Hostname: page-component-6bf8c574d5-n2sc8 Total loading time: 0 Render date: 2025-03-07T10:18:01.578Z Has data issue: false hasContentIssue false
  • English
  • Français

Representability of invariant positive sesquilinear forms on partial *-algebras

Published online by Cambridge University Press:  24 October 2008

J.-P. Antoine  and
A. Inoue
J.-P. Antoine
Affiliation:
Institut de Physique Théorique, Université Catholique de Louvain, B-1348-Louvain-la-Neuve, Belgique
A. Inoue
Affiliation:
Department of Applied Mathematics, Fukuoka University, Fukuoka, Japan
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider invariant positive sesquilinear forms on a (partial) *-algebra A without unit. First we investigate the relationship between extendability and representability for such a form ø; in particular we discuss under which conditions the two concepts are equivalent. Then we introduce the notions of weak representability and strict unrepresentability, and we show that every fully invariant positive sesquilinear form on A × A is uniquely decomposed into a weakly representable part and a strictly unrepresentable part.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 108 , Issue 2 , September 1990 , pp. 337 - 353
Copyright
Copyright © Cambridge Philosophical Society 1990

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Antoine, J.-P. and Karwowski, W.. Partial *-algebras of closable operators in Hilbert space. Publ. Res. Inst. Math. Sci. 21 (1985), 205–236.CrossRefGoogle Scholar
[2]Antoine, J.-P., Mathot, F. and Trapani, C.. Partial *-algebras of closed operators and their commutants I, II. Ann. Inst. H. Poincare 46 (1987), 299–324, 325–351.Google Scholar
[3]Antoine, J.-P., Inoue, A. and Trapani, C.. Partial *-algebras of closable operators. I. General theory. The abelian case. Publ. Res. Inst. Math. Sci. 26 (1990), 359–395. II. States and representations of partial *-algebras. Preprint UCL-LPT-88–26.CrossRefGoogle Scholar
[4]Antoine, J.-P. and Inoue, A.. Strongly cyclic vectors for partial -algebras. Math. Nachr. (To appear.)Google Scholar
[5]Bhatt, S. J.. Representability of positive functionals on abstract star algebras without identity with applications to locally convex *-algebras. Yokohama Math. J. 29 (1981), 7–16.Google Scholar
[6]Gudder, S. P.. A Radon–Nikodym theorem for *-algebras. Pacific J. Math. 80 (1979), 141–149.CrossRefGoogle Scholar
[7]Gudder, S. P. and Scruggs, W.. Unbounded representations of *-algebras. Pacific J. Math. 70 (1977), 369–382.CrossRefGoogle Scholar
[8]Inoue, A.. A Radon–Nikodym theorem for positive linear functionals. J. Operator Theory 10 (1983), 77–86.Google Scholar
[9]Inoue, A.. On regularity of positive linear functionals. Japan. J. Math. (N.S.) 9 (1983), 247–275.CrossRefGoogle Scholar
[10]Inoue, A.. An unbounded generalization of the Tomita–Takesaki theory I, II. Publ. Res. Inst. Math. Sci. 22 (1986), 725–765CrossRefGoogle Scholar
Inoue, A.. An unbounded generalization of the Tomita–Takesaki theory I, II. Publ. Res. Inst. Math. Sci. 23 (1987), 673–726.CrossRefGoogle Scholar
[11]Lassner, G.. Topological algebras of operators. Rep. Math. Phys. 3 (1972), 279–293.CrossRefGoogle Scholar
[12]Lassner, G.. Algebras of unbounded operators and quantum dynamics. Physica 124 (1984), 471–480.CrossRefGoogle Scholar
[13]Nussbaum, A. E.. On the integral representation of positive linear functionals. Trans. Amer. Math. Soc. 128 (1967), 460–473.CrossRefGoogle Scholar
[14]Powell, J. D.. Representations of locally convex *-algebras. Proc. Amer. Math. Soc. 44 (1974), 341–346.Google Scholar
[15]Powers, R. T.. Self-adjoint algebras of unbounded operators. Comm. Math. Phys. 21 (1971), 85–124.CrossRefGoogle Scholar
[16]Rickart, C. E.. General theory of Banach algebras (Van Nostrand, 1960).Google Scholar
[17]Tomita, M.. Foundation of noncommutative Fourier analysis. Japan–U.S. seminar on C*-algebras and applications to physics, Kyoto (1974).Google Scholar