One-parameter Groups

We introduce one-parameter groups, analytic functions with values in Banach spaces, analytic extensions of one-parameter groups and further properties of these objects in this section. The standard reference for this section isCior˘anescu et al.(1976) andKustermans (1997b). See also Kustermans’ notes inApplebaum et al.(2005).

Definition 1.3.1 LetX denote a Banach space, then a one-parameter group on X is a mapσ : RÑ BpXq, where we denote σ^t “ σptq for convenience, such that σ^t`s“ σ^t˝σ^s for all t, s P R, σ⁰ “ id and }σ^t} ď 1 for all t P R. If X is a Banach algebra and σ : R Ñ AutpXq we say it is a one-parameter group of automorphisms on X and similarly ifX is a Banach˚-algebra and σ : R Ñ Aut^˚pXq we say it is a one-parameter group of˚-automorphisms on X.

We sayσ is norm continuous (weak continuous, weak^˚-continuous) if for allxP X the map R Ñ X given by t ÞÑ σ^tpxq is continuous with respect to the norm topology (weak topology, weak^˚-topology) on X. If X is a von Neumann algebra then we can define similar properties with respect to any of the weak operator topologies.

Note it follows that σ_tis an isometry and invertible for all tP R with pσ^tq^´1 “ σ´t. This follows as for all tP R we have σ^t˝ σ´t “ σ⁰ “ id “ σ´t˝ σ^tand

}σ^tpxq} ď }x} “ }σ´tpσ^tpxqq} ď }σ^tpxq}

and thus we have equality throughout.

Remark 1.3.2 We have mentioned several possibilities of continuity for one-parameter groups in the above definition. We can have norm and weak continuity for a Banach space, for a Banach space with a predual we can have weak^˚ continuity and for a von Neumann algebra we haveσ-strong^˚,σ-strong, σ-weak, strong^˚, strong and weak opera-tor continuity. In this thesis we will mostly useσ-weak (or equivalently weak^˚) continuity for von Neumann algebras and norm continuity otherwise, however we mention that the Kustermans and Vaes’ defined one-parameter groups on von Neumann algebraic quantum groups with respect to theσ-strong^˚ topology. We will refer to a one-parameter group in this section when the results are not dependent on the choice of topology, however we will always assume some continuity property on all one-parameter groups.

We wish to discuss the analytic continuation of one-parameter groups, in order to do so we now give details of the analyticity of functions from a complex domain D into a Banach space X. We see from the following lemma that in fact analyticity is the same whether we are working with the norm, weak or weak^˚ topologies on X.

Notation 1.3.3 Letz P CzR, then we denote

Spzq “

$&

%

tw P C | Im w P r0, Im zsu if Im z ą 0 tw P C | Im w P rIm z, 0su if Im z ă 0

andSpzq^ois the interior ofSpzq.

The reader is referred to A.1 inTakesaki(2003b) for a proof of the following lemma.

Lemma 1.3.4 LetX be a Banach space, let D Ă C be a complex domain (i.e. an open connected subset of C) and letf : D Ñ X. Then the following are equivalent:

(i) For allw0 P D and δ ą 0 such that B^δpw⁰q Ă Spzq^o(whereBδpw⁰q is the open ball of radius δ around w0) there is a sequence pxⁿq⁸n“0 Ă X such that for all w with

|w ´ w⁰| ă δ the following is norm convergent

fpwq “ ÿ8 n“0

pw ´ w⁰qⁿxn;

(ii) For allωP X^˚ we have a holomorphic functionDÑ C given by w ÞÑ xfpwq, ωy;

(iii) LetY Ă X^˚be a norm closed subspace such that for allxP X we have

}x} “ sup t|xx, ωy| | ω P Y, }ω} ď 1u .

For eachω P Y we have a holomorphic function D Ñ C given by w ÞÑ xfpwq, ωy.

Definition 1.3.5 Let X be a Banach space and D Ă C be a complex domain, then a functionf : D Ñ X is an analytic function if any of the equivalent conditions in Lemma 1.3.4hold.

The following lemma belongs to complex analysis, we prove it here as it will be useful in this section.

Lemma 1.3.6 LetF : Spzq Ñ C be a function that is continuous, analytic on Spzq^o and Fptq “ 0 for all t P R, then F “ 0 everywhere.

Proof

We may assume without loss of generality that Im z ą 0 and we define a map G : Spzq Y Sp´zq Ñ C by

Gpwq “

$&

%

Fpwq if Im w ă 0 Fpwq if Im w ě 0.

Clearly G is continuous on Spzq and Sp´zq and thus everywhere. We also have Gptq “ Fp0q “ 0 for all t P R and so by the Schwarz reflection principle (see Theorem 11.14 in

Rudin(1987)) this is analytic on DompGq. From Theorem 10.18 inRudin(1987) we have that the set ZpGq “ tw P DompGq | Gpwq “ 0u is either DompGq or there is no limit point of ZpGq in ZpGq. However any t P R is a limit point of ZpGq with t P ZpGq and so we must have ZpGq “ DompGq. So we have shown that Gpwq “ 0 for all w P DompGq and thus F “ 0 as required. ✷

Note that we don’t specify a continuity property for the one-parameter group in the fol-lowing lemma, however for there to exist such an F in the lemma then σ would need to satisfy such a continuity property also.

Lemma 1.3.7 Let X be a Banach space, x P X, σ : R Ñ BpXq a one-parameter group on X and z P CzR. Say there exists a function F : Spzq Ñ X such that (i) F is continuous with respect to either the norm topology, weak topology or (if X has a predual) a weak^˚-topology onX, (ii) F is analytic on Spzq and (iii) F ptq “ σ^tpxq for all tP R. Then F is necessarily unique.

Proof

Fix z P C and let F¹, F2 : Spzq Ñ X be two functions satisfying these conditions. Let F : Spzq Ñ X be the map F “ F¹ ´ F², then clearly F is continuous, analytic and we have Fptq “ 0 for all t P R. Let ω P X^˚ and we consider the map G_ω : Spzq Ñ C given by Gωpwq “ xF pwq, ωy for all w P Spzq. Clearly G^ωptq “ 0 for all t P R. As F is analytic it follows from Lemma1.3.4that G_ωis analytic.

Say F1 and F2 are norm or weak continuous, then Gω is easily seen to be continuous for all ω P X^˚ and so it follows from Lemma1.3.6that G_ω “ 0 for all ω P X^˚ and so F1 “ F² as required. If there exists a predual X_˚ of X such that X –ⁱ pX˚q^˚ then for all ω P X˚ we have G_ωis continuous and as this separates X we can similarly conclude that F1 “ F². ✷

Finally we can introduce the analytic continuation of a one-parameter group.

Definition 1.3.8 LetX be a Banach space and σ a one-parameter group on X. Then we define the following:

(i) For any z P C we define Dompσ^zq as the set of x P X such that there exists a function (necessarily unique by the previous lemma)F : Spzq Ñ X such that (i) F is continuous with respect to the appropriate topology on X, (ii) F is analytic on Spzq^o and (iii)Fptq “ σ^tpxq for all t P R.

(ii) ForxP Dompσ^zq we define σ^zpxq “ F pzq for F : Spzq Ñ X the function in (i).

We sayσz is the analytic extension ofσ at zP C.

Proposition 1.3.9 LetX be a Banach space, σ a one-parameter group on X, z P C and xP Dompσ^zq. Then we have:

(i) For w P Spzq we have Dompσ^wq Ą Dompσ^zq and if Im w “ Im z we have Dompσ^zq “ Dompσ^wq;

(ii) For alltP R we have σ^tpσ^zpxqq “ σ^z`tpxq “ σ^zpσ^tpxqq;

(iii) σzis injective.

Proof (Sketch)

Part (i) follows by considering the restriction of the function from Definition1.3.8from Spzq to Spwq. Part (ii) follows from considering the function G : Spzq Ñ X given by w ÞÑ σ^tpσ^wpxqq ´ σ^t`wpxq. For part (iii) let x, y P Dompσ<sup>z</sup>q such that σ<sup>z</sup>pxq “ σ<sup>z</sup>pyq and let F, G : Spzq Ñ X be the functions that are are continuous, analytic on Spzq<sup>o</sup>, Fptq “ σ<sup>t</sup>pxq for all t P R and Gptq “ σ<sup>t</sup>pyq for all t P R. Then from (ii) we have σt`zpxq “ σ^tpσ^zpxqq “ σ^tpσ^zpyqq “ σ^t`zpyq for all t P R. By considering the function H : Spzq Ñ X given by Hpwq “ F pz ´ wq ´ Gpz ´ wq we find that we must have F “ G, and so x “ F p0q “ Gp0q “ y as required. ✷

Proposition 1.3.10 Let X be a Banach algebra, σ a one-parameter group of automor-phisms on X and z P C, then Dompσ^zq is a subalgebra of X and σ^z acts as a homo-morphism on Dompσ^zq. If in addition X is a Banach ˚-algebra and σ is a one-parameter group of˚-automorphisms then Dompσ^zq^˚ :“ tx P X | x^˚ P Dompσ^zqu “ Dompσ^zq and σzpxq^˚ “ σ^zpx^˚q for all x P Dompσ^zq.

Proof

Let x, y P Dompσ^zq and let F, G : Spzq Ñ X be the functions that are continuous, analytic on Spzq^o, Fptq “ σ^tpxq for all t P R and Gptq “ σ^tpyq for all t P R and thus σzpxq “ F pzq and σ^zpyq “ Gpzq. Let H : Spzq Ñ X be given by Hpwq “ FpwqGpwq for all w P Spzq, then H is a product of continuous functions and a product of analytic functions on Spzq^o and so is continuous and analytic on Spzq^o. Also we have Hptq “ F ptqGptq “ σ^tpxqσ^tpyq “ σ^tpxyq for all t P R. So xy P Dompσ^zq with σzpxyq “ Hpzq “ F pzqGpzq “ σ^zpxqσ^zpyq.

Now let X be a Banach˚-algebra with σ a one-parameter group of ˚-automorphisms.

Let x P Dompσ^zq, then there is some F : Spzq Ñ X that is continuous, analytic on Spzq^o and Fptq “ σ^tpxq for all t P R. Then consider the map G : Spzq Ñ X given by w ÞÑ F pwq^˚. It is easy to show that G is continuous and analytic on Spzq^o and Gptq “ F ptq^˚ “ σ^tpxq^˚ “ σ^tpx^˚q for all t P R. So we have x^˚ P Dompσ^zq with σzpx^˚q “ Gpzq “ F pzq^˚ “ σ^zpxq^˚. ✷

Proposition 1.3.11 Letσ, σ¹ denote two one-parameter groups of˚-automorphisms on a

˚-algebra M. Then σ “ σ¹if and only ifσz “ σ¹zfor anyz “ ti P C with t ‰ 0.

We now examine the tensor product of one-parameter groups to finish this subsection. Fix norm continuous one-parameters groups σ and τ on Banach spaces X and Y respectively throughout this section and fix a subcross norm} ¨ }^µon Xd Y and let X b^µY denote the completion of X d Y with respect to this norm. Then we define a one-parameter group pσ b τq : R Ñ BpX b^µYq in the rest of section. This work is largely influenced by that of Section 4 inKustermans(1997b).

For fixed t P R we have σ^t P BpXq and τ^t P BpY q and so we can consider σ^td τ^t : Xd Y Ñ X d Y for any t P R. We will assume that σ and τ are such that the map σ^td τ^t is continuous with respect to the} ¨ }^µand has norm less than 1.

Definition 1.3.12 For allt P R let pσ b τq^t: Xb Y Ñ X b Y be the unique continuous linear extension ofσtd τ^t.

For any z P C we have (unbounded) linear maps σ^z : X Ñ X and τ^z : Y Ñ Y defined in Definition1.3.8. We consider the map σzd τ^z : Dompσ^zq d Dompτ^zq Ñ X d Y such that xb y ÞÑ σ^zpxq b τ^zpyq and we have the following.

Proposition 1.3.13 For allz P C the map σ^z d τ^z given above is closable with closure equal to the analytic extensionpσ b τq^zofσb τ at z (see Definition1.3.8).