<s>
In	O
mathematics	O
,	O
the	O
Kadison	B-Algorithm
–	I-Algorithm
Singer	I-Algorithm
problem	I-Algorithm
,	O
posed	O
in	O
1959	O
,	O
was	O
a	O
problem	O
in	O
functional	B-Application
analysis	I-Application
about	O
whether	O
certain	O
extensions	O
of	O
certain	O
linear	B-Algorithm
functionals	I-Algorithm
on	O
certain	O
C*	B-Algorithm
-algebras	I-Algorithm
were	O
unique	O
.	O
</s>
<s>
The	O
solution	O
was	O
made	O
possible	O
by	O
a	O
reformulation	O
provided	O
by	O
Joel	O
Anderson	O
,	O
who	O
showed	O
in	O
1979	O
that	O
his	O
"	O
paving	O
conjecture	O
"	O
,	O
which	O
only	O
involves	O
operators	O
on	O
finite-dimensional	O
Hilbert	O
spaces	O
,	O
is	O
equivalent	O
to	O
the	O
Kadison	B-Algorithm
–	I-Algorithm
Singer	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
Consider	O
the	O
separable	O
Hilbert	O
space	O
ℓ2	O
and	O
two	O
related	O
C*	B-Algorithm
-algebras	I-Algorithm
:	O
the	O
algebra	O
of	O
all	O
continuous	O
linear	O
operators	O
from	O
ℓ2	O
to	O
ℓ2	O
,	O
and	O
the	O
algebra	O
of	O
all	O
diagonal	O
continuous	O
linear	O
operators	O
from	O
ℓ2	O
to	O
ℓ2	O
.	O
</s>
<s>
A	O
state	B-Algorithm
on	O
a	O
C*	B-Algorithm
-algebra	I-Algorithm
is	O
a	O
continuous	O
linear	B-Algorithm
functional	I-Algorithm
such	O
that	O
(	O
where	O
denotes	O
the	O
algebra	O
's	O
multiplicative	O
identity	O
)	O
and	O
for	O
every	O
.	O
</s>
<s>
Such	O
a	O
state	B-Algorithm
is	O
called	O
pure	O
if	O
it	O
is	O
an	O
extremal	O
point	O
in	O
the	O
set	O
of	O
all	O
states	O
on	O
(	O
i.e.	O
</s>
<s>
That	O
is	O
,	O
the	O
Kadison	B-Algorithm
–	I-Algorithm
Singer	I-Algorithm
problem	I-Algorithm
consisted	O
in	O
proving	O
or	O
disproving	O
the	O
following	O
statement	O
:	O
</s>
<s>
to	O
every	O
pure	O
state	B-Algorithm
on	O
there	O
exists	O
a	O
unique	O
state	B-Algorithm
on	O
that	O
extends	O
.	O
</s>
<s>
The	O
Kadison	B-Algorithm
–	I-Algorithm
Singer	I-Algorithm
problem	I-Algorithm
has	O
a	O
positive	O
solution	O
if	O
and	O
only	O
if	O
the	O
following	O
"	O
paving	O
conjecture	O
"	O
is	O
true	O
:	O
</s>
<s>
The	O
following	O
"	O
discrepancy	O
"	O
statement	O
,	O
again	O
equivalent	O
to	O
the	O
Kadison	B-Algorithm
–	I-Algorithm
Singer	I-Algorithm
problem	I-Algorithm
because	O
of	O
previous	O
work	O
by	O
Nik	O
Weaver	O
,	O
was	O
proven	O
by	O
Marcus/Spielman/Srivastava	O
using	O
a	O
technique	O
of	O
random	O
polynomials	O
:	O
</s>
