A SIMPLE CHARACTERIZATION OF THE TRACE-CLASS OF OPERATORS PARFENY

The trace-class (τc) of operators on a Hilbert space is characterized in terms of existence of certain centralizers.


INTRODUCTION.
Not long ago Saworotnow [i] characterized the trace-class zA (see [2]) associated wth an arbitrary H*-algebra A (see [3]) as well as the trace-class (Tc) of operators (see [4]).Now, we shall show that, in the second case, there is a much simpler characterization.
We shall use the terminology and the notation of Saworotnow [i].In partic- ular, a trace algebra is a Banach *-algebra with a trace tr and with the follow- ing properties (1) tr(xy)  tr(yx), ( 2) tr(x*x) 4) [tr x[ <_n(x) and (5) x + 0 implies x*x 0 where x, y E B and n( ) denotes the norm of B. It is also assumed that n(xy) < n(x)n(y) for all x, y (B.

MAIN RESULT.
THEOREM.Let B be a simple trace-algebra (see [i]).Assume that for each P. P. SAWOROTNOW there exists a (linear) centralizer U such that Ua > 0(tr x*Uax > 0 for each x B), (Ux) 2 a*a and n(a) trUa.Then there exists a Hilbert space H such that B is isometric to the trace-class (Tc} ,ee [4]) of operators on H.
PROOF.Let A be the H*-algebra associated with B (see [i]) and let Tr de- note the trace on TA induced by A (see [2], p. 97 UxiYi .xiY i follows from the fact that the positive square root of the member ([.xiYi) * .xiYi of A is unique.Thus we may conclude that B is identi- fiable with TA.Now we can complete the proof as in Saworotnow [i], the proof of the corol- lary to Theorem 2.

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).
We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.