Isometry Groups Of Twisted Reduced Group C*-Algebras

An isometry of a unital C*-algebra with respect to a spectral triple is a *-automorphism of the C*-algebra given by the conjugation by a unitary operator which commutes with the Dirac operator. We give a semidirect product topological characterization on the isometry group of a twisted reduced group C*-algebra of a discrete group with respect to the standard spectral triple induced by a length function on the group. We prove that this isometry group is compact in the point-norm topology, and in particular, for a finitely generated discrete group, this isometry group is a compact Lie group in the point-norm topology. We also extend this result to a unital C*-algebra with a filtration, and prove that its isometry group is a compact topological group in the point-norm topology.