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,<ref name="b81">Template:Citation.</ref> and the first example of one was published in 1981 by Bruce Blackadar.<ref name="b81"/><ref>Template:Citation.</ref> 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.
ExamplesEdit
- C, the algebra of complex numbers.
- The reduced group C*-algebra of the free group on finitely many generators.<ref>Template:Citation.</ref>
- The Jiang-Su algebra is simple, projectionless, and KK-equivalent to C.<ref>Template:Citation</ref>
Dimension drop algebrasEdit
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 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>Template:Cite book</ref>