Tensor Products of Operator Algebras and Direct Integrals

Tensor products of C*-algebras and von Neumann algebras will be introduced in this chapter, and direct integrals, a continuous analogue of direct sums, will also be discussed. Unlike the finite dimensional case, tensor products of infinite dimensional vector spaces or algebras are quite nontrivial. It is still considered as one of the tricky parts of the theory. The norms in the tensor product of two Banach spaces are highly nonunique. Fortunately however, the norm in the tensor product of Hilbert spaces which makes it a pre-Hilbert space is unique. This uniqueness makes interesting the tensor products of C*-algebras. However, the norms on the tensor products of C*-algebras which make it a pre-C*-algebra are not unique. In this chapter, the problem of defining a suitable norm on the tensor product of C*-algebras will be attacked. Sections 1 and 2 are just preliminary. In Section 3, completely positive maps are introduced to provide a powerful tool in the succeeding sections. Section 4 is devoted to the problems mentioned above. There we shall show that the tensor product of C*-algebras admits two natural norms: one is the smallest norm among all possible C*-norms and the other is the largest. In Section 5, the tensor product of W*-algebras is discussed, and it is proved that the algebraic type of the tensor product of von Neumann algebras is uniquely determined by those of the component algebras without regard to the underlying Hilbert spaces, Theorem 5.2.