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Let  T  be a locally compact Hausdorff space,  H  a complex Hilbert space, and  A  the      C ∗  {C^\ast }      -algebra      C 0 ( T , B 0 ( H ) ) {C_0}(T,{B_0}(H))      . Let      A 0 {A_0}       be the Pedersen ideal of  A  and      J A {J_A}       the two-sided ideal of  A  consisting of all  x  having compact support, for which      sup { dim ⁡  x ( t ) : t ∈  T } &gt; ∞  \sup \{ \dim x(t):t \in T\} &gt; \infty      . It is known that      A 0 ⊆  J A {A_0} \subseteq {J_A}      , and equality has been conjectured by Pedersen. We give a new characterization of      A 0 {A_0}       which enables us to show that the conjecture is false.

A characterization of the Pedersen ideal of 𝐶𝐷0(𝑇,𝐵₀(𝐻)) and a counter-example