The Blackadar–Handelman theorem for non-unital C⁎-algebras

A well-known theorem of Blackadar and Handelman states that every unital stably finite C⁎-algebra has a bounded quasitrace. Rather strong generalizations of stable finiteness to the non-unital case can be obtained by either requiring the multiplier algebra to be stably finite, or alternatively requiring it to be at least stably not properly infinite. This paper deals with the question whether the Blackadar–Handelman result can be extended to the non-unital case with respect to these generalizations of stable finiteness. Two examples of C⁎-algebras are constructed. The first one is a non-unital, stably commutative C⁎-algebra A that contradicts the weakest possible generalization of the Blackadar–Handelman theorem: The multiplier algebra of A is stably finite, while A has no bounded quasitrace. The second example is a simple C⁎-algebra B with no bounded quasitrace, such that the multiplier algebras of all matrix algebras over B are not properly infinite.