The reduced partial crossed product has the Haagerup property if and only if the C*-algebra and the partial action both have the Haagerup property.
A Hilbert module approach to the Haagerup property.Integral Equations Operator Theory, 73(3):431–454, 2012
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On the Haagerup property for partial crossed products
The reduced partial crossed product has the Haagerup property if and only if the C*-algebra and the partial action both have the Haagerup property.