TY - JOUR

T1 - Decomposable partial actions

AU - Abadie, Fernando

AU - Gardella, Eusebio

AU - Geffen, Shirly

N1 - Funding Information:
The second named author was partially supported by the Deutsche Forschungsgemeinschaft's ( DFG ) eigene Stelle, and by a Postdoctoral Research Fellowship from the Humboldt Foundation . The third named author was supported by a Kreitman Foundation Fellowship, a Minerva Fellowship Programme, and by an Israel Science Foundation grant no. 476/16 . The second and third named authors were partially supported by the DFG through SFB 878 and under Germany's Excellence Strategy – EXC 2044 – 390685587 , Mathematics Münster – Dynamics – Geometry – Structure , and by ERC Advanced Grant 834267 - AMAREC.
Publisher Copyright:
© 2021 Elsevier Inc.

PY - 2021/10/1

Y1 - 2021/10/1

N2 - We define the decomposition property for partial actions of discrete groups on C⁎-algebras. Decomposable partial systems appear naturally in practice, and many commonly occurring partial actions can be decomposed into partial actions with the decomposition property. For instance, any partial action of a finite group is an iterated extension of decomposable systems. Partial actions with the decomposition property are always globalizable and amenable, regardless of the acting group, and their globalization can be explicitly described in terms of certain global sub-systems. A direct computation of their crossed products is also carried out. We show that partial actions with the decomposition property behave in many ways like global actions of finite groups (even when the acting group is infinite), which makes their study particularly accessible. For example, there exists a canonical faithful conditional expectation onto the fixed point algebra, which is moreover a corner in the crossed product in a natural way. (Both of these facts are in general false for partial actions of finite groups.) As an application, we show that freeness of a topological partial action with the decomposition property is equivalent to its fixed point algebra being Morita equivalent to its crossed product. We also show by example that this fails for general partial actions of finite groups.

AB - We define the decomposition property for partial actions of discrete groups on C⁎-algebras. Decomposable partial systems appear naturally in practice, and many commonly occurring partial actions can be decomposed into partial actions with the decomposition property. For instance, any partial action of a finite group is an iterated extension of decomposable systems. Partial actions with the decomposition property are always globalizable and amenable, regardless of the acting group, and their globalization can be explicitly described in terms of certain global sub-systems. A direct computation of their crossed products is also carried out. We show that partial actions with the decomposition property behave in many ways like global actions of finite groups (even when the acting group is infinite), which makes their study particularly accessible. For example, there exists a canonical faithful conditional expectation onto the fixed point algebra, which is moreover a corner in the crossed product in a natural way. (Both of these facts are in general false for partial actions of finite groups.) As an application, we show that freeness of a topological partial action with the decomposition property is equivalent to its fixed point algebra being Morita equivalent to its crossed product. We also show by example that this fails for general partial actions of finite groups.

KW - C-algebras

KW - Crossed products

KW - Fixed point algebras

KW - Partial actions

UR - http://www.scopus.com/inward/record.url?scp=85108220237&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2021.109112

DO - 10.1016/j.jfa.2021.109112

M3 - Article

AN - SCOPUS:85108220237

SN - 0022-1236

VL - 281

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 7

M1 - 109112

ER -