Vol. 30, No. 1, 1969

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On the tensor products of von Neumann algebras

Jun Tomiyama

Vol. 30 (1969), No. 1, 263–270

Let A and B be C-algebras and let AαB be their C-tensor product with Turumaru’s α-norm. The author has previously defined mappings Rφ;A αB B and Lψ : A αB A via bounded linear functionals φ on A and ψ on B, as follows:

    ∑n         ∑n
R φ(   ai ⊗ bi) =   < ai,φ > bi,
i=1          i=1

    n∑          ∑n
L ψ(   ai ⊗ bi) =   < bi,ψ > ai,
i=1         i=1

and has shown how the families {Rφ|φ A} and {Lψ|ψ B} determine the structure of the tensor product of A and B. Moreover, in a joint paper with J. Hakeda the author also proved the existence of these kinds of mappings in tensor products of von Neumann algebras and gave some of their applications. Further applications of these mappings are shown in the present paper.

Mathematical Subject Classification
Primary: 46.65
Received: 8 January 1968
Published: 1 July 1969
Jun Tomiyama
Tokyo Metropolitan University
201 11-10 Nakane 1-chome