<s>
In	O
mathematics	O
,	O
especially	O
functional	B-Application
analysis	I-Application
,	O
a	O
quasitrace	B-Algorithm
is	O
a	O
not	O
necessarily	O
additive	O
tracial	O
functional	O
on	O
a	O
C*	B-Algorithm
-algebra	I-Algorithm
.	O
</s>
<s>
An	O
additive	O
quasitrace	B-Algorithm
is	O
called	O
a	O
trace	O
.	O
</s>
<s>
It	O
is	O
a	O
major	O
open	O
problem	O
if	O
every	O
quasitrace	B-Algorithm
is	O
a	O
trace	O
.	O
</s>
<s>
A	O
quasitrace	B-Algorithm
on	O
a	O
C*	B-Algorithm
-algebra	I-Algorithm
A	O
is	O
a	O
map	O
such	O
that	O
:	O
</s>
<s>
A	O
quasitrace	B-Algorithm
is	O
:	O
</s>
<s>
A	O
1-quasitrace	O
is	O
a	O
map	O
that	O
is	O
just	O
homogeneous	O
,	O
tracial	O
and	O
additive	O
on	O
commuting	O
elements	O
,	O
but	O
does	O
not	O
necessarily	O
extend	O
to	O
such	O
a	O
map	O
on	O
matrix	O
algebras	O
over	O
A	O
.	O
</s>
<s>
If	O
a	O
1-quasitrace	O
extends	O
to	O
the	O
matrix	O
algebra	O
,	O
then	O
it	O
is	O
called	O
a	O
n-quasitrace	O
.	O
</s>
<s>
There	O
are	O
examples	O
of	O
1-quasitraces	O
that	O
are	O
not	O
2-quasitraces	O
.	O
</s>
<s>
One	O
can	O
show	O
that	O
every	O
2-quasitrace	O
is	O
automatically	O
a	O
n-quasitrace	O
for	O
every	O
.	O
</s>
<s>
Sometimes	O
in	O
the	O
literature	O
,	O
a	O
quasitrace	B-Algorithm
means	O
a	O
1-quasitrace	O
and	O
a	O
2-quasitrace	O
means	O
a	O
quasitrace	B-Algorithm
.	O
</s>
<s>
A	O
quasitrace	B-Algorithm
that	O
is	O
additive	O
on	O
all	O
elements	O
is	O
called	O
a	O
trace	O
.	O
</s>
<s>
Uffe	O
Haagerup	O
showed	O
that	O
every	O
quasitrace	B-Algorithm
on	O
a	O
unital	O
,	O
exact	O
C*	B-Algorithm
-algebra	I-Algorithm
is	O
additive	O
and	O
thus	O
a	O
trace	O
.	O
</s>
<s>
As	O
of	O
today	O
(	O
August	O
2020	O
)	O
it	O
remains	O
an	O
open	O
problem	O
if	O
every	O
quasitrace	B-Algorithm
is	O
additive	O
.	O
</s>
<s>
Joachim	O
Cuntz	O
showed	O
that	O
a	O
simple	O
,	O
unital	O
C*	B-Algorithm
-algebra	I-Algorithm
is	O
stably	O
finite	O
if	O
and	O
only	O
if	O
it	O
admits	O
a	O
dimension	O
function	O
.	O
</s>
<s>
A	O
simple	O
,	O
unital	O
C*	B-Algorithm
-algebra	I-Algorithm
is	O
stably	O
finite	O
if	O
and	O
only	O
if	O
it	O
admits	O
a	O
normalized	O
quasitrace	B-Algorithm
.	O
</s>
<s>
An	O
important	O
consequence	O
is	O
that	O
every	O
simple	O
,	O
unital	O
,	O
stably	O
finite	O
,	O
exact	O
C*	B-Algorithm
-algebra	I-Algorithm
admits	O
a	O
tracial	O
state	O
.	O
</s>
<s>
Every	O
quasitrace	B-Algorithm
on	O
a	O
von	O
Neumann	O
algebra	O
is	O
a	O
trace	O
.	O
</s>
