We consider -representations of the unital complex *-algebra generate
d by the identity and two elements, alpha and nu, with nu-nu and one
relation, alpha nu-nu alpha=alpha, the ultra-commutation relations (uc
r). In general, we do not impose any commutation relation between alph
a and alpha. This is a very general scheme, encompassing many importa
nt physical examples, inter alia: the ccr, car, q-deformed bosons and
fermions. The representations of interest in physics have a diagonal n
umber operator pi(nu) whose spectrum is contained in the positive inte
gers (together with some other technical conditions). Our principal re
sult is that every -representation in this class is completely determ
ined, up to unitary equivalence, by the sequence of numbers [n+1]=\[Om
ega(n+1),pi(alpha(+))Omega(n)]\(2) for n greater than or equal to 0, w
ith [0]=0. Here Omega(n) is the normalized eigenvector of pi(nu) corre
sponding to the eigenvalue n if the dimension of that eigenspace is 1.
If the carrier Hilbert space is infinite dimensional, this representa
tion is irreducible if and only if [n]>0 for n greater than or equal t
o 1. Finally, we consider spatial representations of some of these rep
resentations by kernels and differential operators. (C) 1997 American
Institute of Physics.