Projectionless C*-algebra

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,^[1] and the first example of one was published in 1981 by Bruce Blackadar.^[1][2] 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.^[3]
• The Jiang-Su algebra is simple, projectionless, and KK-equivalent to C.^[4]

Dimension drop algebras

Let ${\displaystyle {\mathcal {B}}_{0}}$ be the class consisting of the C*-algebras ${\displaystyle C_{0}(\mathbb {R} ),C_{0}(\mathbb {R} ^{2}),D_{n},SD_{n}}$ for each ${\displaystyle n\geq 2}$, and let ${\displaystyle {\mathcal {B}}}$ be the class of all C*-algebras of the form

${\displaystyle M_{k_{1}}(B_{1})\oplus M_{k_{2}}(B_{2})\oplus ...\oplus M_{k_{r}}(B_{r})}$,

where ${\displaystyle r,k_{1},...,k_{r}}$ are integers, and where ${\displaystyle B_{1},...,B_{r}}$ belong to ${\displaystyle {\mathcal {B}}_{0}}$.

Every C*-algebra A in ${\displaystyle {\mathcal {B}}}$ is projectionless, moreover, its only projection is 0. ^[5]