Preservers on operator systems

Hermann Weyl said, "in order to understand any mathematical structure, one should investigate its group of symmetries." In the class of Banach spaces this leads naturally to the study of isometries. An isometry between two Banach spaces is an operator that preserves the distance between every two elements of the domain. Some simple examples of an isometry are reflection, translation and rotation in Euclidean space; from which we see the beautiful geometric interpretations of isometries. A more sophisticated example is the Fourier transform, which has many applications such as solving PDEs and integral equations.
From the criterion of distance preserving we get many desirable properties for our operator, such as injectivity and continuity. Moreover, if our operator is linear, then being an isometry is equivalent to being norm preserving. What is even more interesting is that the surjective linear isometries are precisely the isometric isomorphisms between Banach spaces. If we establish that our operator is a surjective linear isometry, and therefore an isometric isomorphism, it follows that the two Banach spaces are completely identical in both algebraic and topological structure. It is for this reason that we have a particular interest in researching surjective linear isometries.
Through this project we seek to determine the canonical forms of all surjective linear isometries between two specified Banach spaces. Obtaining such a canonical form gives us a concrete way of writing the isometry, making it less abstract; but it also tells us about the algebraic and geometrical structures of the isometry and Banach spaces. Determining such canonical forms has been an active area of research since the work of Stefan Banach and Marshall Stone, when they determined the canonical form of all surjective linear isometries between the Banach spaces of scalar-valued continuous functions on compact Hausdorff spaces. The canonical form being that of a weighted composition. Furthermore, this result also holds for the Banach space of scalar-valued continuous functions that vanish at infinity on locally compact Hausdorff spaces. This result is known as the Banach-Stone theorem and it astonishingly fits in many other settings. One of the most important generalisations of this theorem was by Richard Kadison, when he extended the result to surjective linear isometries between unital C*-algebras. He determined that the canonical form is a Jordan *-isomorphism, also known as a quantum mechanical isomorphism, followed by a left multiplication of a unitary element.
Our aim in this project is to establish results similar to above, but for the tensor products of two spaces. Here we will consider the projective, injective and Haagerup tensor norms on these tensor product spaces to see how different tensor norms affect the outcomes. The study of tensors is a very active area of research in many areas of mathematics, from abstract harmonic analysis and operator algebra to quantum information theory, and there are many research questions to address. On the other hand, the study of preservers, including isometries, is more classical. With this in mind, this research aims to bridge the gap between the classical study of preservers and the modern study of tensors, which will open up new avenues of research. The research area this project lies in is operator theory and more broadly speaking functional analysis.
Therefore, our first aim is to determine the canonical form of all surjective linear isometries when our Banach spaces in the Banach-Stone theorem are replaced with the tensor product of two Banach spaces. Then similar to how Kadison generalised the Banach-Stone theorem, we next aim to determine the canonical forms of all surjective linear isometries whose domain and codomain are the tensor product of two operator algebras. Once we have established such results we will then look at other closely related preservers to generalise the results.