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The paper starts a series of publications devoted to the theory of topological radicals (TRs) of normed algebras. It contains results on general properties of TRs and their domains, considers TRs close to the Jacobson radical, presents a theory of tensor TRs (they are connected with the problem of calculating the radical of a projective tensor product), introduces and studies TRs related to the notion of joint qusinilpotence.

Topological radicals, I. Basic properties, tensor products and joint quasinilpotence