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abelian von Neumann approximate identity Borel measure class Borel set Borel subset C*-algebra CL C*-ALGEBRAS Edward C*-pseudo-norm canonical measure central decomposition closed sets contains the ideal converges strongly countable open basis decomposition algebra DECOMPOSITION THEORY defined dense direct integral Dixmier Effros Ernest exists a Borel exists a measurable factor representations finite Borel measure Hausdorff space hence Hilbert space homogeneous representations hull-kernel topology ideal center ideal equivalent ideal projections induce isometry JLxd^(x k(supp kernel Lx Lemma locally compact groups LXEX map tt measurable cross-section measurable map metrically regular non-zero projection open set orthogonal measures partial supports prCL primitive ideal space projection ideals projection-valued measure Proof proper separable representations QR(E quasi-equivalence classes quotient map relative Borel structure repre Representations of C*-algebras sentation separable C*-algebra standard measure space strongly disjoint subalgebra Taking a subsequence Theorem 1.8 Theory for Representations topological space weak topology weakly