THE E_{0}-SEMIGROUP SENTINEL

VOLUME 1, NUMBER 1

Bill Arveson, Editorial Staff

13 April 1999

The purpose of these remarks is to report some news concerning type III E_{0}- semigroups. I’ve been working through Bob Powers’ remarkable paper [1], hoping to understand his construction in terms as simple as possible and to make connections with other parts of operator theory (like Wiener-Hopf operators and Hankel operators) where that seemed helpful. The following discussion summarizes what I have learned, and appears to clarify things to some extent.

For notation, ℝ̂ will denote the dual group of the additive group ℝ of real numbers, and I choose normalizations of Lebesgue measure on ℝ and ℝ̂ which cause irritating factors of \sqrt{2π} and its reciprocal, that put in unwanted appearances when taking Fourier transforms, to disappear. This causes the unit interval to have peculiar length; but that problem is secondary since we do no numerical calculations. Following the physicists, I will remind you of where we are by using the letters x, y for elements of ℝ and p, q ∈ ℝ̂ for elements of ℝ̂. C(ℝ̂) will denote all continuous functions f : ℝ̂ → ℂ which have a limit at the point at ∞,

f(∞) = lim_{p→∞} f(p).

For N = 1, 2, …, L^{2}(ℝ) ⊗ ℂ^{N} denotes all vector functions

ξ : ℝ → ℂ^{N}

which are square integrable in the usual sense (and of course we regard ℂ^{N} as an N dimensional Hilbert space in the usual way). L^{2}[0, ∞) ⊗ ℂ^{N} denotes the subspace of functions vanishing almost everywhere on the negative axis, and

P_{+} : L^{2}(ℝ) ⊗ ℂ^{N} → L^{2}[0, ∞) ⊗ ℂ^{N}

is the natural projection. Finally, M_{N}(C(ℝ̂)) denotes the C∗-algebra of all N × N matrices over C(ℝ̂)...whose elements we often consider as matrix-valued functions defined on ℝ̂.

For every Φ ∈ M_{N}(C(ℝ̂)) we define a generalized “convolution” operator C_{Φ} on L^{2}(ℝ) ⊗ ℂ^{N} by

Ĉ_{Φ}ξ(p) = Φ(p)ξ̂(p), p ∈ ℝ̂,

The Sentinel is pleased to acknowledge help with reportage from Schnitzel and Pretzel.

where the hat denotes the Fourier transform

ξ̂(p) = ∫_{−∞}^{∞} e^{−ipx}ξ(x) dx.

C_{Φ} is a singular integral operator, but we require only elementary technology here. Let R be the natural reflection operator defined on Ł^{2}(ℝ) ⊗ ℂ^{N} by Rξ(x) = ξ(−x), x ∈ ℝ.

Definition. For every Φ ∈ M_{N}(C(ℝ̂)), the operator H_{Φ} defined on L^{2}[0, ∞) ⊗ ℂ^{N} by

H_{Φ} = P_{+}RC_{Φ} ↾_{L^{2}[0,∞)⊗ℂ^{N}}

is called the Hankel operator with symbol Φ.

Remarks. I am calling H_{Φ} a Hankel operator because it satisfies the abstract definition of Hankel operator on L^{2}([0, ∞)) ⊗ ℂ^{N}, namely H_{Φ}S_{t} = S_{t}^{∗}H_{Φ}, t ≥ 0, where S = {S_{t} : t ≥ 0} is the shift semigroup

S_{t}ξ(x) = \{ \begin{matrix}ξ(x − t), if x ≥ t \\ 0, if 0 ≤ x < t. \end{matrix}

If Φ happens to be integrable in the sense that ∫ℝ̂ ∥Φ(p)∥ dp < ∞, then one can check that C_{Φ} is an integral operator with a “Hankel” kernel

C_{Φ}ξ(x) = ∫_{0}^{∞} Φ̂(x + y)ξ(y) dy, x ≥ 0,

where Φ̂ ∈ M_{N}(C(ℝ)) is the Fourier transform of Φ

Φ̂(x) = ∫_{−∞}^{∞} e^{ipx}Φ(p) dp.

Moreover, in this case one can verify by routine changes-of-variables that the following are equivalent

(1) H_{Φ} and H_{Φ^{∗}} are Hilbert-Schmidt operators.

(2) ∫_{−∞}^{∞} |x| trace|Φ̂(x)|^{2} dx < ∞ ,

where as usual trace|A|^{2} denotes trace(A^{∗}A). Note that the condition (1) involves H_{Φ^{∗}} and not the adjoint of H_{Φ}.

Here are the two results I was able to squeeze out of Bob’s work.

Theorem A. Let N = 1, 2, …, let Φ ∈ M_{N}(C(ℝ̂)) satisfy 0 ≤ Φ ≤ 1, and let A be the Wiener-Hopf operator defined on the Hilbert space H = L^{2}[0, ∞) ⊗ ℂ^{N} by

A = P_{+}C_{Φ} ↾_{L^{2}[0,∞)⊗ℂ^{N}} .

Let ω_{A} be the gauge-invariant quasifree state defined on the CAR algebra A over the one-particle space H having two-point operator A. For f ∈ H we write c(f) for the creation operator associated with f ∈ H. Then the following are equivalent.

(1) ω_{A} is a type I factor state and for π the GNS representation of A associated with ω_{A}, there is a unique E_{0}-semigroup α = {α_{t} : t ≥ 0} acting on π(A)′′ such that

α_{t}(π(c(f))) = π(c(S_{t}f)), t ≥ 0, f ∈ H.

(2) Φ(p) is a projection for every p ∈ ℝ̂ and the Hankel operator H_{Φ} is of Hilbert-Schmidt class.

Remarks. The Hilbert-Schmidt condition can be characterized as follows. Let ϕ_{ij}(p) be the ijth component of the N × N matrix Φ(p). The Fourier transform ϕ̂_{ij} of ϕ_{ij} is a tempered distribution on ℝ, and H_{Φ} and H_{Φ^{∗}} are Hilbert-Schmidt operators iff for every 1 ≤ i, j ≤ N ϕ̂_{ij}(x) is a function away from x = 0 (more precisely, its singular set is a subset of {0}), and we have

(3)  sup_{ϵ>0} ∫_{ℝ\[−ϵ,ϵ]} |x| · |ϕ̂_{ij}(x)|^{2} dx < ∞.

In [1], Powers takes N = 2 and the symbol

(4)  Φ(p) = 1/2 ( \begin{matrix}1 & e^{iθ(p)} \\ e^{−iθ(p)} & 1 \end{matrix} )

where θ is the function

(5)  θ(p) = \frac{1}{(1 + p^{2})^{1/5}} ,  p ∈ ℝ̂,

and he proves (3) more-or-less directly for this particular Φ, as part of his Lemma 4.2. Note too that the ϕ̂_{ij} associated with (4), (5) are distributions, not functions.

Of course, the real issue here is the existence of units for the E_{0}-semigroup α. Here is my perturbation of what Bob does in section 4 of [1]: it gets rid of the somewhat mysterious contradiction employed in the proof of his Lemma 4.6, and gives a concrete condition that the symbol Φ must satisfy.

Theorem B. Let Φ ∈ M_{N}(C(ℝ̂)) satisfy the conditions of Theorem A (so that we obtain an E_{0}-semigroup α). If α has a unit, then Φ has the following property:

(6)  ∫_{−∞}^{∞} trace|Φ(p) − Φ(∞)|^{2} dp < ∞.

Remarks. Notice that in example (4), (5) we have

Φ(p) − Φ(∞) = 1/2 ( \begin{matrix}0 & e^{iθ(p)} − 1 \\ e^{−iθ(p)} − 1 & 0 \end{matrix} )

so that

trace|Φ(p) − Φ(∞)|^{2} = 1/2|e^{iθ(p)} − 1|^{2} = 1 − cos θ(p).

When |p| is large θ(p) tends to 0 and we have

1 − cos θ(p) ∼ θ(p)^{2}/2 = \frac{1}{2(1 + p^{2})^{2/5}} ∼ \frac{1}{2|p|^{4/5}} .

But since 4/5 < 1, our calculus students know that ∫_{1}^{∞} |p|^{−4/5} dp diverges, hence the integral on the left side of (6) is infinite. Conclusion: By theorem B the associated E_{0}-semigroup has no units.

The condition of Theorem B shows that there are many examples of type III E_{0}-semigroups that arise out of Bob’s construction of Theorem A. But what we are still lacking is a criteria for cocycle conjugacy (in terms of the symbol Φ).

Conjecture. The condition (6) is equivalent to the existence of units for the E_{0}- semigroup α.

The conjucture seems reasonable to me, but after going through all of this in a seminar I have lost some of my steam on these issues. If somebody out there is motivated to attack the conjecture, I can supply the details that are only alluded to above (but only as handwritten lecture notes).

4 BILL ARVESON, EDITORIAL STAFF

References

1. , A non-spatial continuous semigroup os ∗-endomorphisms of B(H), Publ. RIMS (Kyoto University) 23 (1987), 1053–1069.