Frame-Less Hilbert C$$^*$$-modules II

We show that if A is a non-unital $$C^*$$-algebra of compact operators, which is $$ *$$-isomorphic to $$\oplus _{i \in I} K(H_{i})$$, where I is an arbitrary index set and for every $$ i \in I $$, $$H_{i} $$ is a separable Hilbert space, then there exists a Hilbert $$A_1$$-module admitting no frames, where $$A_1$$ is the unitization of A.