Abstract and Applied Analysis

Volume 2011, Article ID 702989, 13 pages

http://dx.doi.org/10.1155/2011/702989

## Completely Dissipative Maps and Stinespring's Dilation-Type Theorem on --Algebras

School of Mathematics, Beijing Institute of Technology, Beijing 100081, China

Received 18 July 2011; Accepted 6 September 2011

Academic Editor: Jean Michel Combes

Copyright © 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.

#### Abstract

The notion of the completely dissipative maps on --algebras is introduced. We show that a completely dissipative map induces a representation of a --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 -algebras, called --algebras. These algebras were introduced in [5] as a generalization of -algebras and were called locally -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 -algebras such as the tangent algebras of -algebras, multipliers of Pedersen's ideal, noncommutative analogues of classical Lie groups, and -theory.

Motivated mainly by the works [11, 12, 15–18], in this paper, we shall introduce the notion of the completely dissipative maps on --algebras consider an abstract version of Stinespring's dilation-type theorem, and explore the natural relation completely dissipative maps to --algebras. We shall show that a completely positive definite map induces a representation of a --algebra. We also get a classification of completely positive definite maps on --algebras. Our approach is motivated by the setting of Banach algebras and -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 of complexes, and the topological spaces are supposed to be Hausdorff.

A locally -convex (-) algebra is a topological algebra whose topology is defined by a family , ( a directed index set), of submultiplicative seminorms. is called an -algebra if, in addition, has an involution * such that , for any , . If moreover, each , , is a -seminorm, that is, for any , , is called to be an --algebra. Given an -algebra denote by the inverse system of Banach algebras corresponding to , that is, is the completion of the normed algebra , , , under norm , with , , . A --algebra is a complete --algebra. In this setting, can be replaced with the collection , the set of all continuous -seminorms on . For a --algebra , and for a , the normed *-algebra , is automatically complete (see [13]), so that is a -algebra. Thus, is an inverse limit of -algebras, that is, (see [6]). A metrizable --algebra is called a --algebra with its topology determined by a countable subfamily of , .

#### 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 --algebras. For this, we need to recall the following theorem in [15, Theorem and Corollary ].

Theorem 2.1 . (see [15]). *Let be a -algebra with unit , a bounded *-map and . Let . Then is completely positive on if and only if is completely dissipative, that is, for all , where , for all , and .*

We can now introduce completely dissipative maps on --algebras as follows.

*Definition 2.2. *Let be a --algebra, a Hilbert space, a continuous map. Set as . If the following conditions are satisfied:(i)(ii)there exists a function such that
for every , , and , then 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]). *Let be a -algebra, a Hilbert space, and a map be given such that, for all ,
**
Then there exists a Hilbert space and a map such that
*

*Definition 2.4. *Let be a --algebra, a Hilbert space, and a map . We say that is completely positive definite if for any , , we have
If there exists a function such that the map satisfies the following additional condition:
for every , , and , then we say, is relatively bounded.

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

To prove the main theorem, the following things have to be elucidated. If and are vector spaces, we denote by their algebraic tensor product. This is linearly spanned by elements (, ). If is a bilinear map, where , and are vector spaces, then there is a unique linear map such that for all and .

If are linear functionals on the vector spaces , respectively, then there is a unique linear functional on such that since the function is bilinear.

Suppose that , where and . If are linearly independent, then . For, in this case, there exist linear functionals such that (Kronecker delta). If is linear, we have Thus, for arbitrary , and this shows that .

Similarly, if , and are linearly independent, then .

Theorem 3.1. *Let be a -algebra with unit , a Hilbert space, be a completely dissipative map, and defined by
**
where , then*(i)* is a relatively bounded completely positive definite map,*(ii)*there exists a Hilbert space , a representation of on , and a map such that
*(iii)* spans a dense subspace of ,*(iv)*the representation associated with is unique up to a unitary equivalence,*(v)*the map is a 1-cocycle with respect to the representation , that is, .*

*Proof. *(i) By the definition, for any , we have
Thus,
From Lemma 2.3, there exists a Hilbert space , and a map such that
It follows that
Therefore, we obtain

Since be a completely dissipative map, then by Definition 2.2(ii), there exists a function such that
where , and , . Thus, we have
It follows that is relatively bounded.

(ii) Let be the algebraic tensor product of , and . The algebraic tensor product of , and consists of elements of the form for all in , in , and in . A map is defined on elementary terms by
where refers to the Hilbert inner product on , and extending by linearity. Hence, we have
for , and in .

It is clear that is linear in the first variable and conjugate linear in the second. For such a form, the polarisation identity
holds. The existence of such sesquilinear form is easily seen if we show that
Choose linearly independent elements in having the same linear span as . Then for unique scalars . Since , we have
and, therefore, , for , because are linearly independent.

Similarly, choose linearly independent elements in having the same linear span as . Then for unique scalars . Since , for , we have
and hence , for , , because are linearly independent. Hence,

It follows from the polarisation identity that a sesquilinear form is hermitian if and only if . Thus, positive sesquilinear forms are hermitian.

Let , and suppose that , then we have the formula
Since is relatively bounded completely positive definite map from (i), it follows that
Then, the sesquilinear form is positive semidefinite. Setting
it follows from the Cauchy-Schwartz inequality that the set is a subspace of . The induced form
on the quotient space is thus positive definite, and it is an inner product on . Thus, the quotient space is a pre-Hilbert space under this inner product. By completing it with respect to the induced inner product, we get a Hilbert space .

For , define a linear map on by , where , , and extending by linearity. Hence for each in , we have

Since is relatively bounded completely positive definite map from (i); hence, there exists a function such that
for every , , and . Thus, for any , by an easy calculation, we have
So leaves invariant, and the induced map
defines a linear operator on the quotient space . Since for all , therefore, extends to a bounded linear operator from to by continuity, which will be also denoted by , thus in . It is easily to see that is representation, and that is a *-representation, that is, . Indeed, for and in , it follows that

For and , we define as follows:
It follows that
So
and for any and , we obtain
Hence,

(iii) For every in and in , we have
So, the linear span of the set is precisely since every element of is the finite sum of with in and . Thus, it follows that spans a dense subspace of .

(iv) Let and be the representations associated with . For each , , and , we have
and so the linear map defined by
extends to an isometry from to , which will be denoted by . Since the range of is dense in ; thus, is a unitary operator of onto .

From the relation , for all , , and , we get
Therefore, . For each and , we have
Since and are bounded, and spans a dense subspace of , we then have for each .

(v) For all , we let . Then, . Thus,
We, therefore, get , that is, . This completes the proof.

Finally, as in cohomology theory of Banach algebras [16], we introduce the notion of -cocycles.

*Definition 3.2. *Let be a --algebra, and a representation of in a Hilbert space . Let be another Hilbert space. For , the group (under pointwise addition) of all -cochains on with values in is denoted by consisting of all continuous -linear maps of into . We set for . We define a group homomorphism (coboundary operator) with respect to by for and, ,
where .

We set
Elements of are called -cocycles with respect to the representation of . We see that a continuous linear map is called 1-cocycle with respect to the representation of if it satisfied . 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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