Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-26T01:40:33.640Z Has data issue: false hasContentIssue false
  • English
  • Français

Positive-Defininte Functions on Free Semigroups

Published online by Cambridge University Press:  20 November 2018

Gelu Popescu
Gelu Popescu*
Affiliation:
Gelu Popescu Division of Mathematics and Statistics, The University of Texas at San Antonio San Antonio, TX 78249 U.S.A., e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

An extension of the Naimark dilation theorem [N], [SzF2] to positive-definite functions on free semigroups is given. This is used to extend the operatorial trigonometric moment problem [A] to a non-commutative setting and to characterize the classes 𝐶ρ (ρ > 0) of all n-tuples of operators that have a p-isometric dilation (see [SzF2] for the case n = 1). It is also shown that 𝐶ρ ⊂ 𝐶ρ′ and 𝐶ρ ≠ 𝐶ρ′ for 0 < ρ < ρ′ < ∞.

The von Neumann inequality [vN], [Po2] is extended to the classes 𝐶p. This is used to prove that any element in 𝐶ρ is simultaneously similar to an element in 𝐶1.

Keywords

47A2047A4547A57
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 48 , Issue 4 , 01 August 1996 , pp. 887 - 896
Copyright
Copyright © Canadian Mathematical Society 1996

References

[A] Akhiezer, N.I., The classical moment problem, Edinburg, Scotland: Olivier and Boyd (1965).Google Scholar
[Cu] Cuntz, J., Simple C*-algebras generated by isometries, Comm. Math., Phys. 57(1977), 173–185.Google Scholar
[E] Evans, D.E., On On, Publ. Res. Int. Math., Sci. 16(1980), 915–927.Google Scholar
[N] Naimark, M.A., Positive definite operator functions on a commutative group, Bulletin Acad. Sci., URSS 7(1943), 237–244.Google Scholar
[PS] Parthasarathy, K.R. and Schmidt, K., Positive-definite kernels, continuous tensor products and central limit theorems of probability theory, Lecture Notes in Mathematics, Springer, Berlin 272 1972.Google Scholar
[P] Paulsen, V.I., Every completely polynomially bounded operator is similar to a contraction, J. Funct., Anal. 55(1984), 1–17.Google Scholar
[Pol] Popescu, G., Isometric dilations for infinite sequences of noncommuting operators, Trans. Amer. Math., Soc. 316(1989), 523–536.Google Scholar
[Po2] Popescu, G., Von Neumann inequality for (B﹛H)n)x, Math., Scand. 68(1991), 292–304.Google Scholar
[Po3] Popescu, G., Functional calculus for noncommuting operators, Michigan Math., J. 42(1995), 345–356.Google Scholar
[Po4] Popescu, G., Non-commutative disc algebras and their representations, Proc. Amer. Math. Soc, to appear.Google Scholar
[S] Stinespring, W.F., Positive functions on C* -algebras, Proc. Amer. Math., Soc. 6(1955).Google Scholar
[SzFl] -Nagy, B.Sz.,Foias, C., Similitude des operators de classe Cp a des contractions, C. R. Acad. Sci. Paris, serie, A, 264(1967), 1063–1065.Google Scholar
[SzF2] -Nagy, B.Sz., Harmonic analysis on operators on Hilbert space, North—Holland, Amsterdam (1970).Google Scholar
[vN] von, J. Neumann, Eine Spectraltheorie fur allgemeine Operatoren eines unitaren Raumes, Math., Nachr. 4(1951), 258–281.Google Scholar