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. Ï• 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, Ï•n is positive for all natural numbers n. Ï•n is completely bounded if
every Ï•n is bounded and the supremum of the norm of Ï•n 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.