Density Theorems for the Centre of a C * - Algebra

where 3 is the centre of 51 and cl denotes closure in the norm topology. When 5t is a von Neumann algebra this result is already known (see [7; Lemma 9, p. 255], where the method requires only trivial modification to deal with the case of a left ideal), and the proof in that situation involves the restriction of an operator in M to a subspace on which it can be inverted. The proof given here for the general case also depends on the " local inversion " of an element A e Jl, but this time in the sense that an element Be51 is found which is an inverse for A modulo every primitive ideal in a neighbourhood of a certain point in the primitive ideal space (for this process we are able to assume that 51 has an identity). As a consequence of (1.1), it is proved that if 51 is a quasicentral C*-algebra then the Pedersen ideal is contained in every dense ideal of 5X. A density question of a different nature concerns us in §4. We say that a C*algebra 51 with centre 3 n a s property (W) if the following holds (where the bar denotes closure in the relevant weak operator topology):