In mathematics, a projectionless C*-algebra is a C*-algebra with no nontrivial projections. For a unital C*-algebra, the projections 0 and 1 are trivial. While for a non-unital C*-algebra, only 0 is considered trivial. The problem of whether simple infinite-dimensional C*-algebras with this property exist was posed in 1958 by Irving Kaplansky,. and the first example of one was published in 1981 by Bruce Blackadar.. For commutative C*-algebras, being projectionless is equivalent to its spectrum being connected. Due to this, being projectionless can be considered as a noncommutative analogue of a connected space.
Examples C, the algebra of complex numbers. The reduced group C*-algebra of the free group on finitely many generators.. The Jiang-Su algebra is simple, projectionless, and KK-equivalent to C.
Dimension drop algebras Let be the class consisting of the C*-algebras for each , and let be the class of all C*-algebras of the form
,
where are integers, and where belong to .
Every C*-algebra A in is projectionless, moreover, its only projection is 0.
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