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The purpose of this article is to initiate a systematic study of dimension-free relations between basic communication and query complexity measures and various matrix norms. In other words, our goal is to obtain inequalities that bound a parameter solely as a function of another parameter. This is in contrast to perhaps the more common framework in communication complexity where poly-logarithmic dependencies on the number of input bits are tolerated. Dimension-free bounds are also closely related to structural results, where one seeks to describe the structure of Boolean matrices and functions that have low complexity. We prove such theorems for several communication and query complexity measures as well as various matrix and operator norms. In several other cases we show that such bounds do not exist. We propose several conjectures, and establish that, in addition to applications in complexity theory, these problems are central to characterization of the idempotents of the algebra of Schur multipliers, and could lead to new extensions of Cohen’s celebrated idempotent theorem regarding the Fourier algebra.