A property of purely infinite simple 𝐶*-algebras

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Abstract
An alternative proof is given for the fact ([13]) that a purely infinite, simple C ∗ {C^*} -algebra has the FS property: the set of self-adjoint elements with finite spectrum is norm dense in the set of all self-adjoint elements. In particular, the Cuntz algebras O n ( 2 ≤ n ≤ + ∞ ) {O_n}(2 \leq n \leq + \infty ) and the Cuntz-Krieger algebras O A {O_A} , if A A is an irreducible matrix, have the FS property. This answers a question raised in [2, 2.10] concerning the structure of projections in the Cuntz algebras. Moreover, many corona algebras and multiplier algebras have the FS property.

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