On a new type of $\ell$-adic regulator for algebraic number fields (the $\ell$-adic regulator without logarithms)

For an algebraic number field $K$ such that a prime $\ell$ splits completely in $K$, we define a regulator $\mathfrak R_\ell(K)\in\mathbb Z_\ell$ that characterizes the subgroup of universal norms from the cyclotomic $\mathbb Z_\ell$-extension of $K$ in the completed group of $S$-units of $K$, where $S$ consists of all prime divisors of $\ell$. We prove that the inequality $\mathfrak R_\ell(K)\ne0$ follows from the $\ell$-adic Schanuel conjecture and holds for some Abelian extensions of imaginary quadratic fields. We study the connection between the regulator $\mathfrak R_\ell(K)$ and the feeble conjecture on the $\ell$-adic regulator, and define analogues of the Gross regulator.