Compact operators and uniform structures in Hilbert \(C^*\)-modules (Q2038441)

Let \(F:M\to N\) be an adjointable operator between Hilbert \(C^*\)-modules over a \(C^*\)-algebra \(A\). In the case when \(N\) is countably generated, a uniform structure on \(N\) was recently presented in [\textit{E. Troitsky}, J. Math. Anal. Appl. 485, No. 2, Article ID 123842, 19 p. (2020; Zbl 1439.46046)], and it was shown there that \(F\) is \(A\)-compact iff \(F(B)\) is totally bounded with respect to this structure, where \(B\subset M\) is the unit ball. In the paper under review, this result is partially generalized to the case of general \(N\): it is shown that \(A\)-compactness implies total boundedness, and the opposite holds if \(N\) is a direct summand in the standard Hilbert \(C^*\)-module \(l_2(A)\) of arbitrary cardinality. The frame property for uniform structures is introduced and the relation to frames is discussed.