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,{{citation

| last = Blackadar | first = Bruce E.

| issue = 1

| journal = Journal of Operator Theory

| mr = 613047

| pages = 63–71

| title = A simple unital projectionless C*-algebra

| volume = 5

| year = 1981}}. and the first example of one was published in 1981 by Bruce Blackadar.{{citation|title=C*-algebras by Example|volume=6|series=Fields Institute Monographs|first=Kenneth R.|last=Davidson|date=1996|publisher=American Mathematical Society|isbn=9780821871898|contribution=IV.8 Blackadar's Simple Unital Projectionless C*-algebra|pages=124–129|url=https://books.google.com/books?id=0TXteNfrzvcC&pg=PA124}}. 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.{{citation

| last1 = Pimsner | first1 = M.

| last2 = Voiculescu | first2 = D.

| issue = 1

| journal = Journal of Operator Theory

| mr = 670181

| pages = 131–156

| title = K-groups of reduced crossed products by free groups

| volume = 8

| year = 1982}}.

The Jiang-Su algebra is simple, projectionless, and KK-equivalent to C.{{citation

| pages = 359–413 | title = On a simple unital projectionless C*-algebra | volume = 121 | year = 1999 | doi = 10.1353/ajm.1999.0012}}

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. {{Cite book|last=Rørdam|first=M.|url=https://www.worldcat.org/oclc/831625390|title=An introduction to K-theory for C*-algebras|date=2000|publisher=Cambridge University Press|others=F. Larsen, N. Laustsen|isbn=978-1-107-36309-0|location=Cambridge, UK|oclc=831625390}}

References

Category:C*-algebras