A New Approach on Tensor Norms and Its Classification

A New Approach on Tensor Norms and Its Classification

Mr. Ajay Kumar1, Dr. Sushil Kumar Jamariar2 , Mr. Alok Kumar Pandey3.

Department Of Mathematics

1. Dr. C.V. Raman University, Bhagwanpur, Vaishali, Bihar,

2. Dr. C.V. Raman University, Bhagwanpur, Vaishali, Bihar,

3. Dr. C.V. Raman University, Kargi Road , Kota, Bilaspur. (C.G)

ABSTRACT – In this paper we are going to establish a new approach on Tensor Norms and its classification with basic properties we discuss five norms on algebric tensor product which are mutually distinct But in

general there are several distinct (usually in complete) C*- norms on algebric tensor product AB-we also

begin with the dual norms and this leads naturally to the Vital Concept of accessibility, which Can be thought

of as an analogue for tensor norms of the approximation property for spaces- Next we have to attempt to the identification of the duals of the chevet – saphar tensor norms in terms of The Classes of p-integral operations.

In final section we conclude with Grothendicks classification of the natural tensor norms.

Keyword : Banach space, Algebric Tensor product, Approximation property. Isometric lonbeding finite dimensional space, C*- Algebra, W*- Algebra,

INTRODUCTION

The Tensors are classified according to their type (n,m) where n is the number of contra variant indices , m is the number of covariant indices and n + m gives the total order of the tensor . Whereas a norm is a function from a real or complex vector space to the non negative real numbers that behaves in certain ways like the distance from the origin it commutes with scaling obeys a from of the triangle in equality and is zero only at the origin ,

In particular the Euclidean distance in a Euclidean space is defined by a norms on the associated Euclidean vector space called Euclidean norm , the 2 norm or some times the magnitudes of the vector . This norm can be defined as the square root of the inner product of a vector with it self. As dual norm. If A and B are finite dimensonal normed

spaces and be a tensor norm then A B is algebrically the dual space of (A* B*) and we may define 1 to be

a dual norm

A 1B = (A* B*)* In other words if U A B

then 1 (u) = Sup {|< , |: , () < 1}

Here we discuss the five norms , v1 ,vr, and V on AA Latter , we will find that all five norms are mutually distinet

.

Let A and B be C*- algebra b with algebric tensor product AB. In general there are serveral distinct C*- norms on AB. Two such norms are of particular interest. The maximal norm Vand the minimal norm .

If 1and 2 are representaves of A and B respectively , on the Hilbort space H { 1 , 2 } is said to be a commuting pair of representatoins of A, B if 1 (a)2 (b) = 2 (b)1 (a) , (a A, b B) The norms v is defined by V( bi ) = Sup() ()

Proposition 1 :-

Let A and B be Banach space with the metric approximation property , then s = 1 on A B . This result does not explain the fact that s = 1 =. This coincidence can be explained by the possession by the injective norm

of a prpoery that is deal to finite generation

Proposition 2:-

Let A = MN , then the five norms , v1 , vr, and v on AA are mutually disfimet More over is normal if and only if 1and 2 are, and for bi M1 B, Cj M2B ( ) ( ) + ( ) () = max

( ( ) ( ) , ( )

( ) )

2 2

The lemma follows easily from this relation and the definitions of the various norms.

PROOF OF PROPOSITION

In view of the lemma, it is Sufficient to check any two of the norms , v1 , r , and V differ on at least one of the tensor products

MM ,MN , NM and NN

1. On MM , = v1 =

In the notation of homomorphisms.

X (x) , (x M)

And Y R ( ) ,(y N)

Constitute a commuting pair of representative of M ,N on H (N), The second representation being normal . Thus the

homomorphism yc (xc) R ( i)j MN

LH(N) is lemma.

Let M1 , M2 and B be w*Algebra then the canonical isomorphism

( M1 M2) B (M1 )(M2 )extends to an isomorphism of ( M1 M2)nB on to ( M1nB)( M1n B )

When n is any of the above five norms.

PROOF OF LEMMA

Let e and f be the identity Projections of M1 and M2respectively , then e + f = 1 ,

Let { ,}be commuting pair of representations of ( M1 M2), B on the Hilbert space H. (e) and( f ) commute with(M1 M2)and (B) so that HI = (e)H and H2 = (f)H are in varient subspaces for and

Let 1 =/Hi , = HI /Hi ( i = 1,2)

1

Then { 1 , l

2

} and { 2 , l

} are commuting pairs of representations of M1M2, B on H1 and H2 respectively.

1. Continuous relative to the norm Vr on MN and also if it is not continious relative to , so that

   vr v on MN.

2. Exactly the same process = vr = v1 on NM

3. The representation yi R ( ) of NN on H (H) is clearly continuous relative to the norm B on NN .

Again by the other relevant proposition, this representation is not a continous relative to .

Thus on NN

Thus the proposition is now complrted Hence the result.

ACKNOWLEDGEMENT

The authors are Thankful to Prof (Dr.) Basant Singh, Provice Chancellor,Dr. C.V. Raman University , Vaishali Bihar, and Prof (Dr.) Dharmendra Kumar Singh, Dean Academic , Dr. C. V. Raman University, Vaishali ,Bihar, India . Thanks to library and its incharge , Dr. C .V . Raman University

,Vaishali, Bihar, India. For extending all facilities in the completion of the present research work

[1] Wilansky ,

REFERENCES

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[7] Tomiyama, J. (1971) Tensor Products and Projection of Norms. One in VON -NEUMANN Algebras. [8] E.G EFFROS and E.C LANGE (1970) Tensor product of operator Algebra.