A note on the minimal tensor product and the C*-envelope of operator systems

TL;DR

This paper proves that the C*-envelope of the minimal tensor product of two operator systems equals the minimal tensor product of their C*-envelopes, and explores implications for the propagation number.

Contribution

It establishes a key isomorphism between the C*-envelope of a tensor product and the tensor product of C*-envelopes, advancing understanding of operator system tensor products.

Findings

The C*-envelope of the minimal tensor product is isomorphic to the tensor product of C*-envelopes.

Identifies the Silov boundary ideal of the tensor product of operator systems.

Shows the propagation number of the tensor product equals the maximum of the factors' propagation numbers.

Abstract

In this article, we show that the $C^*$-envelope of the minimal tensor product of two operator systems is isomorphic to the minimal tensor product of their $C^*$-envelopes. We do this by identifying the Silov boundary ideal of the minimal tensor product of two operator systems. Finally, as an application of this result, we show that the propagation number of the minimal tensor product of operator systems is the maximum of the propagation numbers of the factors.