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

Gillette, R. M.; Taylor, D. C.

doi:10.1090/S0002-9939-1978-0461156-6

A characterization of the Pedersen ideal of $CD0(T,B_{0}(H))$ and a counter-example
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by R. M. Gillette and D. C. Taylor

Proc. Amer. Math. Soc. 68 (1978), 59-63

DOI: https://doi.org/10.1090/S0002-9939-1978-0461156-6

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Abstract:

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

References

Les algèbres, d’opérateurs dans l’espace Hilbertien

Les

algèbres et leur représentations

Dale Husemoller, Fibre bundles, McGraw-Hill Book Co., New York-London-Sydney, 1966. MR 0229247, DOI 10.1007/978-1-4757-4008-0
Kjeld B. Laursen and Allan M. Sinclair, Lifting matrix units in $C^*$-algebras. II, Math. Scand. 37 (1975), no. 1, 167–172. MR 397422, DOI 10.7146/math.scand.a-11598
John W. Milnor and James D. Stasheff, Characteristic classes, Annals of Mathematics Studies, No. 76, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. MR 0440554, DOI 10.1515/9781400881826
Gert Kjaergȧrd Pedersen, Measure theory for $C^{\ast }$ algebras, Math. Scand. 19 (1966), 131–145. MR 212582, DOI 10.7146/math.scand.a-10802

A decomposition theorem for

algebras

22

integrals, an approach to non-commutative measure theory

Gert Kjaergȧrd Pedersen and Nils Holger Petersen, Ideals in a $C^{\ast }$-algebra, Math. Scand. 27 (1970), 193–204 (1971). MR 308797, DOI 10.7146/math.scand.a-10998