  An algebraic structure of matter spectrum is studied. It is shown that a base
mathematical construction, lying in the ground of matter spectrum (introduced
by Heisenberg) , is a two-level Hilbert space. Two-level structure of the
Hilbert space is defined by the following pair: 1) a separable Hilbert space
with operator algebras and fundamental symmetries; 2) a nonseparable (physical)
Hilbert space with dynamical and gauge symmetries, that is, a space of states
(energy levels) of the matter spectrum. The each state of matter spectrum is
defined by a cyclic representation within Gel'fand-Naimark-Segal construction.
A decomposition of the physical Hilbert space onto coherent subspaces is given.
This decomposition allows one to describe the all observed spectrum of states
on an equal footing, including lepton, meson and baryon sectors of the matter
spectrum. Following to Heisenberg, we assume that there exist no fundamental
particles, there exist fundamental symmetries. It is shown also that all the
symmetries of matter spectrum are divided onto three kinds: fundamental,
dynamical and gauge symmetries.
