Editing Projectionless C*-algebra (section)

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]].