## Rocky Mountain Journal of Mathematics

### Construction of globalizations for partial actions on rings, algebras, C$^*$-algebras and Hilbert bimodules

Damián Ferraro

#### Abstract

We give a necessary condition for a partial action on a ring to have globalization. We also show that every partial action on a C$^*$-algebra satisfying this condition admits a globalization and, finally, we use the linking algebra of a Hilbert module to translate our condition to the realm of partial actions on Hilbert modules.

#### Article information

Source
Rocky Mountain J. Math., Volume 48, Number 1 (2018), 181-217.

Dates
First available in Project Euclid: 28 April 2018

https://projecteuclid.org/euclid.rmjm/1524880887

Digital Object Identifier
doi:10.1216/RMJ-2018-48-1-181

Mathematical Reviews number (MathSciNet)
MR3795739

Zentralblatt MATH identifier
06866706

#### Citation

Ferraro, Damián. Construction of globalizations for partial actions on rings, algebras, C$^*$-algebras and Hilbert bimodules. Rocky Mountain J. Math. 48 (2018), no. 1, 181--217. doi:10.1216/RMJ-2018-48-1-181. https://projecteuclid.org/euclid.rmjm/1524880887

#### References

• Fernando Abadie, Enveloping actions and Takai duality for partial actions, J. Funct. Anal. 197 (2003), 14–67.
• Fernando Abadie and Laura Martí Pérez, On the amenability of partial and enveloping actions, Proc. Amer. Math. Soc. 137 (2009), 3689–3693.
• Wagner Cortes and Miguel Ferrero, Globalization of partial actions on semiprime rings, Contemp. Math. 499 (2009), 27.
• John Dauns, Multiplier rings and primitive ideals, Trans. Amer. Math. Soc. 145 (1969), 125–158.
• M. Dokuchaev and R. Exel, Associativity of crossed products by partial actions, enveloping actions and partial representations, Trans. Amer. Math. Soc. 357 (2005), 1931–1952.
• Michael Dokuchaev, Ángel Del Río and Juan Simón, Globalizations of partial actions on nonunital rings, Proc. Amer. Math. Soc. 135 (2007), 343–352.
• Ruy Exel, Circle actions on C$^*$-algebras, partial automorphisms, and a generalized Pimsner-Voiculescu exact sequence, J. Funct. Anal. 122 (1994), 361–401.
• ––––, Twisted partial actions: A classification of regular C$^*$-algebraic bundles, Proc. Lond. Math. Soc. 74 (1997), 417–443.
• ––––, Partial dynamical systems Fell bundles and applications, AMS Math. Surv. Mono. 224 (2017), 321 pages.
• Kevin McClanahan, $K$-theory for partial crossed products by discrete groups, J. Funct. Anal. 130 (1995), 77–117.
• G.K. Pedersen, C$^*$-algebras and their automorphism groups, Academic Press, London, 1979.
• Marc A. Rieffel, Morita equivalence for operator algebras, in Operator algebras and applications, Part I, American Mathematical Society, Providence, RI, 1982.
• Heinrich Zettl, A characterization of ternary rings of operators, Adv. Math. 48 (1983), 117–143.