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, 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 ${ℬ}_0$ be the class consisting of the C*-algebras $C_0(ℝ),C_0({ℝ}^2),D_n,S{D}_n$ for each $n\ge 2$, and let $ℬ$ be the class of all C*-algebras of the form

$M_{k_1}(B_1)\oplus M_{k_2}(B_2)\oplus \mathrm{...}\oplus M_{k_r}(B_r)$,

where $r,k_1,\mathrm{...},k_r$ are integers, and where $B_1,\mathrm{...},B_r$ belong to ${ℬ}_0$.

Every C*-algebra A in $ℬ$ is projectionless, moreover, its only projection is 0.

References

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