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International Journal of Mathematics Vol. 14, No. 2 (2003) 119137
 

Summary: International Journal of Mathematics
Vol. 14, No. 2 (2003) 119­137
c World Scientific Publishing Company
THE STRUCTURE OF SPIN SYSTEMS
WILLIAM ARVESON
Department of Mathematics, University of California, Berkeley CA 94720, USA
GEOFFREY PRICE
Department of Mathematics, US Naval Academy, Annapolis, MD 21402, USA
Received 8 October 2002
A spin system is a sequence of self-adjoint unitary operators U1, U2, . . . acting on a
Hilbert space H which either commute or anticommute, UiUj = ±Uj Ui for all i, j; it is
called irreducible when {U1, U2, . . .} is an irreducible set of operators. There is a unique
infinite matrix (cij ) with 0, 1 entries satisfying
UiUj = (-1)cij UjUi , i, j = 1, 2, . . . .
Every matrix (cij) with 0, 1 entries satisfying cij = cji and cii = 0 arises from a
nontrivial irreducible spin system, and there are uncountably many such matrices.
In cases where the commutation matrix (cij) is of "infinite rank" (these are the ones
for which infinite dimensional irreducible representations exist), we show that the C-
algebra generated by an irreducible spin system is the CAR algebra, an infinite tensor
product of copies of M2(C), and we classify the irreducible spin systems associated with

  

Source: Arveson, William - Department of Mathematics, University of California at Berkeley

 

Collections: Mathematics