Von Neumann algebras | Operator algebras | Hilbert space

Weak trace-class operator

In mathematics, a weak trace class operator is a compact operator on a separable Hilbert space H with singular values the same order as the harmonic sequence.When the dimension of H is infinite, the ideal of weak trace-class operators is strictly larger than the ideal of trace class operators, and has fundamentally different properties. The usual operator trace on the trace-class operators does not extend to the weak trace class. Instead the ideal of weak trace-class operators admits an infinite number of linearly independent quasi-continuous traces, and it is the smallest two-sided ideal for which all traces on it are singular traces. Weak trace-class operators feature in the noncommutative geometry of French mathematician Alain Connes. (Wikipedia).

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Related pages

Trace class | Harmonic series (mathematics) | Compact operator | Singular trace | Singular value | Hilbert space | Spectral triple | Lp space | Calkin correspondence | Ideal (ring theory) | Quasinorm | Separable space | Noncommutative geometry | Dixmier trace