Uniqueness of the multiplicative cyclotomic trace

Making use of the theory of noncommutative motives we characterize the cyclotomic trace map as the unique multiplicative natural transformation from algebraic K-theory to topological Hochschild homology. 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 is lax symmetric monoidal (which implies that E_n-ring spectra give rise to E_{n-1} ring algebraic K-theory spectra). Along the way, we develop a "multiplicative Morita theory", which is of independent interest, establishing a symmetric monoidal equivalence between the infinity category of small idempotent-complete stable infinity categories and the Morita localization of the infinity category of spectral categories.