The main topic of the thesis is approximation properties for locally compact groups with applications to operator algebras. In order to study the relationship between weak amenability and the Haagerup property, the weak Haagerup property and the weak Haagerup constant are introduced. The weak Haagerup property is (strictly) weaker than both weak amenability and the Haagerup property.

We establish a relation between the weak Haagerup property and semigroups of Herz-Schur multipliers. For free groups, we prove that a generator of a semigroup of radial, contractive Herz-Schur multipliers is linearly bounded by the word length function.
In joint work with Haagerup, we show that a connected simple Lie group has the weak Haagerup property if and only if its real rank is at most one. The result coincides with the characterization of connected simple Lie groups which are weakly amenable. Moreover, the weak Haagerup constants of all connected simple Lie groups are determined.

In order to determine the weak Haagerup constants of the rank one simple Lie groups, knowledge about the Fourier algebras of their minimal parabolic subgroups is needed. We prove that for these minimal parabolic subgroups, the Fourier algebra coincides with the elements of the Fourier-Stieltjes algebra vanishing at innity.

In joint work with Li, we characterize the connected simple Lie groups all of whose countable subgroups have the weak Haagerup property. These groups are precisely the connected simple Lie groups locally isomorphic to either SO(3), SL(2;R) or SL(2; C)-