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]
References
- 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, 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
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