Journal of the Mathematical Society of Japan

Structure of locally convex quasi $C^*$-algebras

Fabio BAGARELLO,Maria FRAGOULOPOULOU,Atsushi INOUE, and Camillo TRAPANI

Full-text: Open access

Abstract

The completion of a (normed) $C^*$-algebra $\A_0[\| \cdot \|_0]$  with respect to a locally convex topology $\tau$  on $\A_0$ that makes the multiplication of $\A_0$  separately continuous is, in general, a quasi $*$-algebra, and not a locally convex $*$-algebra [10], [15]. In this way, one is led to consideration of locally convex quasi $C^*$-algebras, which generalize $C^*$-algebras in the context of quasi $*$-algebras. Examples are given and the structure of these relatives of $C^*$-algebras is investigated.

Article information

Source
J. Math. Soc. Japan Volume 60, Number 2 (2008), 511-549.

Dates
First available: 30 May 2008

Permanent link to this document
http://projecteuclid.org/euclid.jmsj/1212156661

Digital Object Identifier
doi:10.2969/jmsj/06020511

Mathematical Reviews number (MathSciNet)
MR2421987

Zentralblatt MATH identifier
1145.47059

Subjects
Primary: 47L60: Algebras of unbounded operators; partial algebras of operators
Secondary: 46K10: Representations of topological algebras with involution 46K70: Nonassociative topological algebras with an involution [See also 46H70, 46L70] 46L05: General theory of $C^*$-algebras

Keywords
quasi *-algebras quasi-positivity locally convex quasi $C^*$-algebras unbounded *-representations

Citation

BAGARELLO, Fabio; FRAGOULOPOULOU, Maria; INOUE, Atsushi; TRAPANI, Camillo. Structure of locally convex quasi C * -algebras. Journal of the Mathematical Society of Japan 60 (2008), no. 2, 511--549. doi:10.2969/jmsj/06020511. http://projecteuclid.org/euclid.jmsj/1212156661.


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References

  • G. R. Allan, A spectral theory for locally convex algebras, Proc. London Math. Soc. (3), 15 (1965), 399–421.
  • G. R. Allan, On a class of locally convex algebras, Proc. London Math. Soc. (3), 17 (1967), 91–114.
  • J.-P. Antoine, F. Bagarello and C. Trapani, Topological partial $*$-algebras: Basic properties and examples, Rev. Math. Phys., 11 (1999), 267–302.
  • J.-P. Antoine, A. Inoue and C. Trapani, Partial $*$-algebras of closable operators, I, The basic theory and the abelian case, Publ. RIMS, Kyoto Univ., 26 (1990), 359–395; II, States and $*$-representations of partial $*$-algebras, ibid., 27 (1991), 399–340.
  • J.-P. Antoine, A. Inoue and C. Trapani, Partial $*$-algebras of closable operators, Rev. Math. Phys., 8 (1996), 1–42.
  • J.-P. Antoine, A. Inoue and C. Trapani, Partial $*$-Algebras and their Operator Realizations, Math. Appl., 553, Kluwer Academic, Dordrecht, 2003.
  • J.-P. Antoine and W. Karwowski, Partial $*$-algebras of closed operators in Quantum Theory of Particles and Fields, (eds. B. Jancewitz and J. Lukerski), World Scientific, Singapore, 1983, pp.,13–30.
  • J.-P. Antoine and W. Karwowski, Partial $*$-algebras of closed linear operators in Hilbert space, Publ. RIMS, Kyoto Univ., 21 (1985), 205–236; Add./Err. ibid., 22 (1986), 507–511.
  • F. Bagarello, Algebras of unbounded operators and physical applications, Rev. Math. Phys., 19 (2007), 231–272.
  • F. Bagarello, M. Fragoulopoulou, A. Inoue and C. Trapani, The completion of a $C^*$-algebra with a locally convex topology, J. Operator Theory, 56 (2006), 357–376.
  • F. Bagarello, A. Inoue and C. Trapani, Unbounded $C^*$-seminorms and $*$-representations of partial $*$-algebras, Z. Anal. Anwend., 20 (2001), 1–20.
  • F. Bagarello and C. Trapani, States and representations of $CQ^*$-algebras, Ann. Inst. H. Poincaré, 61 (1994), 103–133.
  • F. Bagarello and C. Trapani, $CQ^*$-algebras: Structure properties, Publ. RIMS, Kyoto Univ., 32 (1996), 85–116.
  • P. G. Dixon, Generalized $B^*$-algebras, Proc. London Math. Soc., 21 (1970), 693–715.
  • M. Fragoulopoulou, A. Inoue and K.-D. Kürsten, On the completion of a $C^*$-normed algebra under a locally convex algebra topology, Contemporary Math., 427 (2007), 89–95.
  • R. Haag and D. Kastler, An algebraic approach to quantum field theory, J. Math. Phys., 5 (1964), 848–861.
  • A. Inoue, Tomita-Takesaki Theory in Algebras of Unbounded Operators, Lecture Notes in Math., 1699, Springer-Verlag, 1998.
  • G. Lassner, Topological algebras and their applications in quantum statistics, Wiss. Z. KMU-Leipzig, Math. Naturwiss. R., 30 (1981), 572–595.
  • G. Lassner, Algebras of unbounded operators and quantum dynamics, Phys. A, 124 (1984), 471–480.
  • K. Schmüdgen, Unbounded Operator Algebras and Representation Theory, Birkhäuser-Verlag, Basel, 1990.
  • C. Trapani, States and derivations on quasi $*$-algebras, J. Math. Phys., 29 (1988), 1885–1890.
  • C. Trapani, Quasi $*$-algebras of operators and their applications, Rev. Math. Phys., 7 (1995), 1303–1332.
  • C. Trapani, Bounded elements and spectrum in Banach quasi $*$-algebras, Studia Math., 172 (2006), 249–273.