6.4 Equilibrium States
153
(0) Any state ω ∈ K β is τ -invariant: ω ◦ τ t = ω (t ∈ R).
(i) Any K β is a convex W
∗ -compact subset of S(C).
(ii-a) For β = ∞, K β is a simplex in S(C).
(ii-b) K ∞ is a face in S(C).
(iii-a) The set EK β of extrema1 points ω ∈ K β (β = ∞) consists of factor states:
The centers of π ω (C)
are trivial.
(iii-b) The extremal points ω ∈ K ∞ , i.e. ω ∈ EK ∞ , are pure states: ω ∈ ES(C), i.e.
π ω (C)
= L(H ω ).
(iv) ω j ∈ EK β (β = ∞, j = 1, 2) implies either ω 1 = ω 2 , or ω 1 ⊥ ω 2 , i.e. ω 1 and
ω 2 are mutually disjoint, i.e. the central covers s ω 1 and s ω 2 of the corresponding
GNS-representations are mutually orthogonal.
(v) The extremal decomposition of ω ∈ K β (β = ∞) coincides with its central
decomposition, cf. [53, Chap. 4], [235, Chap. 4]. The corresponding probability
measure μ
c
ω on S(C) is pseudosupported (cf. [54, Chap. 6]) by EK β and if the
Hilbert space of the GNS-representation H ω is separable, then μ
c
ω is supported by
EK β : μ
c
ω (EK β ) = μ
c
ω (S(C)) = 1.
6.4.5 Lemma. Let ω ∈ S(C) be a τ -ground state. Let (π ω , H ω , , ω ) be the corresponding GNS representation. Then for the unique selfadjoint operator Q ω on H ω
determined by the relation:
exp(it Q ω ) π ω (y)) ω := π ω (τ t (y))) ω , ∀t ∈ R,
(6.4.9)
the following is valid:
Q ω ≥ 0, and for all t ∈ R one has exp(it Q ω ) ∈ π ω (C)
.
(6.4.10)
Proof. See [54, 5.3.19].
6.4.6 Any (τ , β)-KMS state, according to 6.4.4(i), can be approximated in the w
∗ -
topology by convex combinations of extremal KMS states at the same temperature
β
−1 . The set K β may be void for a genera1 dynamical system and for a given β ∈
(0, ∞]. Occurrence of more than one points in K β means occurrence of several
mutually disjoint states in EK β . Orthogonal central projectors s 1 and s 2 (the central
covers of the corresponding GNS representations) are supporting such disjoint states;
these s j ∈ Z (:= the center of π u (C)
) may be interpreted as corresponding to distinct
values of a macroscopic (global, classical) quantity for distinct j = 1, 2. We interpret
this situation as possibility of existence of several mutually different ‘phases’ of
153
(0) Any state ω ∈ K β is τ -invariant: ω ◦ τ t = ω (t ∈ R).
(i) Any K β is a convex W
∗ -compact subset of S(C).
(ii-a) For β = ∞, K β is a simplex in S(C).
(ii-b) K ∞ is a face in S(C).
(iii-a) The set EK β of extrema1 points ω ∈ K β (β = ∞) consists of factor states:
The centers of π ω (C)
are trivial.
(iii-b) The extremal points ω ∈ K ∞ , i.e. ω ∈ EK ∞ , are pure states: ω ∈ ES(C), i.e.
π ω (C)
= L(H ω ).
(iv) ω j ∈ EK β (β = ∞, j = 1, 2) implies either ω 1 = ω 2 , or ω 1 ⊥ ω 2 , i.e. ω 1 and
ω 2 are mutually disjoint, i.e. the central covers s ω 1 and s ω 2 of the corresponding
GNS-representations are mutually orthogonal.
(v) The extremal decomposition of ω ∈ K β (β = ∞) coincides with its central
decomposition, cf. [53, Chap. 4], [235, Chap. 4]. The corresponding probability
measure μ
c
ω on S(C) is pseudosupported (cf. [54, Chap. 6]) by EK β and if the
Hilbert space of the GNS-representation H ω is separable, then μ
c
ω is supported by
EK β : μ
c
ω (EK β ) = μ
c
ω (S(C)) = 1.
6.4.5 Lemma. Let ω ∈ S(C) be a τ -ground state. Let (π ω , H ω , , ω ) be the corresponding GNS representation. Then for the unique selfadjoint operator Q ω on H ω
determined by the relation:
exp(it Q ω ) π ω (y)) ω := π ω (τ t (y))) ω , ∀t ∈ R,
(6.4.9)
the following is valid:
Q ω ≥ 0, and for all t ∈ R one has exp(it Q ω ) ∈ π ω (C)
.
(6.4.10)
Proof. See [54, 5.3.19].
6.4.6 Any (τ , β)-KMS state, according to 6.4.4(i), can be approximated in the w
∗ -
topology by convex combinations of extremal KMS states at the same temperature
β
−1 . The set K β may be void for a genera1 dynamical system and for a given β ∈
(0, ∞]. Occurrence of more than one points in K β means occurrence of several
mutually disjoint states in EK β . Orthogonal central projectors s 1 and s 2 (the central
covers of the corresponding GNS representations) are supporting such disjoint states;
these s j ∈ Z (:= the center of π u (C)
) may be interpreted as corresponding to distinct
values of a macroscopic (global, classical) quantity for distinct j = 1, 2. We interpret
this situation as possibility of existence of several mutually different ‘phases’ of