Linear representations of bilinear forms on operator algebras
A very nice paper discovering a linear representation of bilinear forms on \(C^\ast\)-algebras and von Neumann algebras. Let \(A\) and \(B\) be \(C^\ast\)-algebras and \(D\) be a bounded bilinear form on \(A\times B\). The author shows that \(D\) need not be induced by a bounded linear functional \(\varphi\) on the minimal \(C^\ast\)- tensor product \(A\otimes B\) in the sense of the equality \(D(x,y)=\varphi(x\otimes y)\), as it is well known in the category of algebraic tensor products, bilinear forms and linear maps. Nevertheless, he proves that the moral of the universal property of tensor product is true also in case of continuous mappings in the following sense: Any bounded bilinear form \(D\) on the product \(A\times B\) of \(C^\ast\)-algebras \(A\) and \(B\) is of the form \[ D(x,y)=\sum_{i\in I}\varphi_i(x)\psi_i(y), \] where \((\varphi_i)\) and \((\psi_i)\) is a family of bounded linear functionals on \(A\) and \(B\), respectively, such that \(\sum_i|\varphi_i(x)|^2<\infty\) and \(\sum_i|\psi_i(y)|^2<\infty\) for each \(x\in A\) and \(y\in B\). In case of \(A\) and \(B\) being von Neumann algebras it is shown that \((\varphi_i)\) and \((\psi_i)\) can be chosen normal, whenever \(D(x,y)\) is normal. Moreover, versions of these results for Hermitian sesquilinear forms are presented. The proofs are based on an elegant use of the Haagerup-Pisier-Grothendieck inequality for bilinear forms on noncommutative algebras. The results have applications to the theory of decoherence functionals in quantum mechanics.
coherence functionals in quantum mechanics
scientific article; zbMATH DE number 1151670