The Fuglede–Putnam–Rosenblum theorem

Let A be a C⋆-algebra, and let a b x : A. The Fuglede–Putnam–Rosenblum theorem states that if a and b are normal and x intertwines a and b (i.e., SemiconjBy x a b, that is, x * a = b * x), then x also intertwines star a and star b. Fuglede's original result [Fug50] was for a = b (i.e., if x commutes with a, then x also commutes with star a), and Putnam [Put51] extended it to intertwining elements.

Rosenblum [Ros58] later gave the elementary proof formalized here using Liouville's theorem which proceeds as follows. Consider the map f : ℂ → A given by z ↦ exp (z • star b) * x * exp (z • star (-a)). When x intertwines a and b (i.e., SemiconjBy x a b), then it also intertwines exp (star z • a) and exp (star z • b). Then the map f can be realized as z ↦ u * x * v for fixed unitaries u and v. In fact, u = exp (I • 2 • ℑ (z • star b)) and v = exp (I • 2 • ℑ (star z • a)); it is here that normality of a and b is used to ensure that exp (star z • a) * exp (- star (z • a)) = exp (I • 2 • ℑ (star z • a)) and likewise for b. Therefore ‖f z‖ = ‖x‖ for all z, and since f is clearly entire, by Liouville's theorem, f is constant. Evaluating at z = 0 proves that f z = x for all z. Therefore, exp (z • star b) * x = x * exp (z • star a). Differentiating both sides and evaluating at z = 0 proves that star b * x = x * star a, as desired.

In a follow-up paper, Cater [Cat61] proved a number of related results using similar techniques. We include one of these below, isStarNormal_iff_forall_exp_mul_exp_mem_unitary, but the proof is independent of the Fuglede–Putnam–Rosenblum theorem.

Main results

• fuglede_putnam_rosenblum: If a and b are normal elements in a C⋆-algebra A which are interwined by x (i.e., SemiconjBy x a b, that is, x * a = b * x), then star a and star b are also intertwined by x.
• isStarNormal_iff_forall_exp_mul_exp_mem_unitary: A characterization of normal elements in a C⋆-algebra in terms of exponentials.

References

• [Fug50] Bent Fuglede, "A commutativity theorem for normal operators"
• [Put51] C. R. Putnam, "On normal operators in Hilbert space"
• [Ros58] M. Rosenblum, "On a theorem of Fuglede and Putnam"
• [Cat61] S. Cater, "Observations on a paper by Rosenblum"

theorem SemiconjBy.star_right {A : Type u_2} [NonUnitalCStarAlgebra A] {a b x : A} (ha : IsStarNormal a) (hb : IsStarNormal b) (h : SemiconjBy x a b) : SemiconjBy x (star a) (star b)

Fuglede–Putnam–Rosenblum: If a and b are normal elements in a C⋆-algebra A which are interwined by x, then star a and star b are also intertwined by x.

theorem fuglede_putnam_rosenblum {A : Type u_2} [NonUnitalCStarAlgebra A] {a b x : A} (ha : IsStarNormal a) (hb : IsStarNormal b) (h : SemiconjBy x a b) : SemiconjBy x (star a) (star b)

Alias of SemiconjBy.star_right.

---

Fuglede–Putnam–Rosenblum: If a and b are normal elements in a C⋆-algebra A which are interwined by x, then star a and star b are also intertwined by x.

theorem IsStarNormal.commute_star_right {A : Type u_2} [NonUnitalCStarAlgebra A] {a x : A} (ha : IsStarNormal a) (h : Commute x a) : Commute x (star a)

Fuglede–Putnam–Rosenblum: If a is a normal element in a C⋆-algebra A which commutes with x, then star a commutes with x.

theorem IsStarNormal.commute_star_left {A : Type u_2} [NonUnitalCStarAlgebra A] {a x : A} (ha : IsStarNormal a) (h : Commute a x) : Commute (star a) x

Fuglede–Putnam–Rosenblum: If a is a normal element in a C⋆-algebra A which commutes with x, then star a commutes with x.

theorem isStarNormal_iff_forall_exp_mul_exp_mem_unitary {A : Type u_1} [CStarAlgebra A] {a : A} : IsStarNormal a ↔ ∀ (x : ℝ), NormedSpace.exp (x • a) * NormedSpace.exp (-x • star a) ∈ unitary A

A characterization of normal elements in a C⋆-algebra in terms of exponentials.