Projectionless C*-algebra explained

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.

Dimension drop algebras

Let

l{B} _{0} be the class consisting of the C*-algebras C _{0(R),} C ^{2 ),} _{0 ( R} D _{n,} SD _{n} for each n \geq 2 , and let l{B} be the class of all C*-algebras of the formM _{k _{1}} (B _{1)} ⊕ M _{k _{2}} (B _{2)} ⊕ ... ⊕ M _{k _{r}} (B _{r)} ,

where

r, k _{1,} ..., k _{r} are integers, and where B _{1,} ..., B _{r} belong to l{B} _{0} .

Every C*-algebra A in

l{B} is projectionless, moreover, its only projection is 0. ^[4]

Notes and References

1. .
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4. Book: Rørdam, M.. An introduction to K-theory for C*-algebras. 2000. Cambridge University Press. F. Larsen, N. Laustsen. 978-1-107-36309-0. Cambridge, UK. 831625390.