On some groups of automorphisms of von Neumann algebras with cyclic and separating vector
Summary (1 min read)
1. Introduction
- It has been shown in [1] that the state of thermal equilibrium of an infinite system is mathematically described by the state ω (over the 0*- algebra of observables 21̂ ) satisfying the KMS boundary conditions.
- In particular, the vector Ω invariant under time and space translations is cyclic for both, 21 and 2Γ.
- In Section 3 the authors deal with the spectrum properties of the group 93#.
- The Theorem 3.5. is the main result of this section.
2. General Discussion
- In this section ψe are concerned with some properties of groups of automorphisms of a von Neumann algebra with a cyclic and separating vector.
- The authors remark that 21' is now also invariant under θ, with a fixed point algebra 91'.
- The first statement follows by [4a, Corollary 2].
- The authors next task is to show that the cyclicity of Ω for 2Γ can in a sence replace the asymptotic abelianness.
- The theorem given below should be compared with [5, Theorem 6].
3. Properties of the Spectrum
- It is of some interest that some group properties of the spectrum, typical for asymptotically abelian systems (see e.g. [10], Theorem 3a) appear also in the situation discussed in the preceding section.
- Let us note that the following theorem is generally true 3.3.
- Moreover, if 93# is abelian, n-parameter and strongly continuous then JF(Δ)J = F{-Δ) for each Borel set Δ C Rn. Hence F(— Jf^) A*Ω — A*Ω and the authors conclude that the spectrum is symmetric.
- Let Q3# and %3Gi satisfy assumptions of Lemma 2.2 (with the same invariant vector Ω).
148 A. Z. JADCZYK:
- Using the same method one can prove the following Theorem.
- It is still possible that the Grand Canonical Ensemble has some additional properties, which follow from KMS for finite systems but not for infinite ones (and which possibly hold true in a thermodynamical limit).
- The above statement is a direct consequence of Theorem 3.3.
- If one applies the above to time translations, then one obtains: V. If 21 \J {Vt} is irreducible then Ω is the only eigenstate of {Vt}. JADCZYK, A. Z., and L. NIKOLOVA: Internal symmetries and observables, preprint (1969).
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Frequently Asked Questions (10)
Q2. what is the eigenvalue of the "Momentum" p?
In particular, for each discrete eigenvalue of the "momentum" p the authors construct two unitary operators p ζ 21 and p ζ 2Γ such thatIt is shown that for each such p, the vector ψP = pΩ also describes the state of thermal equilibrium, with the same temperature as Ω.
Q3. What is the definition of the group G?
Let G -> 23^ be a unitary representation of the group G on a Hubert space Jf. Let Q be the group of all (bounded) characters of the group G. With χ in ύ the authors say that χ is in a point spectrum of 23# if there is 0 =f= ψ ζ 3tf such thatVgψ = χ(g)ψ for all g £G . (*) 10 Commun.math.
Q4. What is the goal of this paper?
The goal of their paper is to study a general situation, when there is given a von Neumann algebra 21, the group Λ-^VgΛV^1 of automorphisms of 21 and the vector Ω invariant under all Vg and cyclic for 21 and 2Γ.
Q5. What is the simplest way to prove the KMS-Algebra?
At = VtA V-1 is in 2ί0 for each A ζ 2ί0 and t ζ R, b) lim \\\\At -A\\\\=0 for each A ζ 2ί0,c) VtΩ = Ω for all t ζR; 5. the involution J such that JAΩ = TA*Ωioτ each A ζ 21 = (2ίo)//, where T = exp(~ jS£Γ/2) and Vt = exp(iHt), 6. The 3-parameter, strongly continuous group of unitary operators {Ux} such thata)
Q6. What is the proof of the KMS?
It is easy to see that the existence of a conjugation Jsatisfying 5. is a necessary and sufficient condition for the state ω : A-^ (Ω, AΩ) to satisfy the KMS boundary conditions (see [2]).
Q7. What is the point spectrum of 93$?
Then the point spectrum of 93$ is a group and for each χ ζσ{^ΰG) there exists a unitary operator χ in 21' such thatχ') for each χ '
Q8. What is the state of thermal equilibrium of an infinite system?
It has been shown in [1] that the state of thermal equilibrium of an infinite system is mathematically described by the state ω (over the 0*- algebra of observables 21̂ ) satisfying the KMS boundary conditions.
Q9. What is the point spectrum of 93#?
if 93# is abelian, n-parameter and strongly continuous then JF(Δ)J = F{-Δ) for each Borel set Δ C Rn.The authors drope an easy proof of this theorem.
Q10. what is the spectrum of vf and 7?
Thena) the spectrum of {Vf} and {̂ 7̂ } is additive (see 3.4); b) the point spectrum of {t^} is a group and for each p in a(Ux)there exist two unitary operators: φ ζ 21 and p ξ 2Γ such that (see 3.5)Uxp = e^ xp Ux ,pE(q)p* = E(p + q) if also qζσ(Ua)9where p denotes either p or p c) for each p ζ σ (Ux) the authors have E (p) < FQ.