### CHARACTERIZATION OF TENSOR NORMS AND CONVERGENCE IN C∗−ALGEBRAS

#### Abstract

A linear map ϕ from a C∗-algebra A to a C∗-algebra B is positive if it maps

positive elements of A to positive elements of B. ϕ is completely positive if

for the corresponding linear maps ϕn from the C∗-algebra of n by n matrices

with entries from A to the C∗-algebra of n by n matrices with entries from

B, ϕn is positive for all natural numbers n. ϕn is completely bounded if

every ϕn is bounded and the supremum of the norm of ϕn is finite for all

natural numbers n. In this paper we have considered the C∗-algebras of

n by n matrices, constructed various maps between the C∗-algebras and

characterized the cross-norms of the C∗-algebras. We have established the

conditions for complete positivity and complete boundedness of the tensor

product of the maps on the C∗-algebras.

Keywords : C∗-algebras, Tensor products and Tensor cross-norms.

positive elements of A to positive elements of B. ϕ is completely positive if

for the corresponding linear maps ϕn from the C∗-algebra of n by n matrices

with entries from A to the C∗-algebra of n by n matrices with entries from

B, ϕn is positive for all natural numbers n. ϕn is completely bounded if

every ϕn is bounded and the supremum of the norm of ϕn is finite for all

natural numbers n. In this paper we have considered the C∗-algebras of

n by n matrices, constructed various maps between the C∗-algebras and

characterized the cross-norms of the C∗-algebras. We have established the

conditions for complete positivity and complete boundedness of the tensor

product of the maps on the C∗-algebras.

Keywords : C∗-algebras, Tensor products and Tensor cross-norms.

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