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

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

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

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

References

^ ^a ^b Blackadar, Bruce E. (1981), "A simple unital projectionless C*-algebra", Journal of Operator Theory, 5 (1): 63–71, MR 0613047.
^ Davidson, Kenneth R., "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.
^ Pimsner, M.; Voiculescu, D. (1982), "K-groups of reduced crossed products by free groups", Journal of Operator Theory, 8 (1): 131–156, MR 0670181.
^ 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
^ 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.

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[b81-1] Blackadar, Bruce E. (1981), "A simple unital projectionless C*-algebra", Journal of Operator Theory, 5 (1): 63–71, MR 0613047.

[2] Davidson, Kenneth R., "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.

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