I REMEMBER CLIFFORD ALGEBRA

1. Clifford C*-algebras Let V be a real Hilbert space and V C be the accompanied *-Hilbert space with the *-operation in V C denoted by v for v ∈ V C and the inner product satisfying (v|w) = (w|v) for v, w ∈ V C. Note that there is a one-to-one correspondence between real Hilbert spaces and *-Hilbert spaces. An element v in a *-Hilbert space is said to be real if v = v. For a linear map L : V C →WC with W another real Hilbert space, its complex conjugate is a linear map L : V C →WC defined by Lx = Lx. If L = L, L is said to be real because it corresponds to a real linear map V →W by restriction and C-linear extension. The Clifford C*-algebra is a unital C*-algebra C(V ) linearly generated by V C with the relations (the canonical anti-commutation relations, the CAR for short) (i) x∗ = x for x ∈ V C and (ii) x∗y + yx∗ = (x|y)1 for x, y ∈ V . Here (x|y) denotes the inner product in V C, which is assumed to be linear in the second variable by our convention. Let O(V ) be the group of orthogonal transformations in V . From structural invariance, there is a natural imbedding of the group O(V ) into Aut(C(V )). If θ is an automorphism of C(V ), we regard it as an element in the crossed product extension of C(V ). Thus θxθ−1 = θ(x) for x ∈ C(V ) and this covariance relation is kept for L objects as well. In particular, the induced unitary on L(C(V )) is denoted by Ad θ, i.e., this is a unitary operator specified by (Ad θ)φ = (φ ◦ θ−1)1/2.