Entire analytic elements of C⋆-algebra actions #
For a point-norm continuous real action α on a possibly nonunital C⋆-algebra A, an element
a is entire analytic when its orbit t ↦ α_t a is the restriction to the real line of an
entire A-valued function. That function is unique, and α.entireExtension a z is its value
α_z a at complex time z.
The entire analytic elements form a nonunital star subalgebra, on which the complex-time
extension is complex linear, multiplicative and satisfies α_z (a⋆) = (α_{conj z} a)⋆. It is
invariant under the action and its complex-time extension: α_w (α_z a) = α_{z + w} a. The
norm of α_z a depends only on the imaginary part of z, so the extension is bounded on
every horizontal strip of bounded width. An eigenvector α_t a = exp (c t) • a is entire
analytic, with α_z a = exp (c z) • a. Entire analytic elements lie in the domain of the
generator δ of α, where δ a is the complex derivative of the extension at 0, and the
subalgebra is invariant under δ.
A continuous linear functional invariant under the action is also invariant in complex time on
entire analytic elements, apply_entireExtension_of_apply_action, so its correlations satisfy
φ (α_z(a) α_w(b)) = φ (a α_{w - z}(b)). For an invariant positive functional ω, the
imaginary-time correlation ω (a⋆ α_{is}(a)) of an entire analytic a is nonnegative,
apply_star_mul_entireExtension_mul_I_nonneg_of_apply_action.
These are the elements on which the analytic form ω(a α_{iβ}(b)) = ω(b a) of the KMS
condition is stated.
References #
Bratteli--Robinson, Operator Algebras and Quantum Statistical Mechanics 1, second edition, Section 2.5.3, for entire analytic elements of one-parameter groups of automorphisms.
An element is entire analytic for a real action when its orbit t ↦ α_t a is the
restriction to the real line of an entire A-valued function.
Instances For
The complex-time action z ↦ α_z a on an entire analytic element a: the unique entire
continuation of its orbit. It is identically zero when a is not entire analytic.
Instances For
The complex-time extension vanishes away from the entire analytic elements.
The complex-time extension of an entire analytic element is entire.
At real times the complex-time extension is the action.
At time zero the complex-time extension is the identity.
For an entire analytic b, the correlation function z ↦ φ (a α_z(b)) is entire.
Uniqueness of continuation. Any entire extension of the orbit of a is the
complex-time extension of a.
An element is entire analytic exactly when its complex-time extension is an entire extension of its orbit.
An eigenvector of the action, α_t a = exp (c t) • a, is entire analytic.
The complex-time extension of an eigenvector α_t a = exp (c t) • a is
α_z a = exp (c z) • a.
Time shift. Translating complex time by a real time t is the action of t.
The values of the complex-time extension are again entire analytic.
Group law in complex time. α_w (α_z a) = α_{z + w} a.
The action at a real time preserves entire analyticity.
The complex-time extension of a real translate is the translated extension.
The norm of α_z a depends only on the imaginary part of z.
Strip bound. The complex-time extension is bounded on every horizontal strip whose imaginary parts form a bounded set.
An element is entire analytic exactly when one of its real translates is.
Algebraic operations #
Zero is entire analytic.
The complex-time extension of zero vanishes.
Sums of entire analytic elements are entire analytic.
The complex-time extension is additive.
Scalar multiples of entire analytic elements are entire analytic.
The complex-time extension is complex linear.
Negatives of entire analytic elements are entire analytic.
The complex-time extension commutes with negation.
Differences of entire analytic elements are entire analytic.
The complex-time extension commutes with subtraction.
Products of entire analytic elements are entire analytic.
The complex-time extension is multiplicative: α_z (a b) = α_z a α_z b.
Adjoints of entire analytic elements are entire analytic.
Star relation. α_z (a⋆) = (α_{conj z} a)⋆.
Star preserves entire analyticity in both directions.
The entire analytic elements of a real action, as a nonunital star subalgebra.
Instances For
Membership in the entire analytic subalgebra is entire analyticity.
The generator #
Entire analytic elements lie in the domain of the generator.
On an entire analytic element the generator is the complex derivative at time zero of the complex-time extension.
The complex-time extension of the generator value is the derivative of the extension.
The generator at a complex-time translate is the complex derivative of the extension there:
δ(α_z(a)) = (d/dz) α_z(a).
The generator maps entire analytic elements to entire analytic elements.
The entire analytic subalgebra is contained in the generator domain.
Invariant functionals #
Invariant functionals are invariant in complex time. If a continuous linear functional
φ satisfies φ (α_t a) = φ a for all real t, then φ (α_z b) = φ b for every entire
analytic b and every complex time z.
For an invariant continuous linear functional φ and entire analytic a, b,
φ (α_z(a) α_w(b)) = φ (a α_{w - z}(b)).
For an invariant continuous linear functional φ and an entire analytic a,
φ ((α_z(a))⋆ α_w(a)) = φ (a⋆ α_{w - conj z}(a)).
For an invariant continuous linear functional φ and an entire analytic a, imaginary-time
correlations add: φ ((α_{is}(a))⋆ α_{it}(a)) = φ (a⋆ α_{i(s + t)}(a)) for real s, t.
For an invariant positive functional ω and an entire analytic a, the imaginary-time
correlation ω (a⋆ α_{is}(a)) is nonnegative for every real s.
Examples #
For the trivial action every element is entire analytic.
For the trivial action the complex-time extension is constant.
In a unital algebra the identity is entire analytic.
The complex-time extension of the identity is constant.