Noncommutative Fourier transforms of bounded bilinear forms and completely bounded multilinear operators
type: Item
instance of: scholarly article
title: Noncommutative Fourier transforms of bounded bilinear forms and completely bounded multilinear operators
zbMATH Open document ID: 0663.46053
DOI: 10.1016/0022-1236(88)90034-1
author: K. Ylinen
published in: Journal of Functional Analysis
publication date: 1988-01-01T00:00:00Z
review text: 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…
Mathematics Subject Classification ID: 46L10
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46L51
Mathematics Subject Classification ID: 46L53
Mathematics Subject Classification ID: 46L54
zbMATH DE Number: 4084359
zbMATH Keywords: locally compact groups
zbMATH Keywords: group \(C^*\)-algebra
zbMATH Keywords: enveloping \(W^*\)- algebras
zbMATH Keywords: unitary representations
zbMATH Keywords: universal representations
zbMATH Keywords: Jordan morphisms
zbMATH Keywords: Grothendieck-Pisier-Haagerup inequality
zbMATH Keywords: n-linear map
zbMATH Keywords: dilation techniques
zbMATH Keywords: representation of the Fourier transform
zbMATH Keywords: tensor product
zbMATH Keywords: Banach algebra of completely bounded n-linear operators
zbMATH Keywords: contractive unital morphisms
zbMATH Keywords: noncommutative probability theory
reviewed by: Herbert Halpern
MaRDI profile type: Publication
full work available at URL: 0022-1236(88)90034-1
OpenAlex ID: W2081043731
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description: scientific article; zbMATH DE number 4084359