Uniqueness of the multiplicative cyclotomic trace

Making use of the theory of noncommutative motives, we characterize the topological Dennis trace map as the unique multiplicative natural transformation from algebraic K-theory to topological Hochschild homology (THH) and the cyclotomic trace map as the unique multiplicative lift through topological cyclic homology (TC). Moreover, we prove that the space of all multiplicative structures on algebraic K-theory is contractible.We also show that the algebraic K-theory functor from small stable ∞-categories to spectra is lax symmetric monoidal, which in particular implies that En ring spectra give rise to En−1 ring algebraic K-theory spectra. Along the way, we develop a “multiplicative Morita theory”, establishing a symmetric monoidal equivalence between the ∞-category of small idempotent-complete stable ∞-categories and the Morita localization of the ∞-category of spectral categories.