embedding $L(H_1) \hookrightarrow L(H_1 \otimes H_2)$

Question

Let $H_1, H_2$ be Hilbert spaces. Let $L(H_i)$ ,$L(H_1 \otimes H_2)$ denote the bounded linear operators on the spaces.

There is a canonical map $$L(H_1) \rightarrow L(H_1 \otimes H_2), \quad T\mapsto T \otimes I$$ then this map isometric $*$-homomorphism.

I wonder if there is a reference for the proof of the above statement.

Answer 1

This is not true - this map is an isometric $*$-homomorphism, but it is not an isomorphism - it is not surjective.

Isometricity easily follows from the fact, that injective $*$-homomorphisms between $C^*$-algebras are isometric. This theorem can be found in most books on $C^*$-algebras, e.g. Davidson's book "$C^*$-algebras by example", theorem I.5.5.