We analyze the algebraic, topological, and order properties of
I*-algebras: complex unital topological *-algebras for which Σ
Jxj*xj=0 implies
xj=0 (
j∈
J),
J⊂
N any finite subset. We consider the ergodic properties of states on an
I*-algebra with a distinguished group of automorphisms. Particular attention is given to
I*-algebras of the form
E=Σ
N⊕\bar{⊗}
nE where
E is a nuclear
LF-space. When
E=\mathscr{S}(
R4) (\mathscr{D}(
R3)⊕\mathscr{D}(
R3) respectively) then
E has applications to relativistic quantum field theory (the canonical anti-commutation or commutation relations, respectively).
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