W*-Algebras as a complete axiomatisation of Von-Neumann algebras

TL;DR

This paper proves a stronger version of Sakai's theorem, establishing that W*-algebras provide a complete and unique axiomatisation of Von-Neumann algebras, analogous to C*-algebras.

Contribution

It offers a strengthened proof of Sakai's theorem, demonstrating the completeness and uniqueness of W*-algebras as axiomatisations of Von-Neumann algebras.

Findings

W*-algebras form a complete axiomatisation of Von-Neumann algebras.

The predual of a W*-algebra is unique up to isometric isomorphism.

The proof simplifies Sakai's original proof using the strengthened theorem.

Abstract

By the Gelfand-Naimark theorem, any C*-algebra is isometrically isomorphic to a *-algebra of bounded operators on a Hilbert space which is closed with respect to the topology induced by the operator norm. Hence, the C*-algebras furnish an axiomatisation of the structure of those operator algebras. In fact, the axiomatisation is complete in the sense that the entire structural information considered on the closed *-algebra -- namely addition, scalar multiplication, adjunction and composition of operators as well as the operator norm -- is reflected in the C*-algebra. From this point of view, we treat Von-Neumann algebras, another class of operator algebras also defined as closed *-algebras, but with respect to another topology -- one possibility is the weak operator topology. The problem asking for an axiomatisation of Von-Neumann algebras was solved by S. Sakai who introduced so-called W*-algebras. They are C*-algebras which, considered as a Banach space, are isometrically isomorphic to the dual space of another Banach space called the predual. Sakai's theorem asserts that the W*-algebras provide an axiomatisation of Von-Neumann algebras. In this master's thesis, we give a proof of Sakai's theorem. In fact, we provide a stronger version of Sakai's theorem which shows that W*-algebras indeed form a complete axiomatisation analogous to the situation for C*-algebras. Additionally, we show uniqueness of the predual up to isometric isomorphism, simplifying Sakai's original proof by using our strengthened version of Sakai's theorem.