*-Regularity of exponential Lie groups

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Inventiones math. 56, 231-238 (1980)

mathematicae ~ by Springer-Verlag 1980

,-Regularity of Exponential Lie Groups J. Boidol UniversitSt Bielefeld, Fakult/it fiir Mathematik, UniversitMsstrasse 1, D-4800 Bielefeld I, Federal Republic of Germany

Let ~r be a (*-semisimple) Banach-*-algebra and let C*(,~) be its C*-hull. Let Prim C*(.~t') be the primitive ideal space of C*(,4) and P r i m , , ~ the space of kernels of topological irreducible *-representations of ~_J in Hilbert space, both equipped with the Jacobson topology. Then we have a canonical mapping

7/: Prim C * ( , ~ ) ~ Prim,,~' ..r

J c~'

which is continuous and surjective. We want to call a (,-semisimple) Banach-*algebra ~J *-regular, if 7j is even a homeomorphism. Similarly we call a locally compact group G *-regular if its group algebra L~(G) is ,-regular. In [2] the problem was posed to determine the class [7j] of all *-regular locally compact groups. The main results were (A) I f G is *-regular, then it is amenable. (B) All groups G with polynomial growth are *-regular. (C) All semidirect products G =-A ~ N with abelian separable A and N such that IV/A is a To-space are *-regular. Furthermore it was shown that all connected simply connected solvable (real) Lie groups G with dirn G < 4, G 4=G4, 9(0) = exp 94.9 (0) in the notation of [1, p. 180] are *-regular. Then in [3] it was shown following a suggestion of D. Poguntke that G=G4,9(O ) is not *-regular. Thus there is exactly one connected simply connected solvable Lie group G with dim G _k

m I (X) = m1(X ) + (ad X) J mo(X ) = (ad X) .i re(J). Since ad X is nilpotent m o d u l o p there exists k o >_ k such that for all j>= k o

(adX)Jm(f)~_p,

hence m l ( X ) = [ X , mdX)]c_[p,p ].

We conclude that

(m1(X)) = m(./)~' _ [p, p] c_ k e r n f which contradicts our assumption. N o w let X e g ( f ) such that a d X is not nilpotent m o d u l o p. Let i o be the largest index such that ad X operates with nontrivial eigenvalue e on [io+l/tio. We get that tr ad X - tr (ad X)1% = tr (ad X)I%+ 1 - tr (ad X)I% 4=0, since Reck4=0 also, by the fact that .q is exponential. Thus with h=[io the proposition is proved. Proposition 4. The set {m(f)~'~'[feg *} is a finite set oJ ideals (~f g.

Proof. re(f) is an ideal of 9 and m ( f y a characteristic ideal of re(f), hence the re(f) ~' are ideals of 9 for all f e g * . We prove that even the set M TM

= {a~l[g, 9 ] - a