Abstract

We study the noncommutative geometry of algebras of Lipschitz continuous and Hölder continuous functions, where nonclassical and novel differential geometric invariants arise. Indeed, we introduce a new class of Hochschild and cyclic cohomology classes that pair nontrivially with higher algebraic $K$-theory yet vanish when restricted to the algebra of smooth functions. Said cohomology classes provide additional methods to extract numerical invariants from Connes–Karoubi’s relative sequence in $K$-theory.

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