Abstract
We say that a locally compact groupG hasT 1 primitive ideal space if the groupC *-algebra,C *(G), has the property that every primitive ideal (i.e. kernel of an irreducible representation) is closed in the hull-kernel topology on the space of primitive ideals ofC *(G), denoted by PrimG. This means of course that every primitive ideal inC *(G) is maximal. Long agoDixmier proved that every connected nilpotent Lie group hasT 1 primitive ideal space. More recentlyPoguntke showed that discrete nilpotent groups haveT 1 primitive ideal space and a few month agoCarey andMoran proved the same property for second countable locally compact groups having a compactly generated open normal subgroup. In this note we combine the methods used in [3] with some ideas in [9] and show that for nilpotent locally compact groupsG, having a compactly generated open normal subgroup, closed prime ideals inC *(G) are always maximal which implies of course that PrimG isT 1.
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Ludwig, J. Prime ideals in the C*-algebra of a nilpotent group. Monatshefte für Mathematik 101, 159–165 (1986). https://doi.org/10.1007/BF01298928
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DOI: https://doi.org/10.1007/BF01298928