468 D. Positive Definite FunctionsCocycles of unitary representations. Let (n, 11) be a unitary represen-
tation of r. Then, a 1-cocycle on r (with coefficients in ( n, 11)) is a function
b: r ~ 1{ such that
b(st) = b(s) + n(s)b(t)
for every s, t Er. We note that b(e) = 0 and that the closed linear span of
b(I') is a n(I')-invariant subspace of 11. We often do not bother to mention
the unitary representation (n, 11). Let bi, i = 1, 2, be 1-cocycles on r with
coefficients in ( 7ri, 1-li) such that bi (r) have dense linear span in 1-li. If there
exists a unitary operator u from 111onto112 such that ub1(s) = b2(s) for all
s EI', then un1(s)u* = n2(s) for alls EI'.
We observe that a function b: r ~ 1i is a 1-cocycle if and only if the
map e from r into the group of affine isometries on 11, defined by
B(s): 1i 3 e f-7 n(s)e + b(s) E 1i
for s E I' and e E 11, is a group homomorphism. Note that any group
homomorphism from r into the group of (affine) isometries on a real Hilbert
space 1i"JE. is of the above form, where 7r is an orthogonal representation of
r on 1i"JE..
A 1-cocycle b on I' is called a 1-coboundary if there exists e E 1i such
that b(s) = ~ -n(s)e for alls EI' (or equivalently B(s)e = e for alls EI').
Lemma D.10. A 1-cocycle is a 1-coboundary if and only if it is bounded.
Proof. Let b: r ~ 1i be a bounded 1-cocycle. Since b(I') is a bounded
subset in a Hilbert space, there exists a unique circumcenter e E 1i of
b(I') (see Exercise D.1). Since b(I') is invariant under the affine isometry
B(s), the vector e is invariant under B(s) for every s E I'. It follows that
b(s) = e - n(s)( D
We observe that if bis a 1-cocycle on a group r, then
llb(s) - b(t)ll =II - n(s)b(s-^1 t)11 = llb(s-^1 t)ll
for every s, t Er. Let us forget for a moment the group structure of r. We
say a kernel k: I' x r ~ IR is conditionally negative definite if there exists a
function b from r into a Hilbert space 1i such that k(s, t) = llb(s)-b(t) 112. It
is well known and not hard to see that a real-valued kernel k is conditionally
negative definite if and only if k is symmetric (i.e., k(s, t) = k(t, s)), k(s, s) =
0 for every s E r and I:~j=l k(si, sj)aiaj :S: 0 for any finite sequences
s1, ... 'Sn Er and a1, ... 'O'.n E IR with I:~=l O'.i = 0.
Theorem D.11 (Schoenberg). Let k be a conditionally negative definite
kernel. Then the kernel I' x I' 3 ( s, t) f-7 exp ( -k ( s, ·t)) E IR is positive