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

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
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### 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.
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