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The tensor product of commutative algebras is exactly their coproduct in the category of commutative algebras. In other words, if A and B are two commutative algebras, then the covariant functor that represents A⊗B assigns to an algebra Z the set of all pairs of morphisms f: A→Z and g: B→Z.

Tensor product of noncommutative algebras also admits a categorical characterization. Namely, if A and B are two noncommutative algebras, then the functor that represents A⊗B assigns to an algebra Z the set of all pairs of morphisms f: A→Z and g: B→Z whose images commute in Z, i.e., m(f⊠g)=ms(f⊠g), where m is the multiplication Z⊗Z→Z, s is the symmetry Z⊗Z→Z⊗Z, and ⊠ is the external tensor product: f⊠g: A⊗B→Z⊗Z.

The category of commutative von Neumann algebras also admits a coproduct, which therefore can be thought of as the categorical tensor product of von Neumann algebras. This tensor product can be extended to noncommutative von Neumann algebras in the same way as described above. Apparently this product was first described by Alain Guichardet in his 1966 paper.

The categorical tensor product is much bigger than the spatial tensor product. The difference for commutative algebras is explained in this answer: http://mathoverflow.net/questions/49426/is-there-a-category-structure-one-can-place-on-measure-spaces-so-that-category-th/49542#49542

Is there a categorical characterization of the spatial tensor product of von Neumann algebras?

By the universal property of the categorical tensor product for any two von Neumann algebras A and B there is a canonical morphism Q: C→S of von Neumann algebras from the categorical tensor product C to the spatial tensor product S. This morphism is an epimorphism, i.e., it is surjective. However, unless one of the algebras is finite-dimensional, it has non-trivial kernel.

Hence, the algebra S is represented by a subfunctor of the covariant functor that assigns to a von Neumann algebra Z the set of all pairs of morphisms f: A→Z and g: B→Z with commuting images.

Can we characterize categorically the pairs (f,g) that belong to this subfunctor?

Alternatively, the kernel of the morphism Q is a σ-weakly closed two-sided ideal of C, which corresponds to a central projection of C. Can we characterize this central projection categorically?

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# Can we characterize the spatial tensor product of von Neumann algebras categorically?

The tensor product of commutative algebras is exactly their coproduct in the category of commutative algebras. In other words, if A and B are two commutative algebras, then the covariant functor that represents A⊗B assigns to an algebra Z the set of all pairs of morphisms f: A→Z and g: B→Z.

Tensor product of noncommutative algebras also admits a categorical characterization. Namely, if A and B are two noncommutative algebras, then the functor that represents A⊗B assigns to an algebra Z the set of all pairs of morphisms f: A→Z and g: B→Z whose images commute in Z, i.e., m(f⊠g)=ms(f⊠g), where m is the multiplication Z⊗Z→Z, s is the symmetry Z⊗Z→Z⊗Z, and ⊠ is the external tensor product: f⊠g: A⊗B→Z⊗Z.

The category of commutative von Neumann algebras also admits a coproduct, which therefore can be thought of as the categorical tensor product of von Neumann algebras. This tensor product can be extended to noncommutative von Neumann algebras in the same way as described above.

The categorical tensor product is much bigger than the spatial tensor product. The difference for commutative algebras is explained in this answer: http://mathoverflow.net/questions/49426/is-there-a-category-structure-one-can-place-on-measure-spaces-so-that-category-th/49542#49542

Is there a categorical characterization of the spatial tensor product of von Neumann algebras?

By the universal property of the categorical tensor product for any two von Neumann algebras A and B there is a canonical morphism Q: C→S of von Neumann algebras from the categorical tensor product C to the spatial tensor product S. This morphism is an epimorphism, i.e., it is surjective. However, unless one of the algebras is finite-dimensional, it has non-trivial kernel.

Hence, the algebra S is represented by a subfunctor of the covariant functor that assigns to a von Neumann algebra Z the set of all pairs of morphisms f: A→Z and g: B→Z with commuting images.

Can we characterize categorically the pairs (f,g) that belong to this subfunctor?

Alternatively, the kernel of the morphism Q is a σ-weakly closed two-sided ideal of C, which corresponds to a central projection of C. Can we characterize this central projection categorically?