- Source: 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.
Examples
C, the algebra of complex numbers.
The reduced group C*-algebra of the free group on finitely many generators.
The Jiang-Su algebra is simple, projectionless, and KK-equivalent to C.
Dimension drop algebras
Let
B
0
{\displaystyle {\mathcal {B}}_{0}}
be the class consisting of the C*-algebras
C
0
(
R
)
,
C
0
(
R
2
)
,
D
n
,
S
D
n
{\displaystyle C_{0}(\mathbb {R} ),C_{0}(\mathbb {R} ^{2}),D_{n},SD_{n}}
for each
n
≥
2
{\displaystyle n\geq 2}
, and let
B
{\displaystyle {\mathcal {B}}}
be the class of all C*-algebras of the form
M
k
1
(
B
1
)
⊕
M
k
2
(
B
2
)
⊕
.
.
.
⊕
M
k
r
(
B
r
)
{\displaystyle M_{k_{1}}(B_{1})\oplus M_{k_{2}}(B_{2})\oplus ...\oplus M_{k_{r}}(B_{r})}
,
where
r
,
k
1
,
.
.
.
,
k
r
{\displaystyle r,k_{1},...,k_{r}}
are integers, and where
B
1
,
.
.
.
,
B
r
{\displaystyle B_{1},...,B_{r}}
belong to
B
0
{\displaystyle {\mathcal {B}}_{0}}
.
Every C*-algebra A in
B
{\displaystyle {\mathcal {B}}}
is projectionless, moreover, its only projection is 0.
