**Journal of Applied Mathematics and Physics**

Vol.03 No.03(2015), Article ID:55165,15 pages

10.4236/jamp.2015.33045

Tensor-Product Representation for Switched Linear Systems

Dengyin Jiang, Lisheng Hu

Department of Automation, Shanghai Jiao Tong University, Shanghai, China

Email: lshu@sjtu.edu.cn

Copyright © 2015 by authors and Scientific Research Publishing Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

Received 8 February 2015; accepted 25 March 2015; published 30 March 2015

ABSTRACT

This paper presents the method for the construction of tensor-product representation for multivariate switched linear systems, based on a suitable tensor-product representation of vectors and matrices. We obtain a representation theorem for multivariate switched linear systems. The stability properties of the tensor-product representation are investigated in depth, achieving the important result that any stable switched systems can be constructed into a stable tensor-product representation of finite dimension. It is shown that the tensor-product representation provides a high level framework for describing the dynamic behavior. The interpretation of expressions with- in the tensor-product representation framework leads to enhanced conceptual and physical un- derstanding of switched linear systems dynamic behavior.

**Keywords:**

Switched Linear Systems, Tensor-Product Representation, Stability

1. Introduction

Switched systems as an important class of hybrid systems [1] have drawn considerable attention in recent years. This is mainly due to the switched and hybrid nature of many physical processes and the growing use of computers in the control of physical plants [2] [3] . And, many dynamical systems encountered in engineering practice exhibit switching between several subsystems that is dependent on various environmental factors, for example, sampled-data control systems [4] -[9] , Markov jump systems [10] -[12] , and references therein. The extensive engineering applications of switched systems are also motivated by the better performance that can be achieved using a controller switching strategy [13] [14] . Currently, there is enormous growth in the literature in the study of switched systems [15] -[17] . These dynamical systems combine behaviors that are typical of continuous-time dynamical systems with behaviors that are typical of discrete-time dynamical systems. Hybrid systems [1] , in which continuous dynamics and discrete events coexist, are considered to be a natural modeling framework for switched systems. Thus, a switched system is a dynamical system that consists of a finite number of subsystems described by differential or difference equations and a logical rule that orchestrates switching between these subsystems [2] [15] [16] [18] -[20] .

On the other hand, switched systems cannot be represented in the framework of linear operators due to the logical switching space. For multivariate systems, the relationship between the rows vectors of the matrix space and that between the column vectors might be important for finding a projection to represent the systems operator. Motivated by this idea, switched systems may be represented in the framework of multi-linear algebra. Multi- linear algebra, the algebra of higher-order tensors, may be applied for analyzing the multi-factor structure of switched systems, including input space and logical switching space. Based on the considerations of multi-linear algebra, tensor-product representation is used to investigate the switched systems expression. From the theoretical point of view, this approach extended the subspace method using matrix technique to the multivariate switch- ed systems. The multilinear algebra, the algebra of higher-order tensors (also being a multidimensional or N- way array), which are the higher-order equivalents of vectors (first order) and matrices (second order), i.e., quan- tities of which the elements are addressed by more than two indices, may be applied for analyzing the multifac- tor structure of multivariate switched systems or such class of nonlinear systems.

In this paper, we investigated the more general problem of tensor-product representations for multivariate switched linear systems. We obtain the tensor-product representation theorem for globally table or globally asymptotically stable multivariate switched linear systems which are only defined on subspaces of algebras. This generalization allows us to obtain a Hahn-Banach extension theorem for such maps, which leads to further results on the Haagerup tensor-product norm applied into more general class of nonlinear systems. The stability properties of the tensor-product representation are investigated in depth, which any stable switched system admits a stable tensor-product representation of finite dimension. It is shown that the tensor-product representation provides a high level framework for describing the dynamic behavior of switched systems. The interpretation of expressions within the tensor-product representation framework leads to enhanced conceptual and physical understanding of switched linear systems dynamic behavior.

The main contribution of this paper is to obtain a general expression of tensor-product representation for multivariate switched systems, and the tensor-product representation is conceptually simple and relatively easy to express on the Haagerup tensor-product norm for multivariate switched systems. We focus on the correspondence between completely bounded multilinear maps and completely bounded linear maps on tensor products endowed with the Haagerup norm [21] -[23] . An abstract characterization of operator systems allows us to embed these tensor product spaces into C^{*}-algebras, after which the techniques developed in [24] lead to the extension and representation theorem. The idea of matrix multiplication spread from the definition of a completely bounded multilinear operator to the Haagerup tensor product and matricial operator spaces has implemented, see [25] [26] for a detailed discussion of completely bounded linear operators, and [22] [27] for the multilinear case. And more details can be founded in related review paper [28] .

The rest of this work is organized as follows. Section 2 contains the preliminaries on the tensor-product of vectors and matrices and the basic algebraic tools are provided. Section 3 gives the problem formulation for multivariate switched linear systems expression and its relation with bilinear algebra expression. In Section 4, the tensor-product representation for multivariate switched linear systems are investigated, and the stability pro- perties of the tensor-product representation based on the proposed methods are also investigated, which is followed by concluding conclusion in Section 5.

Notations: The notations throughout this paper are standard. R^{n} denotes the n dimensional Euclidean space. R^{n}^{×}^{m} is the set of all n × m real matrices. G denotes the systems function. I is the identity matrix with appropriate dimensions,
is the inner product, and
is the Haagerup norm. The symbol
denotes the direct sum of vectors or spaces, as well as
and
denote the matching elementwise Kronecker product and Kronecker product, respectively. And, tensors are written as calligraphic letters.

2. Preliminaries

In this section, we give some necessary preliminaries on the tensor-product of vectors and matrices, mainly concerning with basic algebraic tools on definitions and the correspondence between completely bounded bilinear maps and completely bounded linear maps on Haagerup tensor products norm, which will be used in this paper. And, the C^{*}-algebraic technique is used to investigate the tensor-product representation of multivariate switched linear systems.

Let A and B denote unital C^{*} algebras and E and F will denote subspaces. E^{*} and F^{*} will denote the subspaces consisting of the adjoints of elements in E and F, respectively. The Haagerup norm on the algebraic tensor pro- duct
is defined as follows,

(1)

where the infimum is taken over all possible representations. The resulting normed space is written. It is clear from the definitions that is a matrix normed space in the sense of the re- ference [29] .

Let
denote the C^{*}-algebra of n × n matrices over the field of complex numbers. If V is a vector space then the space
of n × n matrices over V is naturally a vector space; indeed
is a bimodule over the ring. If
and
then
denotes the matrix in
with x and y in the diagonal blocks (in that order) and zero in the off-diagonal blocks.

Definition 1 [30] Let be in and be in; then is the ele- ment of given by

(2)

It is valuable to note that if L is a matrix of scalars, then.

If and are matrix normed spaces then we define the Haagerup matrix norm on by setting for,

(3)

where the infimum is taken over all such expressions for with in, in. It is not difficult to see that with these definitions becomes a matrix normed space, which we denote.

Definition 2 [28] Let A and B be C^{*}-algebras, and let
be a linear operator. Define
from
into, where I_{n} is the identity operator on M_{n}, for each; that is,

for all. An operator is completely bounded if is bounded,

and the completely bounded norm is defined by

(4)

Proposition 1 [28] Let be a matricial normed space. Then V is completely isometrically isomorphic to a matricial operator space if and only if for all, and.

Let denote the bounded linear operators on a Hilbert space H. If is a linear map, then linear maps can be defined, for each positive integer n, by

. If is a bilinear map then V is defined to be the smallest constant K satisfying. Any bilinear map induces bilinear maps

where the entry of is

(5)

It is easiest to visualize this as formal matrix multiplication of and. If is finite then we say that V is completely bounded and will denote this supremum. These definitions were first given in [27] .

Lemma 1 [22] The Haagerup norm on is equal to the quotient norm induced by the Haagerup norm on.

Proposition 2 [22] Let be a bilinear map and let be the associated linear map. Then V is completely bounded if and only if is completely bounded and.

3. Problem Formulation

We consider a collection of discrete-time linear systems described by the equations with parametric uncertainties of process transfer functions and disturbance transfer functions with a class of piecewise constant functions, , which serves as the switching signal between the collection of discrete-time systems (6) given as

(6)

where is the input vector, is the output vector, is a family of discrete pro- cess transfer functions that is parametrized by some index set S, is the Gaussian distributed external disturbance that is a zero mean white noise satisfying, , , where is a positive definite diagonal matrix, is a family of disturbance transfer functions that also is parametrized by some index set S. And denotes the switching signal taking value from the finite index set, and s represents the integer of the all finite switched subsystems of switched systems and S is the set of positive integers. Furthermore, the switching signal is assumed to be generated by

(7)

where. In other words, the discrete mode is determined by a form of static output feedback, and the parametric uncertainty is assumed to be piecewise constant time-varying. The logical switching signal may be determined by the systems output signals.

It is assumed that the multivariate switched linear systems is globally table or globally asymptotically stable. This assumption essentially is to guarantee the entire switched systems are completely bounded, which is the very key condition for our study here. For this assumption, an interesting question is: under what conditions it is possible to transform the switched linear system in (6) into descriptive multilinear systems framework and further develop the tensor-product representation theorem. It is also desirable that the tensor-product representation property should promote efficiently numerical computation with respect to the multidimensional time series data analysis for estimating the controller performance of multivariate switched linear systems. The main goal of this paper is to obtain a necessary and sufficient condition for the existence of tensor-product representation for (6) based on such multilinear algebra framework. The problem can be formulated as follows.

Problem 1 Given the switched linear system described in (6), derive necessary and sufficient conditions for tensor-product representation of systems operators, which show that the algebraic structure of switched linear systems can abstractly represented in tensor space.

In order to let the developed results focus on the tensor-product representation framework of multivariate switched linear systems in this paper, we consider the simplified systems description form with neglecting the effects of disturbances a_{t} in the following,

(8)

Then, we can obtain the switched linear systems operator, and the equivalent bilinear expression V is given as

(9)

where U and Y represent the inputs space and outputs space of subsystems, respectively and the L represents the logical switching space. And the logic switching signals function in (7) can be expressed as

(10)

where T denotes the logical operators with the same meaning in (7). From the expression (10), we can see that there exists a map from the switched linear systems outputs to the switching signals. Obviously, the expression (9) exactly describes the map of multivariate switched linear systems from bilinear algebra framework. The systems operator description in (8) can be interpreted as dynamical systems behavior in the physical processes, and the equivalent bilinear form in (9) is expressed from the point of view of algebra structure, which is suitable for the technical development of tensor-product representation approach for controller performance measure of multivariate switched linear systems or such more complex class of multivariate nonlinear systems.

4. Tensor-Product Representations

In this section, we discuss the tensor-product representation of multivariate switched linear systems from the point of view of systems operator by employing the theory of multilinear algebra. We first give a representation theorem for complete bounded linear operator on.

Theorem 3 [22] If is completely bounded then there exist representations, , isometries, and a contraction such that on elementary tensors

(11)

where.

Corollary 1 If is a completely bounded bilinear map then there exist representations, , and contractions, such that

(12)

Proof. Combine Proposition 2 with Theorem 3, the result is followed.

Now, we focus on studying the tensor-product representation of completely bounded bilinear operators from A × B into. Let f and g be linear functional from A into and B into, respectively. We define a linear function by

(13)

for all, ,. Let and be linear operators from into and into, respectively. Then we can define a linear operator by

(14)

For a linear operator, the adjoint linear operator of can be written as. The linear operator F is completely bounded if and only if each is a completely bounded linear functional on. If F is completely bounded, then so is with.

Let be a completely bounded linear operator and. For all

, , we have

(15)

in.

Lemma 2 Let and be the linear operators. If the linear operators F and G are completely bounded, then the corresponding linear operator is completely bounded. In this case, we have

.

Proof. First of all, we define completely bounded maps, by

(16)

For completely bounded operators, from Theorem 3 we can obtain

(17)

And more details can be found in Theorem 4. If F and G are completely bounded, we have

(18)

where, ,. Hence is completely bounded with and this completes the proof.

Lemma 3 If has norm one then there exist representations, , a contraction, and unit vectors, such that

(19)

Proof. By Proposition 2, is completely bounded, and so we may apply Theorem 3. The result now follows by setting,.

If and are bounded then there is a linear map de- fined by

(20)

This map need not be bounded unless both and are completely bounded.

Consider the simplified form of multivariate switched linear systems described in (9), we give the result of the representation theorem as in the following.

Theorem 4 Consider the multivariate switched systems described in (9), the associated linear map is completely bounded if and only if there are completely bounded linear functionals and on E such that, where

The linear operators and are completely bounded with.

The proof can be found in the Appendix.

Remark 1 There can be no extension theorem for general bilinear maps. If P

(21)

is another completely bounded linear operator from into such that, then there is a complex number with such that.

Suppose that is another completely bounded linear operator such that. Since, we may assume that and. Then there is an element

such that. Thus for all, we have

(22)

If we let, then we get for all, i.e. we get. Since

, then.

Proposition 5 Consider multivariate switched linear systems described in (9) and (10) with the associated linear map:, if the open or closed-loop switched linear systems is globally stable or globally asymptotically stable, then there exist representations, , isometries: and, and a contraction such that on elementary tensors

(23)

where.

Proof. Applying Theorem 3, the result is followed.

From the expressions (11) in Theorem 3 and (23) in Proposition 5, it is shown that the bilinear systems and switched linear systems are the same with the linear systems representation form. However, from (7) of the switched linear systems description in previous section, it can be found that systems outputs determine the logical switching signals described in (10), which shows the contraction T in Proposition 5 is determined in the operator itself (or switched linear systems itself). Furthermore, the logical functions outputs values are the logical values 0 or 1, so the logical functions outputs values are determined for the entire dynamical systems, of course, for the active subsystem, the logical value is 1. More details will be shown in the proof of Theorem 7.

Theorem 6 [22] Let H be a fixed infinite dimensional separable Hilbert space. The norm in satisfies

(24)

where the supremum is taken over all complete contractions,.

Lemma 4 Consider multivariate switched linear systems described in (9) and (10). Let

be the linear functional and be the logical functional. Then, if the linear functional F is completely bounded and the logical functional G is nonzero projection with

, the corresponding tensor-product representation is

completely bounded. In this case, we have.

Proof. Applying Theorem 6, the result is followed.

A necessary and sufficient conditions for tensor-product representation of multivariate switched linear systems can now be presented, under the assumption that the open or closed-loop switched linear systems is globally stable or globally asymptotically stable.

Theorem 7 Consider multivariate switched linear systems described in (9) and (10), the tensor-product representation is globally stable or globally asymptotically stable if and only if there are completely bounded linear functionals on and the logical switching functionals on such that, where

(25)

the linear function is completely bounded, and

(26)

the logical function is nonzero logical value projection with

.

The proof can be found in the Appendix.

It is noted that the F in Theorem 4 is defined to be taken over all representations for the linear operators, but the F in Theorem 7 is defined to be taken over all representations of the subsystems operators for multivariate switched linear systems. This difference naturally lead to the different tensor-product representations for the general bilinear systems and the multivariate switched linear systems, and the n in M_{n} can be interpreted as the integer of the all finite switched subsystems of multivariate switched linear systems. More details can be found in the Examples 1 and 2.

Remark 2 Theorem 7 mainly considers the logic switching space, which exactly describes the tensor-product algebra representation for multivariate switched linear systems, which can extend into the scenario of the time- varying disturbance dynamics. And Theorem 4 just considers the tensor-product algebra representation structure for general bilinear systems operators, which can also extend into more general class of multivariate nonlinear systems in Theorem 10 in this paper. In addition, Theorem 4 shows the tensor-product representation structure for general bilinear operator and Theorem 7 shows the tensor-product representation structure for multi- variate switched linear systems in the framework of bilinear algebra. Furthermore, multivariate switched linear systems essentially can be interpreted as the special class of general bilinear operator with logical linear operator substituting the general linear operators.

Next, we give an example (including operator description and unfolding form) of tensor-product representa- tion for multivariate switched linear systems from the perspective of the structured systems models subject to the explicitly bounded disturbance which guarantees the entire systems are completely bounded.

Example 1 Let the k-th subsystem transfer function be given as

(27)

For the dynamical systems, the tensor-product representation for the operator systems can be obtained as

(28)

where, or, also,. The input space is and the logical switching space is. For the entire switched linear systems, the tensor-product representation for the operator systems is given as

(29)

where, , , or, , also,. The input space is and the logical switching sequence is. Here, we assume that each subsystem just performs only once for the entire dy- namical systems. Consequently, the tensor-product representation for the input space can be obtained as and the corresponding output space is.

Intuitively, the systems description can be interpreted as considering a linear combination of the linear operators of the original switched linear system that evolves in an asymptotically stable manner in a subspace. The tensor-product representation algebra system evolves in a higher dimensional space to which the original system can be projected under the stability condition of the entire multivariate switched linear systems. From Example 1, we can see that the multivariate switched dynamical systems can be expressed in the form of tensor-product extending the matrix description of multivariate linear systems.

Remark 3 For the entire switched linear systems, the tensor-product representation can be interpreted as the linear systems form expressed as. Therefore, under the stability conditions of the systems, the control systems performance measure is attainable. It is shown that the multivariate switched linear systems can be transformed into the linear systems expression form via tensor-product representation technique, which is convenient to develop the controller performance assessment methods for such class of nonlinear systems.

Proposition 8 Let be multi-linear maps and let be its associated linear map, then the.

Proof. The proof follows that of Proposition 2.

Theorem 9 [22] Let be subspaces of -algebras, and let be a completely bounded multilinear map. Then there exist Hilbert spaces, representations, , , isometries and contractions, such that,

(30)

The construction of efficient representations to multivariate functions and related operators plays a crucial role in the numerical analysis of higher dimensional problems arising in a wide range of modern applications. We generalize our results in Theorem 4 to the k-linear operators.

Theorem 10 Let, Then is completely bounded if and only if there are completely bounded linear functionals with on U such that, where

(31)

The i-th linear operator F_{i} is completely bounded with.

The proof can be found in the Appendix.

Remark 4 It is difficult to describe the multivariate nonlinear systems in practical industrial applications. The analytic methods developed in [31] [32] for tensor-product approximations to multidimensional integral operators play an important role in the nonlinear control systems performance assessment. For the case of collocation schemes it focuses on the construction of tensor decompositions which are exponentially convergent in the separation rank. Separable approximation of functions and tensors can be derived by using a corresponding separable expansion of the generating function, including a separable approximation of multivariate functions, and the tensor-product approximation of some analytic matrix-valued functions. In practical applications, the general error estimation should mainly consider in the tensor-product approximations for multivariate switched linear systems. In the near future, we will apply analytically-based representation methods, which are efficient for a special class of function-related operators/tensors. The key observation is that there is a natural duality between separable approximation of the multivariate generating function and the tensor-product decomposition of the related multidimensional array which is suitable for geometric quantization for control systems performance assessment.

Corollary 2 A k-linear operator from A^{k} into
is representable if and only if it is completely bounded; when
is completely bounded,
, and the representable norm
is attained.

Proof. From the Theorem 9 and Theorem 10, the results is followed.

Next, we extend the multivariate switched linear systems to the case of time varying disturbance scenario. We also assume that disturbance dynamics are piecewise linear time varying. This type of time varying disturbances are reflected as different output trajectories in different subprocess, each subprocess having the same linear time invariant disturbance models. In other words, it is assumed that the disturbance models are the same within each subprocess, and it is required that the switching sequence of disturbance models is also the same within each subprocess. We will give the tensor-product algebraic representation from the point of view of operator theory in the following.

Theorem 11 Consider multivariate switched linear systems described in (9) and (10) with time varying disturbance dynamics, the tensor-product representation is globally stable or globally asymptotically stable if and only if there are completely bounded linear functionals on, the logical switching functionals on and on; the linear function is completely bounded, and the logical function

with and the disturbance logic switching function with are nonzero value projections, then with.

From the Lemma 4 and Theorem 9, the results is followed.

Now, we will give an example of tensor-product representation for multivariate switched linear systems with time-varying disturbance dynamics. The representation results is constructive for practical industrial process systems modeling framework.

Example 2 A possible tensor expression of system (6) can be described for all the K MIMO subsystems in the tensor-product representation form. And, the j-th disturbance transfer function is given as

(32)

For the dynamical systems, the tensor-product representation for the operator systems can be obtained as

(33)

where, or, ,;, , or, ,; also, , , ,. The input space is, the logical switching space is, and the disturbance logic switching space is.

For the entire switched systems, the tensor-product representation for the operator systems is given as

(34)

where, and

, , , or, ,;, , or, ,; also, , , ,. The k-subsystem input under j-dis- turbance mode is, the subsystems logical switching sequence is and the disturbance switching sequence. Here, we assume that each subsystem just performs only once for the entire dynamical systems. Consequently, the tensor-product representation for the input space can be obtained as and the corresponding output space is. The tensor space expression form for multivariate switched linear systems with time-varying disturbance dynamics is given as

(35)

where is the three-order tensor, and and are the four-order tensors.

5. Conclusion

A new tensor-product representation for describing the dynamic behavior of switched linear systems has been presented. The tensor-product representation makes it easy to see the relationship between abstractly algebraic expressions that propagate spatial quantities from subsystem-to-subsystem. Abstract dynamic equations arising from a hybrid systems analysis, can be reinterpreted as equivalent operator-formulated equations. This algebra expression can model the behavior of the switched linear systems by the use of a tensor-product representation, which shows that the algebraic structure can simplify switched linear systems expression. The proposed technique is well suited for real and complex systems, and is capable of providing the tensor-product representation of the class of stable nonlinear system. Furthermore, the interpretation of expressions within the tensor-product representation framework can be enhanced conceptual and physical understanding of switched linear systems, or more general of multivariate nonlinear systems, dynamic behavior expression.

Cite this paper

Dengyin Jiang, Lisheng Hu,, (2015) Tensor-Product Representation for Switched Linear Systems. *Journal of Applied Mathematics and Physics*, **03**, 322-337. doi: 10.4236/jamp.2015.33045

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Appendix

A.1. Proof of Theorem 4

Proof. The necessity and sufficiency proofs are in the following.

Necessity of proof: Let, be completely bounded. By applying Theorem 3 and Lemma 3, there are representations, , unit vectors , , and, such that

(36)

for,. Let P and Q be, respectively, the projections onto the closures of

and . Then(37)

Note that if X is any unitary matrix in M_{n}, then

(38)

Such elements are annihilated by, and this leads to

(39)

for all unitaries X. This forces the matrix to have the diagonal form, where,

.

The element can be represented using only a finite number of vectors

and. Define and. Choose a separable Hilbert

space H and norm one operators,
which are isometric on J_{1} and J_{2} and define completely bounded maps,
by

(40)

With these definitions it is easy to check that

(41)

where, , this implies that. It follows from Lemma 2 that

and are completely bounded with. This completes the necessity proof.

Sufficiency of proof: It follows, from Lemma 2, that is a completely bounded linear operator from into with. It suffices to show that the associated bilinear operator is completely bounded from into.

Since, by applying Theorem 3 and Lemma 3, there are repre-

sentations, , unit vectors,

, and, such that

(42)

for,.

The element can be represented using only a finite number of vectors

and. Define and. Choose a separable Hilbert

space H and norm one operators,
which are isometric on J_{1} and J_{2} and define com- pletely bounded maps,
by

(43)

Since

(44)

It follows that f and g are well-defined and we have

(45)

This shows that. This completes the sufficiency proof.

A.2. Proof of Theorem 7

Proof. To avoid technical complications, we only discuss the case n = 2. In the other words, consider the multivariate switched linear systems with only two subsystems. The calculations are in the same spirit for general subsystems.

Necessity of proof: Let,
be completely bounded. By Proposition 5, there are representations
of U on a Hilbert space H_{1} and
of L on a Hilbert space H_{2}, a bounded linear operator
with
and
with, and a logical projection T from
into
such that

(46)

for all,
, with,. Let
be a unit vector in H_{2} such that

and let be the standard basis for H. For, there

are logical functional g_{i} on H_{1} such that

(47)

Let
be a unit vector in H such that
and the linear functional f_{i} on U such that

(48)

for all. Then we have

(49)

for all and for all. This implies that, where and. It follows from Lemma 4 that and (Here,) are completely bounded with. This completes the necessity proof.

Sufficiency of proof: It follows, from Lemma 4, that the is a completely bounded from

into with. It suffices to show that the associated bilinear representation V is completely bounded from into. Here we only prove the case as in the necessity proof.

Since, by Theorem 3, there are representations
of U on a Hilbert space H_{1} and
of L on a Hilbert space H_{2}, a bounded linear operator
with
and
with, and a logical operator T from
into
with

such that

(50)

for all, with.

Since very element can be written as

(51)

for some, , we define a linear functional by

(52)

for all.

And very element can be written as

(53)

for some, , we define a linear functional by

(54)

for all. Since

(55)

where, it follows that and are well-defined and we have

(56)

lead to

(57)

for all. Therefore,

(58)

where.

Since is a dense subspace of, there is a unique norm preserving linear extension of from to the whole Hilbert space, still denoted by. It is shown that with

. This completes the sufficiency proof.

A.3. Proof of Theorem 10

Proof. We first consider the case. By Proposition 1, is completely isometrically isomorphic to a subspace F of some -algebra φ. The linear map is completely bounded, which corresponds to a trilinear map V from Proposition 2. We identify with F, then the map is completely bounded. Thus, by the bilinear case there exist Hilbert spaces and, representations and isometries, , and a contraction, such that

(59)

Since the inclusion is completely bounded, the map is completely bounded. Applying the representation theorem 4 for completely bounded bilinear maps to the map and substituting this expression into the formula for, then we can get.

From Lemma 2, we can obtain. Then, for the map

combining with inclusion, we have. Consequently, by substituting this

expression into the formula for, we can obtain. When, the results can be directly derived as the same line of this idea.