<s>
In	O
noncommutative	O
geometry	O
and	O
related	O
branches	O
of	O
mathematics	O
and	O
mathematical	O
physics	O
,	O
a	O
spectral	B-Algorithm
triple	I-Algorithm
is	O
a	O
set	O
of	O
data	O
which	O
encodes	O
a	O
geometric	O
phenomenon	O
in	O
an	O
analytic	O
way	O
.	O
</s>
<s>
Some	O
authors	O
refer	O
to	O
this	O
notion	O
as	O
unbounded	B-Algorithm
K-cycles	I-Algorithm
or	O
as	O
unbounded	O
Fredholm	B-Algorithm
modules	I-Algorithm
.	O
</s>
<s>
A	O
motivating	O
example	O
of	O
spectral	B-Algorithm
triple	I-Algorithm
is	O
given	O
by	O
the	O
algebra	O
of	O
smooth	O
functions	O
on	O
a	O
compact	O
spin	O
manifold	O
,	O
acting	O
on	O
the	O
Hilbert	O
space	O
of	O
L2-spinors	O
,	O
accompanied	O
by	O
the	O
Dirac	O
operator	O
associated	O
to	O
the	O
spin	O
structure	O
.	O
</s>
<s>
An	O
odd	O
spectral	B-Algorithm
triple	I-Algorithm
is	O
a	O
triple	O
(	O
A	O
,	O
H	O
,	O
D	O
)	O
consisting	O
of	O
a	O
Hilbert	O
space	O
H	O
,	O
an	O
algebra	O
A	O
of	O
operators	O
on	O
H	O
(	O
usually	O
closed	O
under	O
taking	O
adjoints	O
)	O
and	O
a	O
densely	O
defined	O
self	O
adjoint	O
operator	O
D	O
satisfying	O
‖[ a	O
,	O
D ]‖	O
<	O
∞	O
for	O
any	O
a	O
∈	O
A	O
.	O
</s>
<s>
An	O
even	O
spectral	B-Algorithm
triple	I-Algorithm
is	O
an	O
odd	O
spectral	B-Algorithm
triple	I-Algorithm
with	O
a	O
Z/2Z	O
-grading	O
on	O
H	O
,	O
such	O
that	O
the	O
elements	O
in	O
A	O
are	O
even	O
while	O
D	O
is	O
odd	O
with	O
respect	O
to	O
this	O
grading	O
.	O
</s>
<s>
One	O
could	O
also	O
say	O
that	O
an	O
even	O
spectral	B-Algorithm
triple	I-Algorithm
is	O
given	O
by	O
a	O
quartet	O
(	O
A	O
,	O
H	O
,	O
D	O
,	O
γ	O
)	O
such	O
that	O
γ	O
is	O
a	O
self	O
adjoint	O
unitary	O
on	O
H	O
satisfying	O
a	O
γ	O
=	O
γ	O
a	O
for	O
any	O
a	O
in	O
A	O
and	O
D	O
γ	O
=	O
-	O
γ	O
D	O
.	O
</s>
<s>
A	O
finitely	O
summable	O
spectral	B-Algorithm
triple	I-Algorithm
is	O
a	O
spectral	B-Algorithm
triple	I-Algorithm
(	O
A	O
,	O
H	O
,	O
D	O
)	O
such	O
that	O
a.D	O
for	O
any	O
a	O
in	O
A	O
has	O
a	O
compact	O
resolvent	O
which	O
belongs	O
to	O
the	O
class	O
of	O
Lp+	O
-operators	O
for	O
a	O
fixed	O
p	O
(	O
when	O
A	O
contains	O
the	O
identity	O
operator	O
on	O
H	O
,	O
it	O
is	O
enough	O
to	O
require	O
D−1	O
in	O
Lp+( H	O
)	O
)	O
.	O
</s>
<s>
A	O
spectral	B-Algorithm
triple	I-Algorithm
is	O
said	O
to	O
be	O
θ-summable	O
when	O
e−tD2	O
is	O
of	O
trace	O
class	O
for	O
any	O
t	O
>	O
0	O
.	O
</s>
<s>
Let	O
δ(T )	O
denote	O
the	O
commutator	O
of	O
|D|	O
with	O
an	O
operator	O
T	O
on	O
H	O
.	O
A	O
spectral	B-Algorithm
triple	I-Algorithm
is	O
said	O
to	O
be	O
regular	O
when	O
the	O
elements	O
in	O
A	O
and	O
the	O
operators	O
of	O
the	O
form	O
[	O
a	O
,	O
D ]	O
for	O
a	O
in	O
A	O
are	O
in	O
the	O
domain	O
of	O
the	O
iterates	O
δn	O
of	O
δ	O
.	O
</s>
<s>
When	O
a	O
spectral	B-Algorithm
triple	I-Algorithm
(	O
A	O
,	O
H	O
,	O
D	O
)	O
is	O
p-summable	O
,	O
one	O
may	O
define	O
its	O
zeta	O
function	O
ζD(s )	O
=	O
Tr( |D|−s	O
)	O
;	O
more	O
generally	O
there	O
are	O
zeta	O
functions	O
ζb(s )	O
=	O
Tr( b|D|−s	O
)	O
for	O
each	O
element	O
b	O
in	O
the	O
algebra	O
B	O
generated	O
by	O
δn(A )	O
and	O
δn( [a	O
,	O
D ]	O
)	O
for	O
positive	O
integers	O
n	O
.	O
They	O
are	O
related	O
to	O
the	O
heat	O
kernel	O
exp( 	O
-t	O
|D|	O
)	O
by	O
a	O
Mellin	O
transform	O
.	O
</s>
<s>
A	O
real	O
spectral	B-Algorithm
triple	I-Algorithm
is	O
a	O
spectral	B-Algorithm
triple	I-Algorithm
(	O
A	O
,	O
H	O
,	O
D	O
)	O
accompanied	O
with	O
an	O
anti-linear	O
involution	O
J	O
on	O
H	O
,	O
satisfying	O
[	O
a	O
,	O
JbJ ]	O
=	O
0	O
for	O
a	O
,	O
b	O
in	O
A	O
.	O
</s>
<s>
Given	O
a	O
spectral	B-Algorithm
triple	I-Algorithm
(	O
A	O
,	O
H	O
,	O
D	O
)	O
,	O
one	O
can	O
apply	O
several	O
important	O
operations	O
to	O
it	O
.	O
</s>
<s>
When	O
the	O
spectral	B-Algorithm
triple	I-Algorithm
is	O
finitely	O
summable	O
,	O
one	O
may	O
write	O
the	O
above	O
indexes	O
using	O
the	O
(	O
super	O
)	O
trace	O
,	O
and	O
a	O
product	O
of	O
F	O
,	O
e	O
(	O
resp.u	O
)	O
and	O
commutator	O
of	O
F	O
with	O
e	O
(	O
resp.u	O
)	O
.	O
</s>
