Dixmier trace for Toeplitz operators on symmetric domains

For Toeplitz operators on bounded symmetric domains of higher rank, there is no obvious way to define the Dixmier trace within the Toeplitz C⁎-algebra, since commutators are in general not compact. In this paper we solve this problem by constructing a Hilbert quotient module, corresponding to partitions of length 1, which leads to commutators in the Macaev class Ln,∞. We also obtain an explicit formula for the Dixmier trace of such commutators in terms of the underlying boundary geometry, which involves a new type of flag manifold defined in Jordan theoretic terms.