Multivariable moment problems

Multivariable moment problems In this paper we solve moment problems for Poisson transforms and, more generally, for completely positive linear maps on unital C *-algebras generated by “universal” row contractions associated with $$\mathbb{F}_{n}^{+}$$ , the free semigroup with n generators. This class of C*-algebras includes the Cuntz-Toeplitz algebra $$C^{*} (S_{1}, \ldots, S_{n})$$ (resp. $$C^{*} (B_{1}, \ldots, B_{n})$$ ) generated by the creation operators on the full (resp. symmetric, or anti-symmetric)) Fock space with n generators. As consequences, we obtain characterizations for the orbits of contractive Hilbert modules over complex free semigroup algebras such as $$\mathbb{CF}_{n}^{+}, \mathbb{C}[z_{1}, \ldots, z_{n}]$$ ,and, more generally, the quotient algebra $$\mathbb{CF}_{n}^{+}$$ , where J is an arbitrary two-sided ideal of $$\mathbb{CF}_{n}^{+}$$ . All these results are extended to the generalized Cuntz algebra $$\cal{O} (x_{i=1}^{n} G_{i}^{+})$$ , where G i + are the positive cones ofdiscrete subgroups G i + of the real line $$\mathbb{R}$$ . Moreover, we characterize the orbits of Hilbert modules over the quotient algebra $$\mathbb{C}_{i=1}^{*n} G_{i}^{+}/J$$ , where J is an arbitrary two-sided ideal ofthe free semigroup algebra $$\mathbb{C}_{i=1}^{*n} G_{i}^{+}$$ . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Multivariable moment problems

Positivity , Volume 8 (4) – Sep 11, 2004
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Publisher
Kluwer Academic Publishers
Copyright
Copyright © 2004 by Kluwer Academic Publishers
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-004-7398-1
Publisher site
See Article on Publisher Site

Abstract

In this paper we solve moment problems for Poisson transforms and, more generally, for completely positive linear maps on unital C *-algebras generated by “universal” row contractions associated with $$\mathbb{F}_{n}^{+}$$ , the free semigroup with n generators. This class of C*-algebras includes the Cuntz-Toeplitz algebra $$C^{*} (S_{1}, \ldots, S_{n})$$ (resp. $$C^{*} (B_{1}, \ldots, B_{n})$$ ) generated by the creation operators on the full (resp. symmetric, or anti-symmetric)) Fock space with n generators. As consequences, we obtain characterizations for the orbits of contractive Hilbert modules over complex free semigroup algebras such as $$\mathbb{CF}_{n}^{+}, \mathbb{C}[z_{1}, \ldots, z_{n}]$$ ,and, more generally, the quotient algebra $$\mathbb{CF}_{n}^{+}$$ , where J is an arbitrary two-sided ideal of $$\mathbb{CF}_{n}^{+}$$ . All these results are extended to the generalized Cuntz algebra $$\cal{O} (x_{i=1}^{n} G_{i}^{+})$$ , where G i + are the positive cones ofdiscrete subgroups G i + of the real line $$\mathbb{R}$$ . Moreover, we characterize the orbits of Hilbert modules over the quotient algebra $$\mathbb{C}_{i=1}^{*n} G_{i}^{+}/J$$ , where J is an arbitrary two-sided ideal ofthe free semigroup algebra $$\mathbb{C}_{i=1}^{*n} G_{i}^{+}$$ .

Journal

PositivitySpringer Journals

Published: Sep 11, 2004

References

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