Banach partial \(^*\)-algebras: an overview. (English) Zbl 1420.46040

Antoine, J.-P.; Trapani, C.

Adv. Oper. Theory 4, No. 1, 71-98 (2019).

Summary: A Banach partial \(^*\)-algebra is a locally convex partial \(^*\)-algebra whose total space is a Banach space. A Banach partial \(^*\)-algebra is said to be of type (B) if it possesses a generating family of multiplier spaces that are also Banach spaces. We describe the basic properties of these objects and display a number of examples, namely, \(L^p\)-like function spaces and spaces of operators on Hilbert scales or lattices. Finally, we analyze the important cases of Banach quasi \(^*\)-algebras and \(CQ^*\)-algebras.

MSC:

46J10	Banach algebras of continuous functions, function algebras
47L60	Algebras of unbounded operators; partial algebras of operators

Keywords:

partial \(^*\)-algebra; Banach partial \(^*\)-algebra; \(CQ^*\)-algebra; partial inner product space; operators on Hilbert scale

References:

[1]	J.-P. Antoine, F. Bagarello, and C. Trapani, Topological partial \(*\)-algebras: Basic properties and examples, Rev. Math. Phys. 11 (1999), 267–302. · Zbl 0966.47048
[2]	J.-P. Antoine, A. Inoue, and C. Trapani, Partial \(*\)-algebras and their operator rRealizations, Kluwer, Dordrecht, 2002.
[3]	J.-P. Antoine and W. Karwowski, Interpolation theory and refinement of nested Hilbert spaces, J. Math. Phys. 22 (1981), 2489–2496. · Zbl 0494.46023
[4]	J.-P. Antoine and C. Trapani, Partial inner product spaces – Theory and applications, Lecture Notes in Mathematics, vol. 1986, Springer-Verlag, Berlin, Heidelberg, 2009. · Zbl 1195.46001
[5]	J.-P. Antoine and C. Trapani, A note on Banach partial \(*\)-algebras, Mediterr. J. Math. 3 (2006), 67–86.
[6]	J.-P. Antoine, C. Trapani, and F. Tschinke, Continuous \(*\)-homomorphisms of Banach partial \(*\)-algebras, Mediterr. J. Math. 4 (2007), 357–373. · Zbl 1178.46050
[7]	F. Bagarello, A. Inoue, and C. Trapani, Some classes of topological quasi \(*\)-algebras, Proc. Amer. Math. Soc. 129 (2001), 2973–2980. · Zbl 0979.46039
[8]	F. Bagarello and C. Trapani, \(CQ^*\)-algebras: Structure properties, Publ. RIMS, Kyoto Univ. 32 (1996), 85–116.
[9]	F. Bagarello and C. Trapani, \(L^p\)-spaces as quasi \(*\)-algebras, J. Math. Anal. Appl. 197 (1996,) 810–824.
[10]	F. Bagarello and C. Trapani, States and representations of \(CQ^*\)-algebras, Ann. Inst. H. Poincaré 61 (1994), 103–133.
[11]	J. Bergh and J. Löfström, Interpolation spaces, Springer-Verlag, Berlin, 1976.
[12]	G. Birkhoff, Lattice theory, 3rd ed., Amer. Math. Soc., Coll. Publ., Providence, RI., 1966. · JFM 66.0100.04
[13]	J. J. F. Fournier and J. Stewart, Amalgams of \(L^p\) and \(ℓ^q\), Bull. Amer. Math. Soc. 13 (1985), 1–21.
[14]	G. G. Gould, On a class of integration spaces, J. London Math. Soc. 34 (1959), 161–172. · Zbl 0099.09503
[15]	R. Haag and D. Kastler, An algebraic approach to quantum field theorem, J. Math. Phys. 5 (1964), 848–861. · Zbl 0139.46003
[16]	T. Kato, Perturbation theory for linear operators, Springer-Verlag, Berlin, 1976. · Zbl 0342.47009
[17]	G. Köthe, Topological Vector Spaces, I and II, Springer-Verlag, Berlin, 1966, 1979.
[18]	G. Lassner, Algebras of unbounded operators and quantum dynamics, Physica A 124 (1984), 471–480. · Zbl 0599.47072
[19]	G. Lassner, Topological algebras and their applications in quantum statistics, Wiss. Z. KMU-Leipzig, Math.-Naturwiss. R. 30 (1981), 572–595. · Zbl 0483.47027
[20]	E. Nelson, Note on non-commutative integration, J. Funct. Anal. 15 (1974), 103–116 · Zbl 0292.46030
[21]	H. H. Schaefer, Topological vector spaces, Springer-Verlag, Berlin, 1971. · Zbl 0212.14001
[22]	I. E. Segal, A noncommutative extension of abstract integration, Ann. Math. 57 (1953), 401–457. · Zbl 0051.34201
[23]	S. Str\v atil\v a and L. Zsidó, Lectures on von Neumann algebras, Editura Academiei, Bucharest and Abacus Press, Tunbridge Wells, Kent, 1979.
[24]	C. Trapani, Bounded elements and spectrum in Banach quasi \(*\)-algebras, Studia Math. 172 (2006), 249–273. · Zbl 1101.46035
[25]	C. Trapani, Quasi \(*\)-algebras of operators and their applications, Reviews Math. Phys. 7 (1995), 1303–1332. · Zbl 0839.46074
[26]	C. Trapani, States and derivations on quasi \(*\)-algebras, J. Math. Phys. 29 (1988), 1885–1890. · Zbl 0649.47037
[27]	C. Trapani and M. Fragoulopoulou, Locally convex quasi \(*\)-algebras and their representations, 2018 (in preparation).
[28]	A. C. Zaanen, Integration, North-Holland, Amsterdam, 1967.