Nonmeagre ideals from traces on corona algebras

We show that a tracial state on the corona of a separable nonunital C*-algebra gives a proper nonmeagre ideal in its multiplier algebra. Here nonmeagreness is the block condition introduced by Vignati and Yilmaz: every good approximate identity and every disjoint sequence of finite intervals admit an infinite block sum in the ideal. More generally, an ideal containing the original algebra is nonmeagre whenever its quotient has only countable families of pairwise orthogonal nonzero positive elements. Applying this criterion to Goodearl's simple AF example answers Question 4.13 of Vignati and Yilmaz affirmatively in ZFC. Goodearl's examples also yield a simple AF algebra with uncountably many maximal proper nonmeagre multiplier ideals.

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