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 B 0 {\displaystyle {\mathcal {B}}_{0}} be the class consisting of the C*-algebras C 0 ( R ) , C 0 ( R 2 ) , D n , S D n {\displaystyle C_{0}(\mathbb {R} ),C_{0}(\mathbb {R} ^{2}),D_{n},SD_{n}} for each n ≥ 2 {\displaystyle n\geq 2} , and let B {\displaystyle {\mathcal {B}}} be the class of all C*-algebras of the form
M k 1 ( B 1 ) ⊕ M k 2 ( B 2 ) ⊕ . . . ⊕ M k r ( B r ) {\displaystyle M_{k_{1}}(B_{1})\oplus M_{k_{2}}(B_{2})\oplus ...\oplus M_{k_{r}}(B_{r})} , where r , k 1 , . . . , k r {\displaystyle r,k_{1},...,k_{r}} are integers, and where B 1 , . . . , B r {\displaystyle B_{1},...,B_{r}} belong to B 0 {\displaystyle {\mathcal {B}}_{0}} . Every C*-algebra A in B {\displaystyle {\mathcal {B}}} is projectionless, moreover, its only projection is 0.
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