2
L. I. Nicolaescu
Definition 1.1 (a) A superspace is a vector space H together with a distinguished
direct sum decomposition
H * fl0eft.
R=diag(lHQ1 lj/j) is called the grading of the superstructure. If H is a Hilbert space
and the decomposition above is orthogonal H is called a Hilbert superspace.
(b) A superalgebra is a Z2-graded algebra A = AQ^A\. The elements in A0 (resp.Ai)
are called even (resp.odd).
(c) The supercommutator in a superalgebra is the bilinear map [•,-]5 : A x A A
defined on homogeneous elements by the formula
[x,y]8 = xy - (-l)Wylyx
where | | {0,1} denotes the degree of a homogeneous element.
Example 1.2 If V = VQ 0 Vi is a superspace then the algebra of endomorphisms of
V has a natural Z2-grading. The even operators preserve the grading while the odd
ones switch it. We will write End (V) to emphasize this super structure of End (V).
§1.2 Clifford modules The notion of Clifford module plays a crucial role in
the definition of Karoubi's Kp,g-theory. We collect here some facts and definitions
concerning these objects.
Definition 1.3 (a) A Hilbert (p,q)- modul e is a Hilbert space H together with a
morphism of C*-algebras
p : Cp'9 -4 B{H).
(B(H)= bounded linear operators on H).
(b) A (p,q) s modul e is a superspace V = Vo 0 V\ together with a morphism of
superalgebras
/
9:C p ' 9 -End(V) .
A Hilbert (p,q) s-module is defined in the obvious way.
Example 1.4 A Hilbert (p,q) module (H,p) is uniquely defined by a choice of op-
erators Ji = p(ti) and Cj = p{e3) satisfying
J* = -Ji , C* = Cj V i , j
J- = -i , c) = i v*,i
{Ji,Cj} = 0 Vt\ j
{J
t l
,J
l 2
} = 0 = {C3l,CJ2} Vii y^i2ji +32.
where for any linear operators A, B the bracket {A, B} denotes their anticommutator
AB + BA. Note that any finite dimensional Cp'9-module can be given a structure of
Hilbert module constructing a metric using the averaging trick (average with respect
to the action of the finite group generated by the e's and e's).
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