Editing Projectionless C*-algebra

In [[mathematics]], a  '''projectionless C*-algebra''' is a [[C*-algebra]] with no nontrivial [[projection (linear algebra)#Orthogonal projections|projection]]s. For a [[Unital ring|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 algebra|simple]] [[dimension (vector space)|infinite-dimensional]] C*-algebras with this property exist was posed in 1958 by [[Irving Kaplansky]],<ref name="b81">{{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}}.</ref> and the first example of one was published in 1981 by [[Bruce Blackadar]].<ref name="b81"/><ref>{{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}}.</ref> For [[commutative property|commutative]] C*-algebras, being projectionless is equivalent to its [[spectrum of a C*-algebra|spectrum]] being [[connected space|connected]]. Due to this, being projectionless can be considered as a [[noncommutative topology|noncommutative]] analogue of a connected [[Topological space|space]].

== Examples ==
* '''C''', the algebra of [[complex number]]s.
* The [[Group algebra of a locally compact group#The reduced group C.2A-algebra Cr.2A.28G.29|reduced group C*-algebra]] of the [[free group]] on finitely many generators.<ref>{{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}}.</ref>
* The [[Jiang-Su algebra]] is simple, projectionless, and [[KK-theory|KK-equivalent]] to '''C'''.<ref>{{citation
| last1 = Jiang | first1 = Xinhui | last2 = Su | first2 = Hongbing | issue = 2 | journal = American Journal of Mathematics
| pages = 359–413 | title = On a simple unital projectionless C*-algebra | volume = 121 | year = 1999 | doi = 10.1353/ajm.1999.0012}}</ref>

== Dimension drop algebras ==
Let <math>\mathcal{B}_0</math> be the class consisting of the C*-algebras <math>C_0(\mathbb{R}), C_0(\mathbb{R}^2), D_n, SD_n</math> for each <math>n \geq 2</math>, and let <math>\mathcal{B}</math> be the class of all C*-algebras of the form

<math>M_{k_1}(B_1) \oplus M_{k_2}(B_2) \oplus ... \oplus M_{k_r}(B_r) </math>,

where <math>r, k_1, ..., k_r </math> are [[Integer|integers]], and where <math>B_1, ..., B_r </math> belong to <math>\mathcal{B}_0 </math>.

Every C*-algebra A in <math>\mathcal{B}</math> is projectionless, moreover, its only projection is 0. <ref>{{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}}</ref>

==References==
{{reflist}}

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[[Category:C*-algebras]]


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