Definitions
Let be a second countable locally compact group. The following properties are all equivalent, and any of them may be taken to be definitions of the Haagerup property:
- There is a proper continuous conditionally negative definite function .
- has the Haagerup approximation property, also known as Property : there is a sequence of normalized continuous positive-definite functions which vanish at infinity on and converge to 1 uniformly on compact subsets of .
- There is a strongly continuous unitary representation of which weakly contains the trivial representation and whose matrix coefficients vanish at infinity on .
- There is a proper continuous affine isometric action of on a Hilbert space.
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