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.

我们定义离散组的部分动作的分解属性 $C^{⁎}$-代数。可分解的局部系统在实践中自然出现，许多常见的局部动作可以分解为具有分解性质的局部动作。例如，有限群的任何部分动作都是可分解系统的迭代扩展。

具有分解特性的部分动作总是可全球化和服从的，而不管动作组是什么，它们的全球化可以用某些全局子系统来明确描述。还进行了它们交叉乘积的直接计算。我们展示了具有分解特性的部分动作在许多方面表现得像有限群的全局动作（即使动作群是无限的），这使得他们的研究特别容易。例如，在不动点代数上存在一个规范的忠实条件期望，而且它是自然方式的交叉乘积中的一个角。（对于有限群的部分作用，这两个事实通常都是错误的。）作为一个应用，我们证明了具有分解性质的拓扑部分动作的自由度等价于其不动点代数是 Morita 等价于其交叉积。我们还通过例子表明这对于有限群的一般部分动作是失败的。