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]

References

1. Blackadar, Bruce E. (1981), "A simple unital projectionless C*-algebra", Journal of Operator Theory, 5 (1): 63–71, MR 0613047.
2. Davidson, Kenneth R. (1996), "IV.8 Blackadar's Simple Unital Projectionless C*-algebra", C*-algebras by Example, Fields Institute Monographs, vol. 6, American Mathematical Society, pp. 124–129, ISBN 9780821871898.
3. Pimsner, M.; Voiculescu, D. (1982), "K-groups of reduced crossed products by free groups", Journal of Operator Theory, 8 (1): 131–156, MR 0670181.
4. Jiang, Xinhui; Su, Hongbing (1999), "On a simple unital projectionless C*-algebra", American Journal of Mathematics, 121 (2): 359–413, doi:10.1353/ajm.1999.0012
5. Rørdam, M. (2000). An introduction to K-theory for C*-algebras. F. Larsen, N. Laustsen. Cambridge, UK: Cambridge University Press. ISBN 978-1-107-36309-0. OCLC 831625390.