Noncommutative Fourier transforms of bounded bilinear forms and completely bounded multilinear operators
Let \(G_ i\) \((i=1,...,n)\) be locally compact groups, let H be a Hilbert space, and let B be an n-linear map of \(C^*(G_ 1)\times...\times C^*(G_ n)\) into L(H). Here \(C^*(G)\) is the group \(C^*\)-algebra of G. First let \(n=2\) and \(L(H)={\mathbb{C}}\). Then B is a bilinear form and B has a unique extension \(\tilde B\) to a separately continuous bilinear map of \(W^*(G_ 1)\times W^*(G_ 2)\) of the product of the enveloping \(W^*\)-algebras \(C^*(G_ 1)\) and \(C^*(G_ 2)\). The Fourier transform \(\hat B\) of B is the map \^B(s,t)\(=\tilde B(\omega_ 1(s),\omega_ 2(t))\) where \(\omega_ i\) are the unitary representations of \(G_ i\) associated with the universal representations. The author shows that a function F: \(G_ 1\times G_ 2\to {\mathbb{C}}\) is the Fourier transform of some bounded bilinear form on \(C^*(G_ 1)\times C^*(G_ 2)\) if and only if there is a Hilbert space H and two continuous Jordan morphisms (i.e., the direct sum of a continuous unitary representation and antirepresentation) \(\pi_ i: G_ i\to L(H)\) and two vectors \(\xi\) and \(\eta\) in H such that \(F(s,t)=(\pi_ 1(s)\pi_ 2(t)\xi,\eta)\) for all (s,t) in \(G_ 1\times G_ 2\). The proof uses the Grothendieck-Pisier-Haagerup inequality in the following form: if \(A_ 1\) and \(A_ 2\) are \(C^*\)-algebras and B: \(A_ 1\times A_ 2\to {\mathbb{C}}\) is a bounded bilinear form, then there are Jordan morphisms \(\pi_ i: A_ i\to L(H)\) for some Hilbert space H and two vectors \(\xi\) and \(\eta\) in H such that \(B(x,y)=(\pi_ 1(x)\pi_ 2(y)\xi,\eta)\) for all (x,y) in \(A_ 1\times A_ 2.\) Now let \(A_ i\) \((i=1,...,n)\) be \(C^*\)-algebras. An n-linear map \(\Phi\) of a product \(A_ 1\times...\times A_ 2\) into the algebra L(H) is completely bounded according to the characterization of \textit{E. Christensen} and \textit{A. M. Sinclair} [J. Funct. Anal. 72, 151-181 (1987; Zbl 0622.46040)] if and only if there exists Hilbert spaces \(H_ i\) \((i=0,...,n+1)\) with \(H_ 0=H=H_{n+1}\), operators \(V_ i\in L(H_ i,H_{i+1})\) and representations \(\theta_ i\) of \(A_ i\) on \(H_ i\) such that \(\Phi (x_ 1,...,x_ n)=V_ 0\theta_ 1(x_ 1)V_ 1\theta_ 2(x_ 2)\cdot \cdot \cdot V_ n\theta (x_ n)V_{n+1}\) for all \(x_ i\in A_ i\). In this case the norm \(\| \Phi \|_{cb}=glb\| V_ 0\| \| V_ 1\| \cdot \cdot \cdot \| V_{n+1}\|\) is attained, and in fact, is attained for a set of \(V_ i's\) with \(\| V_ i\| =\cdot \cdot \cdot =\| V_ n\| =1\). Using the existence of the representations \(\pi_ i\), the author defines the Fourier transform of a completely bounded n-linear function in the same manner as he defined the Fourier transform of a bounded bilinear form. Then, using dilation techniques, the author shows that given a completely bounded n-linear form \(\Phi\) into L(H), there is a second Hilbert space K and \(R\in L(K,H)\), \(S\in L(H,K)\), and Jordan morphisms \(\pi_ i: A_ i\to L(K)\) such that \(\Phi (x_ 1,...,x_ n)=R\pi_ 1(x_ 1)\cdot \cdot \cdot \pi_ n(x_ n)S\) for all \(x_ i\in A\). Then a function F: \(G_ 1\times \cdot \cdot \cdot \times G_ n\to L(H)\) is the Fourier transform of a completely bounded n-linear map of \(\Phi\) : \(A_ 1\times \cdot \cdot \cdot \times A_ 2\to L(H)\) if and only if F has the form \(F(s_ 1,...,s_ n)=R\pi_ 1(s_ 1)\cdot \cdot \cdot \pi_ n(s_ n)S\) where R, S, \(\pi_ i\) are as before. From this representation of the Fourier transform, the author can assert that the tensor product (and in particular, the product) of two Fourier transforms is a Fourier transform. The author studies the Banach algebra of completely bounded n-linear operators. The fact that multiplication (convolution) is defined and that the (completely bounded) norm satisfies the Banach algebra condition follows from the work just described. Here the author shows that there are unique contractive unital morphisms \(\alpha\) and \(\beta\) of \(B(G_ 1){\hat \otimes}\times \cdot \cdot \cdot \times {\hat \otimes}B(G_ 1)\to B(G_ 1\times \cdot \cdot \cdot \times G_ n)\) and of \(B(G_ 1\times \cdot \cdot \cdot \times G_ n)\to CB(C^*(G_ 1)\times \cdot \cdot \cdot \times C^*(G_ n))\) respectively such that \(\alpha (f_ 1\otimes \cdot \cdot \cdot \otimes f_ n)(s_ 1,...,s_ n)=\prod f_ i(s_ i)\) for all \(f_ i\in B(G_ i)\) and such that the Fourier transform of \(\beta\) (f) is f for all \(f\in B(G_ 1\times \cdot \cdot \cdot \times g_ n)\). Here B(G) is the commutative Banach algebra of all linear combinations of positive definite functions on G equipped with the norm derived from the identification of B(G) with \(C^*(G)^*.\) The author presents some applications to noncommutative probability theory.
Banach algebra of completely bounded n-linear operators
scientific article; zbMATH DE number 4084359
scientific article