Abstract
This survey aims to highlight some of the consequences that representable (and continuous) functionals have in the framework of Banach quasi *-algebras. In particular, we look at the link between the notions of *-semisimplicity and full representability in which representable functionals are involved. Then, we emphasize their essential role in studying *-derivations and representability properties for the tensor product of Hilbert quasi *-algebras, a special class of Banach quasi *-algebras.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
M.S. Adamo, The interplay between representable functionals and derivations on Banach quasi *-algebras, in Proceedings of the International Conference on Topological Algebras and Their Applications – ICTAA 2018, ed. by M. Abel. Mathematics Studies (Tartu), 7 (Estonian Mathematical Society, Tartu, 2018), pp. 48–59
M.S. Adamo, About tensor product of Hilbert quasi *-algebras and their representability, (accepted for publication in the Proceedings of OT27, Theta 2020)
M.S. Adamo, M. Fragoulopoulou, Tensor products of normed and Banach quasi *-algebras. J. Math. Anal. Appl. 490, 2 (2020)
M.S. Adamo, C. Trapani, Representable and continuous functionals on a Banach quasi ∗-algebra. Mediterr. J. Math. 14, 157 (2017)
M.S. Adamo, C. Trapani, Unbounded derivations and *-automorphisms groups of Banach quasi *-algebras. Ann. Mat. Pura Appl. (4) 198, 1711–1729 (2019)
J.-P. Antoine, A. Inoue, C. Trapani, O*-dynamical systems and *-derivations of unbounded operator algebras. Math. Nachr. 204, 5–28 (1999)
J.-P. Antoine, A. Inoue, C. Trapani, Partial *-Algebras and Their Operator Realizations (Kluwer Academic, Dordrecht, 2003)
F. Bagarello, C. Trapani, CQ∗-algebras: structure properties. Publ. RIMS Kyoto Univ. 32, 85–116 (1996)
F. Bagarello, C. Trapani, The Heisenberg dynamics of spin systems: a quasi *-algebra approach. J. Math. Phys. 37, 4219–4234 (1996)
F. Bagarello, C. Trapani, L p-spaces as quasi *-algebras. J. Math. Anal. Appl. 197, 810–824 (1996)
O. Bratteli, D.W. Robinson, Unbounded derivations of C*-algebras. Commun. Math. Phys. 42, 253–268 (1975)
L. Chambadal, J.L. Ovaert, Algèbre Linéaire et Algèbre Tensorielle (Dunod Université, Paris, 1968)
A. Defant, K. Floret, Tensor Norms and Operator Ideals (North-Holland, Amsterdam, 1993)
M. Fragoulopoulou, Topological Algebras with Involution (North-Holland, Amsterdam, 2005)
M. Fragoulopoulou, C. Trapani, S. Triolo, Locally convex quasi *-algebras with sufficiently many *-representations. J. Math. Anal. Appl. 388, 1180–1193 (2012)
M. Fragoulopoulou, A. Inoue, M. Weigt, Tensor products of generalized B ∗-algebras. J. Math. Anal. Appl. 420, 1787–1802 (2014)
M. Fragoulopoulou, A. Inoue, M. Weigt, Tensor products of unbounded operator algebras. Rocky Mt. 44, 895–912 (2014)
W.-D. Heinrichs, Topological tensor products of unbounded operator algebras on Frèchet domains. Publ. RIMS Kyoto Univ. 33, 241–255 (1997)
A.Ya. Helemskii, Lectures and Exercises on Functional Analysis (American Mathematical Society, Providence, 2006)
E. Hille, R. Phillips, Functional Analysis and Semi-groups (American Mathematical Society, Providence, 1996)
G. Lassner, Topological algebras and their applications in quantum statistics. Wiss. Z. KMU-Leipzig, Math. Naturwiss. R. 30, 572–595 (1981)
G. Lassner, Algebras of unbounded operators and quantum dynamics. Physica A 124, 471–480 (1984)
K.B. Laursen, Tensor products of Banach algebras with involution. Trans. Am. Math. Soc. 136, 467–487 (1969)
A. Mallios, Topological Algebras. Selected Topics (North-Holland, Amsterdam, 1986)
G.J. Murphy, C*-Algebras and Operator Theory (Academic, Boston, 2014)
A.M. Sinclair, Automatic Continuity of Linear Operators. London Mathematical Society, Lecture Notes Series 21 (Cambridge University Press, Cambridge, 1976)
M. Takesaki, Theory of Operator Algebras I (Springer, New York, 1979)
C. Trapani, Quasi *-algebras of operators and their applications. Rev. Math. Phys. 7, 1303–1332 (1995)
C. Trapani, Bounded elements and spectrum in Banach quasi *-algebras. Stud. Math. 172, 249–273 (2006)
C. Trapani, *-Representations, seminorms and structure properties of normed quasi *-algebras. Stud. Math. 186, 47–75 (2008)
M. Weigt, Derivations of τ-measurable operators. Oper. Theory Adv. Appl. 195, 273–286 (2009)
M. Weigt, I. Zarakas, Unbounded derivations of GB*-algebras. Oper. Theory Adv. Appl. 247, 69–82 (2015)
M. Weigt, I. Zarakas, Derivations of Fréchet nuclear GB*-algebras. Bull. Aust. Math. Soc. 92, 290–301 (2015)
Acknowledgements
The author is grateful to the organizers of the International Workshop on Operator Theory and its Applications 2019, especially to the Organizers of the section entitled “Linear Operators and Function Spaces”, for this interesting and delightful conference and the Instituto Superior Técnico of Lisbon for its hospitality. The author was financially supported by the ERC Advanced Grant no. 669240 QUEST “Quantum Algebraic Structures and Models”.
The author wishes to thank the anonymous referees for their useful suggestions that improved the presentation of this manuscript.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Adamo, M.S. (2021). On Some Applications of Representable and Continuous Functionals of Banach Quasi ∗-Algebras. In: Bastos, M.A., Castro, L., Karlovich, A.Y. (eds) Operator Theory, Functional Analysis and Applications. Operator Theory: Advances and Applications, vol 282. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-51945-2_2
Download citation
DOI: https://doi.org/10.1007/978-3-030-51945-2_2
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-51944-5
Online ISBN: 978-3-030-51945-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)