On the Spectral Type of One-parameter Groups on Operator Algebras1

We show that two unitary representations of R of very different spectral type can give rise to the same one-parameter group of "-automorphisms of a C*-algebra. Arveson [1] has introduced a spectral theory for automorphism groups on C*-algebras. However, little seems to be known about the spectral type of such groups. The fact that two unitary representations of R of very different spectral properties can give rise to the same one-parameter group of automorphisms of a C*-algebra is not surprising as can be seen by the following simple example, due to the referee: Let H = /2(N) and let h G B(H) be the compact positive operator "multiplication by {l/«}"-i'• Let {r„}"_i be an enumeration of the rationals in [0, 1] and let k G B(H) be the positive operator "multiplication by {/„}"=i". Then, the operator (h ® 1) + (1 <8> k) on H ® H has spectrum equal to [0, 2] while h <8> 1 has spectrum equal to {1/'n}TM=x u {0}. However, the unitary representations of R on H ® H defined by: t _> e'<(A®l> = gllh ® j