Operator Isomorphisms on Hilbert Space Tensor Products

TL;DR

This paper introduces an isomorphism between operator algebras on Hilbert-Schmidt operators and tensor product spaces, extending Choi's finite-dimensional result to infinite dimensions while preserving operator products.

Contribution

It presents a new isomorphism applicable to infinite-dimensional Hilbert spaces, unlike Choi's theorem, and maintains operator product structure.

Findings

The isomorphism applies to infinite-dimensional Hilbert spaces.

It preserves operator products, unlike Choi's isomorphism.

It offers potential applications in operator algebra analysis.

Abstract

This article presents an isomorphism between two operator algebras $L_1$ and $L_2$ where $L_1$ is the set of operators on a space of Hilbert-Schmidt operators and $L_2$ is the set of operators on a tensor product space. We next compare our isomorphism to a well-known result called Choi's isomorphism theorem. The advantage of Choi's isomorphism is that it takes completely positive maps to positive operators. One advantage of our isomorphism is that it applies to infinite dimensional Hilbert spaces, while Choi's isomorphism only holds for finite dimensions. Also, our isomorphism preserves operator products while Choi's does not. We close with a brief discussion on some uses of our isomorphism.