Equilibrium states on right LCM semigroup C*-algebras revisited Nadia S. Larsen Equilibrium states on right LCM semigroup C*-algebras revisited Nadia S. Larsen University of Oslo 4 December 2017 joint with N. Brownlowe, J. Ramagge and N. Stammeier
Equilibrium The KMS condition for finite systems states on right LCM semigroup C*-algebras revisited Finite quantum systems: a time evolution on M n ( C ) is given by Nadia S. Larsen a one-parameter group of automorphisms σ t ( a ) = e itH ae − itH , where t ∈ R , a ∈ M n ( C ) and H is a self-adjoint matrix.
Equilibrium The KMS condition for finite systems states on right LCM semigroup C*-algebras revisited Finite quantum systems: a time evolution on M n ( C ) is given by Nadia S. Larsen a one-parameter group of automorphisms σ t ( a ) = e itH ae − itH , where t ∈ R , a ∈ M n ( C ) and H is a self-adjoint matrix. The Gibbs state at β > 0 is ϕ G ( a ) = Tr( ae − β H ) Tr( e − β H ) . It satisfies ϕ G ( ab ) = ϕ G ( b σ i β ( a )) , (1) for a , b ∈ M n ( C ) analytic , i.e. t �→ σ t ( a ) extends to an entire function on C .
Equilibrium The KMS condition for finite systems states on right LCM semigroup C*-algebras revisited Finite quantum systems: a time evolution on M n ( C ) is given by Nadia S. Larsen a one-parameter group of automorphisms σ t ( a ) = e itH ae − itH , where t ∈ R , a ∈ M n ( C ) and H is a self-adjoint matrix. The Gibbs state at β > 0 is ϕ G ( a ) = Tr( ae − β H ) Tr( e − β H ) . It satisfies ϕ G ( ab ) = ϕ G ( b σ i β ( a )) , (1) for a , b ∈ M n ( C ) analytic , i.e. t �→ σ t ( a ) extends to an entire function on C . Partition function of ( M n ( C ) , σ ) is β �→ Tr( e − β H ).
Equilibrium The KMS condition for finite systems states on right LCM semigroup C*-algebras revisited Finite quantum systems: a time evolution on M n ( C ) is given by Nadia S. Larsen a one-parameter group of automorphisms σ t ( a ) = e itH ae − itH , where t ∈ R , a ∈ M n ( C ) and H is a self-adjoint matrix. The Gibbs state at β > 0 is ϕ G ( a ) = Tr( ae − β H ) Tr( e − β H ) . It satisfies ϕ G ( ab ) = ϕ G ( b σ i β ( a )) , (1) for a , b ∈ M n ( C ) analytic , i.e. t �→ σ t ( a ) extends to an entire function on C . Partition function of ( M n ( C ) , σ ) is β �→ Tr( e − β H ). (1) - the KMS condition , cf. Haag-Hugenholtz-Winnick (1967): equilibrium for a state on a C ∗ -algebra with time evolution.
Equilibrium KMS states states on right LCM semigroup C*-algebras revisited By analogy with finite systems and the Gibbs state, extend the Nadia S. Larsen notions of KMS β state, partition function, inverse temperature.
Equilibrium KMS states states on right LCM semigroup C*-algebras revisited By analogy with finite systems and the Gibbs state, extend the Nadia S. Larsen notions of KMS β state, partition function, inverse temperature. A C ∗ -algebra, σ : R → Aut( A ) time evolution, ϕ a state on A . 1 ϕ is KMS β (at inverse temperature β ∈ [0 , ∞ )) if ϕ ( ab ) = ϕ ( b σ i β ( a )) for all a , b ∈ A a , the dense ∗ -subalgebra of analytic elements.
Equilibrium KMS states states on right LCM semigroup C*-algebras revisited By analogy with finite systems and the Gibbs state, extend the Nadia S. Larsen notions of KMS β state, partition function, inverse temperature. A C ∗ -algebra, σ : R → Aut( A ) time evolution, ϕ a state on A . 1 ϕ is KMS β (at inverse temperature β ∈ [0 , ∞ )) if ϕ ( ab ) = ϕ ( b σ i β ( a )) for all a , b ∈ A a , the dense ∗ -subalgebra of analytic elements. 2 A state ϕ is a ground state if for all a , b with b analytic, the function z → ϕ ( a σ z ( b )) is bounded in the upper-half plane.
Equilibrium KMS states states on right LCM semigroup C*-algebras revisited By analogy with finite systems and the Gibbs state, extend the Nadia S. Larsen notions of KMS β state, partition function, inverse temperature. A C ∗ -algebra, σ : R → Aut( A ) time evolution, ϕ a state on A . 1 ϕ is KMS β (at inverse temperature β ∈ [0 , ∞ )) if ϕ ( ab ) = ϕ ( b σ i β ( a )) for all a , b ∈ A a , the dense ∗ -subalgebra of analytic elements. 2 A state ϕ is a ground state if for all a , b with b analytic, the function z → ϕ ( a σ z ( b )) is bounded in the upper-half plane. 3 KMS ∞ if ϕ = w ∗ lim ϕ n as β n → ∞ and ϕ n is KMS β n .
Equilibrium KMS states states on right LCM semigroup C*-algebras revisited By analogy with finite systems and the Gibbs state, extend the Nadia S. Larsen notions of KMS β state, partition function, inverse temperature. A C ∗ -algebra, σ : R → Aut( A ) time evolution, ϕ a state on A . 1 ϕ is KMS β (at inverse temperature β ∈ [0 , ∞ )) if ϕ ( ab ) = ϕ ( b σ i β ( a )) for all a , b ∈ A a , the dense ∗ -subalgebra of analytic elements. 2 A state ϕ is a ground state if for all a , b with b analytic, the function z → ϕ ( a σ z ( b )) is bounded in the upper-half plane. 3 KMS ∞ if ϕ = w ∗ lim ϕ n as β n → ∞ and ϕ n is KMS β n . References: Bratteli-Robinson, Pedersen, Connes-Marcolli.
Equilibrium KMS ∞ strictly subset of ground states states on right LCM semigroup C*-algebras revisited Theorem (Laca-Raeburn (2010)) Nadia S. Larsen C ∗ ( N ⋊ N × ) is the universal C ∗ -algebra generated by isometries s and { v p | p prime } , subject to the relations 1 v p s = s p v p ; 2 v p v q = v q v p , 3 v ∗ p v q = v q v ∗ p when p � = q, 4 s ∗ v p = s p − 1 v p s ∗ , and p s k v p = 0 for 1 ≤ k < p. 5 v ∗ Dynamics: σ t ( s ) = s and σ t ( v p ) = p it v p .
Equilibrium KMS ∞ strictly subset of ground states states on right LCM semigroup C*-algebras revisited Theorem (Laca-Raeburn (2010)) Nadia S. Larsen C ∗ ( N ⋊ N × ) is the universal C ∗ -algebra generated by isometries s and { v p | p prime } , subject to the relations 1 v p s = s p v p ; 2 v p v q = v q v p , 3 v ∗ p v q = v q v ∗ p when p � = q, 4 s ∗ v p = s p − 1 v p s ∗ , and p s k v p = 0 for 1 ≤ k < p. 5 v ∗ Dynamics: σ t ( s ) = s and σ t ( v p ) = p it v p .Then, for β < 1 , there are no KMS states, if β ∈ [1 , 2] , there is a unique KMS β state;
Equilibrium KMS ∞ strictly subset of ground states states on right LCM semigroup C*-algebras revisited Theorem (Laca-Raeburn (2010)) Nadia S. Larsen C ∗ ( N ⋊ N × ) is the universal C ∗ -algebra generated by isometries s and { v p | p prime } , subject to the relations 1 v p s = s p v p ; 2 v p v q = v q v p , 3 v ∗ p v q = v q v ∗ p when p � = q, 4 s ∗ v p = s p − 1 v p s ∗ , and p s k v p = 0 for 1 ≤ k < p. 5 v ∗ Dynamics: σ t ( s ) = s and σ t ( v p ) = p it v p .Then, for β < 1 , there are no KMS states, if β ∈ [1 , 2] , there is a unique KMS β state; if β ∈ (2 , ∞ ] , the KMS β states are parametrised by probability measures on T while the ground states are parametrised by states on the Toeplitz C ∗ -algebra generated by a single isometry.
Equilibrium KMS ∞ strictly subset of ground states states on right LCM semigroup C*-algebras revisited Theorem (Afsar-Brownlowe-L-Stammeier (2016)) Nadia S. Let S be a right LCM monoid and N : S → N × homomorphism Larsen such that S is admissible. Consider the time evolution σ t ( v s ) = N it s v s . If β c ∈ R is such that the function � n − ( β − 1) , ζ N ( β ) := n ∈ Irr ( N ( S )) converges for β ≥ β c , then for ( C ∗ ( S ) , R , σ ) we have
Equilibrium KMS ∞ strictly subset of ground states states on right LCM semigroup C*-algebras revisited Theorem (Afsar-Brownlowe-L-Stammeier (2016)) Nadia S. Let S be a right LCM monoid and N : S → N × homomorphism Larsen such that S is admissible. Consider the time evolution σ t ( v s ) = N it s v s . If β c ∈ R is such that the function � n − ( β − 1) , ζ N ( β ) := n ∈ Irr ( N ( S )) converges for β ≥ β c , then for ( C ∗ ( S ) , R , σ ) we have 1 β ∈ [0 , 1) : no KMS β state;
Equilibrium KMS ∞ strictly subset of ground states states on right LCM semigroup C*-algebras revisited Theorem (Afsar-Brownlowe-L-Stammeier (2016)) Nadia S. Let S be a right LCM monoid and N : S → N × homomorphism Larsen such that S is admissible. Consider the time evolution σ t ( v s ) = N it s v s . If β c ∈ R is such that the function � n − ( β − 1) , ζ N ( β ) := n ∈ Irr ( N ( S )) converges for β ≥ β c , then for ( C ∗ ( S ) , R , σ ) we have 1 β ∈ [0 , 1) : no KMS β state; 2 β ∈ [1 , β c ] : unique KMS β if action S c � S / S c essentially free, where S c ⊂ S subsemigroup of elements having LCM with any t in S;
Equilibrium KMS ∞ strictly subset of ground states states on right LCM semigroup C*-algebras revisited Theorem (Afsar-Brownlowe-L-Stammeier (2016)) Nadia S. Let S be a right LCM monoid and N : S → N × homomorphism Larsen such that S is admissible. Consider the time evolution σ t ( v s ) = N it s v s . If β c ∈ R is such that the function � n − ( β − 1) , ζ N ( β ) := n ∈ Irr ( N ( S )) converges for β ≥ β c , then for ( C ∗ ( S ) , R , σ ) we have 1 β ∈ [0 , 1) : no KMS β state; 2 β ∈ [1 , β c ] : unique KMS β if action S c � S / S c essentially free, where S c ⊂ S subsemigroup of elements having LCM with any t in S; 3 β ∈ ( β c , ∞ ] : KMS β states parametrised by normalised traces on C ∗ ( S c ) ;
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