An exponentiation theorem for unbounded derivations

A NNALES DE L ’I. H. P., SECTION A

J. F. G ^ILLE

An exponentiation theorem for unbounded derivations

Annales de l’I. H. P., section A, tome 13, n

^o

3 (1970), p. 215-220

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p. 215

An exponentiation ^theorem

for unbounded derivations

J. F. GILLE

(*)

ABSTRACT - We

give

^a sufficient

(and necessary)

^{condition to}

^define

the

exponential

^{of unbounded} derivations in

C*-algebras.

^’

Ann. Inst. Henri Poincaré,

Vol. XIII, n° 3, 1970, Physique théorique.

1. DEFINITIONS

Let j~ be ^a Banach

algebra,

^a derivation is a linear function D from ^a dense

sub-algebra

^of

~,

^into

d,

^{such that}

For a *-Banach

algebra d,

^the derivation D is said to be hermitian if:

The set of the elements x in d such that the function

exists and is

analytic

^{in some}

neighbourhood

^of

0,

^{is called « the set of the}

analytic

elements » with

respect

^{to this} derivation and is written

d(a).

(*) ^{Postal Address : Centre de} Physique Théorique, ^{C. N. R.} S., 31, ^{chemin J.} Aiguier,

13-Marseille (9~), ^France.

216 J. F. GILLE

2. THEOREM

Let j~ be a

C*-algebra,

^{D an} hermitian closed derivation of

~,

^such

as

d(a)

is dense in

d,

^{then D induces a}

strongly

continuous

group {03B1t|

^te

[?}

of

automorphisms

^{of ~.}

Proof

^{- If} &#x3E; 0 such that we can define :

which is

absolutely convergent

^{in d.}

ar(x)

^E

j~~B

^since

for I t’ I I

^we shall show that:

We write

now

absolutelv and uniformlv convereins on the same interval

J

We write zJ ⁼

1 (x~

t , n

^then

(z~)~

^{is a}

^Cauchy

^sequence

for ~ ~ - ~ ~

^ar.

n=0

So

D(Yj - yk)

^{converges to 0}

as j

and k go ^to

infinity.

^Let

EXPONENTIATION THEOREM FOR UNBOUNDED DERIVATIONS

hence

and

consequently

which is

absolutely converging

^as I goes ^to

infinity,

^{so we can rearrange}

the terms:

Through elementary calculations, taking advantage

^{of the}

absolutely

convergence of the series and of the

continuity

^{of * one}

gets :

for t E R

sufficiently

^small.

Moreover

dt E R, It 1

mtx ; ^{we write}

at is now well defined for all t E R on

d(a)

and fulfils

(2 .1 )

^and

(2 . 2)

^for every x

in ^.

ar is a

*-algebra isomorphism applying ^d(a)

^into

^d(a)

^and

^Vx

^E

t ^-

at(x)

^{is an}

analytic

^{function. We} shall extend _at to j~. We

can

assume

that .91 has a unit

element, for,

^if not, ^{we can} define D

on A

⁼ C x

.91,

the

algebra

^{obtained from .91}

by adjunction

^{of a unit}

^element,

Moreover,

we can assume that e E

~~ 1 ~ ;

because if not one settles :

D(~)=0.

218 J. F. GILLE

Note that ⁼ e because

D(e)

^{= 0. If} y ⁼

ao(y)

^is

invertible,

^there

exists a

neighbourhood

^of 0 such that is invertible. Now if t ^-

at(y)

is

analytic,

^{then t - is also}

analytic.

^{We can}

put

^{= 1}

so ==&#x3E;

y -1 E ~{a~

^for

x - ~,e invertible

==&#x3E; 3~

^and

(x -

^{= e}

=&#x3E; ar is well defined on y and

[Xf(x) 2014

^{= e}

therefore is

invertible ;

^hence

Spec’ at(x)

^c

Spec’

^x.

On the other

hand,

^{for an} hermitian element _y of ~ :

([7], 15 . 4 .14 .1 ) ;

^{hence :}

and

finally = II

^{on We extend} at ^to ~

(2 .1 )

^to

(2 . 5)

still hold

3 ( yn)n,

^{and x = lim} yn. Therefore

n

t ^-

rxt(x)

^is continuous as a uniform limit of continuous functions. So that the

one-parameter unitary group {03B1t|t

^E

R }

^is

^strongly

continuous.

Comment. We

get

^{an extension to}

C*-algebras

of the work of E. Nelson

on Hilbert spaces

([5]).

3. CONVERSE PROPOSITION

We

give

^{a new}

proof

of the result of Kastler-Pool-Poulsen

[4],

^which

improves

^{some one of I. Guelfand}

[3].

Let ð be a Banach

space, {03B1t}t~R

^a

^strongly

continuous

one-parameter

group of

uniformly

^{bounded linear}

operators,

^{i. e.}

~+oo

Vx

Vp

^~

W8(R) ;

^let

a(p)x

⁼

J+

^{which exists} in the Boch- J 2014 ⁰⁰

ner’s sense

since [) at(x)p(t) ~ ~ M ~ x I I /?(f)! I

^{and one has that :}

PROPOSITION.

~~e~ ( _ ~ x E ~ ~ t E ~ -~ rxlx)

^is

entire })

^{is dense in ~.}

Proof.

^{2014 Let} p be a function in

CR

so

that

^~ D. Then p ^e

03C6,

^and

~+00

Moreover,

suppose that

J-30

^We notice that

Vs

&#x3E;

0, 3~

&#x3E; 0 so that

Now, VB

&#x3E;

0, 3no

^{such that} no

ar(x) -

^B.

n

On the other hence

II [2(M

⁺

1)!! x II

⁺

I]B

^{and x = lim}

n

We prove that ^E

y

x E Indeed :

where

h (r)

⁼

ar(x).

_~

A

Now, h being

continuous and

bounded, and p h

⁼

p"h

^is

a. distribution

(cf. [6])

^with

compact support,

^{hence due to the}

Paley-

Wiener theorem 03C1n* h is an

entire

^function.

ACKNOWLEDGEMENTS

The author is _very indebted to Professor D. Kastler for his _encourag-

ments and for his constant interest in this work. Thanks also are due to Dr. J.

Manuceau,

^who

suggested

^the

problem

and for his constant aid.

M. A.

Messager

^{made us able to} illuminate some delicate

points.

^{We are}

indebted to Mr. M.

Sirugue

^et Mr. D. Testard for

reading

^the

manuscript.

220 J. F. GILLE

REFERENCES

[1] ^J. DIEUDONNÉ, Éléments d’analyse. Gauthier-Villars, Paris, 1965.

[2] ^J. DIXMIER, ^Les C*-algèbres ^{et leurs} représentations. Gauthier-Villars, ^{Paris, 1964.}

[3] ^I. GUELFAND, ^{C. R.} (Doklady) ^{Acad. Sc.} URSS, ^N. S., 25, 1939, 713.

[4] ^D. KASTLER, ^{J. C. T. POOL and E. T.} POULSEN, Commun. Math. Phys., 12, 1969, ^175.

[5] ^E. NELSON, Annals of Math., 70, 3, 1959, ^572.

[6] ^L. SCHWARTZ, Théorie des distributions. Hermann, Paris, 1966.

Manuscrit _reçu le 4 mars 1970.