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

Fabio BAGARELLO, Maria FRAGOULOPOULOU, Atsushi INOUE, and Camillo TRAPANI
Source: J. Math. Soc. Japan Volume 60, Number 2 (2008), 511-549.

#### 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.

First Page:
Primary Subjects: 47L60
Secondary Subjects: 46K10, 46K70, 46L05
Full-text: Open access

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

### References

G. R. Allan, A spectral theory for locally convex algebras, Proc. London Math. Soc. (3), 15 (1965), 399–421.
Mathematical Reviews (MathSciNet): MR176344
Digital Object Identifier: doi:10.1112/plms/s3-15.1.399
G. R. Allan, On a class of locally convex algebras, Proc. London Math. Soc. (3), 17 (1967), 91–114.
Mathematical Reviews (MathSciNet): MR205102
Zentralblatt MATH: 0147.33503
Digital Object Identifier: doi:10.1112/plms/s3-17.1.91
J.-P. Antoine, F. Bagarello and C. Trapani, Topological partial $*$-algebras: Basic properties and examples, Rev. Math. Phys., 11 (1999), 267–302.
Mathematical Reviews (MathSciNet): MR1688375
Digital Object Identifier: doi:10.1142/S0129055X99000106
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.
Mathematical Reviews (MathSciNet): MR1047417
Digital Object Identifier: doi:10.2977/prims/1195171084
J.-P. Antoine, A. Inoue and C. Trapani, Partial $*$-algebras of closable operators, Rev. Math. Phys., 8 (1996), 1–42.
Mathematical Reviews (MathSciNet): MR1372514
Digital Object Identifier: doi:10.1142/S0129055X96000020
J.-P. Antoine, A. Inoue and C. Trapani, Partial $*$-Algebras and their Operator Realizations, Math. Appl., 553, Kluwer Academic, Dordrecht, 2003.
Mathematical Reviews (MathSciNet): MR1947892
Zentralblatt MATH: 1023.46004
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.
Mathematical Reviews (MathSciNet): MR772779
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.
Mathematical Reviews (MathSciNet): MR780895
Digital Object Identifier: doi:10.2977/prims/1195179844
F. Bagarello, Algebras of unbounded operators and physical applications, Rev. Math. Phys., 19 (2007), 231–272.
Mathematical Reviews (MathSciNet): MR2316534
Digital Object Identifier: doi:10.1142/S0129055X07002961
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.
Mathematical Reviews (MathSciNet): MR2282687
Zentralblatt MATH: 1115.46044
F. Bagarello, A. Inoue and C. Trapani, Unbounded $C^*$-seminorms and $*$-representations of partial $*$-algebras, Z. Anal. Anwend., 20 (2001), 1–20.
Mathematical Reviews (MathSciNet): MR1846603
F. Bagarello and C. Trapani, States and representations of $CQ^*$-algebras, Ann. Inst. H. Poincaré, 61 (1994), 103–133.
Mathematical Reviews (MathSciNet): MR1303188
Zentralblatt MATH: 0820.46053
F. Bagarello and C. Trapani, $CQ^*$-algebras: Structure properties, Publ. RIMS, Kyoto Univ., 32 (1996), 85–116.
Mathematical Reviews (MathSciNet): MR1384752
Digital Object Identifier: doi:10.2977/prims/1195163181
P. G. Dixon, Generalized $B^*$-algebras, Proc. London Math. Soc., 21 (1970), 693–715.
Mathematical Reviews (MathSciNet): MR278079
Zentralblatt MATH: 0205.42604
Digital Object Identifier: doi:10.1112/plms/s3-21.4.693
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.
Mathematical Reviews (MathSciNet): MR2326354
Zentralblatt MATH: 1123.46039
R. Haag and D. Kastler, An algebraic approach to quantum field theory, J. Math. Phys., 5 (1964), 848–861.
Mathematical Reviews (MathSciNet): MR165864
Zentralblatt MATH: 0139.46003
Digital Object Identifier: doi:10.1063/1.1704187
A. Inoue, Tomita-Takesaki Theory in Algebras of Unbounded Operators, Lecture Notes in Math., 1699, Springer-Verlag, 1998.
Mathematical Reviews (MathSciNet): MR1725388
G. Lassner, Topological algebras and their applications in quantum statistics, Wiss. Z. KMU-Leipzig, Math. Naturwiss. R., 30 (1981), 572–595.
Mathematical Reviews (MathSciNet): MR655241
G. Lassner, Algebras of unbounded operators and quantum dynamics, Phys. A, 124 (1984), 471–480.
Mathematical Reviews (MathSciNet): MR759198
Digital Object Identifier: doi:10.1016/0378-4371(84)90263-2
K. Schmüdgen, Unbounded Operator Algebras and Representation Theory, Birkhäuser-Verlag, Basel, 1990.
Mathematical Reviews (MathSciNet): MR1056697
C. Trapani, States and derivations on quasi $*$-algebras, J. Math. Phys., 29 (1988), 1885–1890.
Mathematical Reviews (MathSciNet): MR955193
Zentralblatt MATH: 0649.47037
Digital Object Identifier: doi:10.1063/1.527840
C. Trapani, Quasi $*$-algebras of operators and their applications, Rev. Math. Phys., 7 (1995), 1303–1332.
Mathematical Reviews (MathSciNet): MR1369745
Digital Object Identifier: doi:10.1142/S0129055X95000475
C. Trapani, Bounded elements and spectrum in Banach quasi $*$-algebras, Studia Math., 172 (2006), 249–273.
Mathematical Reviews (MathSciNet): MR2204560
Zentralblatt MATH: 1101.46035
Digital Object Identifier: doi:10.4064/sm172-3-4