We introduce an abstract theory of the principal symbol mapping for pseudodifferential operators extending the results of Sukochev and Zanin (J Oper Theory, 2019) and providing a simple algebraic approach… Click to show full abstract
We introduce an abstract theory of the principal symbol mapping for pseudodifferential operators extending the results of Sukochev and Zanin (J Oper Theory, 2019) and providing a simple algebraic approach to the theory of pseudodifferential operators in settings important in noncommutative geometry. We provide a variant of Connes’ trace theorem which applies to certain noncommutative settings, with a minimum of technical preliminaries. Our approach allows us to consider operators with non-smooth symbols, and we demonstrate the power of our approach by extending Connes’ trace theorem to operators with non-smooth symbols in three examples: the Lie group $$\mathrm {SU}(2)$$SU(2), noncommutative tori and Moyal planes.
               
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