Operator space tensor products and inductive limits

Abstract

We investigate whether the λ -tensor product of operator spaces which includes the operator space projective, Haagerup and Schur tensor products, and the λ -tensor product of operator systems commute with inductive limits, in the sense that if ( X n , ϕ n ) is an inductive system of operator spaces (or operator systems) with an inductive limit X , and if Y is any operator space (or operator system), then whether ( X n ⨂ λ Y , ϕ n ⊗ id ) becomes an inductive system, and if so, whether the inductive limit of this system is same as X ⨂ λ Y , where ⨂ λ denotes the λ -tensor product of operator spaces (or operator systems). We also discuss about the bidual of such an inductive limit and obtain embeddings lim → ( X n ⁎ ⁎ ⨂ λ Y ⁎ ⁎ ) ⊆ ( lim → X n ⨂ λ Y ) ⁎ ⁎ under certain restrictions.