2 L. I. Nicolaescu Definition 1.1 (a) A superspace is a vector space H together with a distinguished direct sum decomposition H * fl 0 eft. 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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