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, and the first example of one was published in 1981 by Bruce Blackadar. 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. Dimension drop algebras. Let  formula_1 be the class consisting of the C*-algebras formula_2 for each formula_3, and let formula_4 be the class of all C*-algebras of the form formula_5, where formula_6 are integers, and where formula_7 belong to formula_8. Every C*-algebra A in formula_4 is projectionless, moreover, its only projection is 0.