.26
H
n
n
ρa a
ai ⊗ bi ⊗ xi , aj ⊗ bj ⊗ xj
∗
i1
ρa∗ aξ, ξ.
j1
So ϕA leaves N invariant, and the induced map
ϕaξ N ϕa
n
ai ⊗ bi ⊗ xi N
ϕaξ N
i1
n
3.27
aai ⊗ bi ⊗ xi N
i1
defines a linear operator on the quotient space A⊗A⊗H/N. Since ϕa ≤ ρa∗ a1/2 for all
a ∈ A, therefore, ϕa extends to a bounded linear operator from X to X by continuity, which
will be also denoted by ϕa, thus ϕa in BX. It is easily to see that ϕ is representation, and
that ϕ is a ∗ -representation, that is, ϕa∗ ϕa∗ , for all a ∈ A. Indeed, for ξ ni1 ai ⊗ bi ⊗ xi
m
and η j1 cj ⊗ dj ⊗ yj in A ⊗ A ⊗ H, it follows that
ξ, ϕa∗ η ϕaξ, η
n
m
ϕa
ai ⊗ b1 ⊗ xi , cj ⊗ dj ⊗ yj
i1
j1
n
m
aai ⊗ bi ⊗ xi , cj ⊗ dj ⊗ yj
i1
j1
d∗ , c∗ aai , bi xi , yj
D
j j
i,j
i,j
H
∗ , a∗ cj ∗ ai , bi xi , yj
Dd
j
H
Abstract and Applied Analysis
9
n
m
ai ⊗ bi ⊗ xi , a∗ cj ⊗ dj ⊗ yj
i1
j1
⎛
⎞
n
m
ai ⊗ bi ⊗ xi , ϕa∗ ⎝ cj ⊗ dj ⊗ yj ⎠
i1
j1
ξ, ϕa∗ η .
3.28
For a ∈ A and x ∈ H, we define V : A → BH, X as follows:
V ax e ⊗ a ⊗ x N.
3.29
It follows that
∗ , e, ax, x
V ax, V ax Da
H
∗
≤ Da
, e, a · x2 .
3.30
So
1/2
∗
, e, a ,
V a ≤ Da
3.31
and for any a, b, c ∈ A and x, y ∈ H, we obtain
V a∗ ∗ ϕbV cx, y H ϕbV cx, V a∗ y H Da,
b, cx, y .
H
3.32
Hence,
Da,
b, c V a∗ ∗ ϕbV c,
∀a, b, c ∈ A.
3.33
iii For every a, b in A and x in H, we have
ϕaV bx ϕae ⊗ b ⊗ x N a ⊗ b ⊗ x N.
3.34
So, the linear span of the set {ϕaV bx : 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 {ϕaV bx : a, b ∈ A, x ∈ H} spans a dense subspace of X.
10
Abstract and Applied Analysis
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
2 n
n
n
ϕai V bi xi , ϕai V bi xi
ϕai V bi xi i1
i1
i1
X
n n V bj ∗ ϕa∗j ai V bi xi , xj
i1 j1
n n b∗ , a∗ ai , bi xi , xj
D
j
j
i1 j1
n n H
H
Wbj ∗ ψa∗j ai Wbi xi , xj
i1 j1
n
H
n
ψai Wbi xi , ψai Wbi xi
i1
i1
2
n
ψai Wxi i1
Y
3.35
and so the linear map defined by
n
n
U0
ϕai V bi xi ψai Wbi xi
i1
3.36
i1
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 relation Da,
b, c V a∗ ∗ ϕbV c Wa∗ ∗ ψbWc, 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
n
m
n
m
U
ϕai V bi xi , ψ cj W dj yj ψai Wbi xi , ψ cj W dj yj
i1
j1
i1
m n j1
Wdj ∗ ψcj∗ ai Wbi xi , yj
i1 j1
H
n m d∗ , c∗ ai , bi xi , yj
D
j
j
H
i1 j1
n
m
ϕai V bi xi , ϕ cj V dj yj .
i1
j1
3.37
Abstract and Applied Analysis
11
m
Therefore, U∗ m
j1 ψcj Wdj yj j1 ϕcj V dj yj . For each a, ai , bi ∈ A, i 1, 2, . . . , n
and x1 , . . . , xn ∈ H, we have
UϕaU
∗
n
i1
ψai Wbi xi
U
n
ϕaai V bi xi
n
ψa
ψai Wbi xi . 3.38
i1
i1
Since UϕaU∗ and ψa are bounded, and {ψaWbx : a, b ∈ A, x ∈ H} spans a dense
subspace of Y , we then have UϕaU∗ ψa for each a ∈ A.
v For all a, b ∈ A, we let T ϕaV b − V ab V a. Then, T ∈ BH, X. Thus,
∗ T ∗ T ϕaV b − V ab V a ϕaV b − V ab V a
V b∗ ϕa∗ − V ab∗ V a∗ ϕaV b − V ab V a
V b∗ ϕa∗ aV b − V b∗ ϕa∗ V ab V b∗ ϕa∗ a − V ab∗ ϕaV b
V ab∗ V ab − V ab∗ V a V a∗ ϕaV b − V a∗ V ab V a∗ V a
∗ , a∗ a, b − Db
∗ , a∗ , ab Db
∗ , a∗ , a − Db
∗ a∗ , a, b
Db
∗ , a, b Db
∗ a∗ , e, ab − Db
∗ a∗ , e, a − Da
∗ , e, ab Da
∗ , e, a
Da
Db∗ a∗ a, b − Da∗ a, b − Db∗ a∗ , ab Da∗ , ab
Db∗ a∗ , a − Da∗ , a − Db∗ a∗ a, b Da, b
Da∗ a, b − Da, b Db∗ a∗ , ab − De, ab
− Db∗ a∗ , a De, a − Da∗ , ab De, ab Da∗ , a − De, a
0.
3.39
We, therefore, get T 0, that is, Δ1ϕ V ϕaV 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. Let A 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 BK, H is denoted by Cn A; BK, H consisting of all continuous n-linear
12
Abstract and Applied Analysis
maps of An into BK, H. We set Cn A; BK, H 0 for n ≤ 0. We define a group homomorphism coboundary operator Δnϕ : Cn A; BK, H → Cn1 A; BK, H with respect to ϕ by
Δnϕ 0 for n ≤ 0 and, n ≥ 1,
Δnϕ fa1 , a2 , . . . , an1 ϕa1 fa2 , a3 , . . . , an1 n
−1i fa1 , a2 , ai−1 , ai ai1 , ai2 , . . . , an1 3.40
i1
−1n1 fa1 , a2 , . . . , an ,
where a1 , a2 , . . . , an1 ∈ A.
We set
Zϕn A; BK, H f ∈ Cn A; BK, H : Δnϕ f 0 .
3.41
Elements of Zϕn A; BK, H are called n-cocycles with respect to the representation ϕ of A.
We see that a continuous linear map f : A → BK, H is called 1-cocycle with respect to the
representation ϕ of A if it satisfied Δ1ϕ f ϕafb − fab fa 0. The 1-cocycle appears
at v in Theorem 3.1. For further, results will report in another paper.
Acknowledgments
The author would like to thank the area editor Prof. Jean Michel Combes and the referees for
giving useful suggestions for the improvement of this paper. The research of the author is
supported by the National Natural Science Foundation of China 60972089, 11171022.
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