EXTENDED.

C*-ALGEBRASAND

W*-ALGEBRAS

byJ. B. COOPERIntroduction : One of the most successful and attractive branches of modern
functional analysis is. the theory of Banach algebras and its application to thetheory of bounded operators in Banach and Hilbert spaces. As is well known, it isnecessary, for many applications, to consider unbounded ‘operators and, as far
as we know, there is no completely satisfactory axiomifation of this theory, corresponding to that for bounded operators. The starting point for generalisationsof the theory of Banach algebras has usually been a topological algebra withsome additional properties (e.g. locally convexity). In order to define a spectrum of an element of the algebra which would correspond to the classicalspectrum of an unbounded operator, it is necessary to introduce a subalgebra
whoseelements

correspondtoboundedoperators.

Essentiallyequivalentdefinitions of such bounded elements (in slightly different contexts) were given
by Warner,

Waelbroeck, and Allan.

In this lecture,

we would

like to follow a

different course- we begin with a “’core”Banach algebra of bounded elements”
and consider certain algebraic extension. We then use the normed structure of
the core algebra to define a suitable structure on its extendsion. It turns outthat the convenient structure is a bornology and not a topology. Our results andmethods are close to those of Allan and indeed it was the reading of his workthat motivated us. However, by abandoning the original topology used by Allan(which we believe to be irrelevant), one obtains a certain simplification" and anumber of results become more precise.
In this lecture, we consider only commutative *-algebras. The methods weuse are essentially a formalisation of familiar techniques used to extend the spectral theory for bounded operators to unbounded operators. We have coticentrated on obtaining representationsof our algebras as algebras of functions.

One

can then easily develop the usual theory (definition of the spectrum of an element,a functional calculus, spectral mapping theorems, etc.). We have also given what
we believe to be the natural extension of the concept of a. W*-algebra (algebraof operators) to algebras of unbounded operators and have given, in this context,
an abstract version of the spectral theorem for unbounded operators in Hilbert
space.

76

COOPER, J. B.; “Extended C* -Algebras and W* -Algebras”

Notation: Let E be a vector space over €. A subset B of E is a ball if it isabsolutely convex and contains no line. We then define Ep to be the linear
--]

span

of

B(:=

U

nB) and regard it as a normed

space under the norm

n=1||| 8: x1—>inf {MA : xe) B}

B is a Banach ball if (Eg, || || 8) is a Banach space.
A convex bornological space (abbreviated CBS) is a vector space E together
with a family ®B of balls which is directed on the right by inclusion and whose
union is E. If (E, B) is-a CBS, BSFE is defined to be bounded if B< B, for some
B,in 8. (E,®) is a complete CBS if there is a subset B; ‘of B consisting of
Banachballs which

CBS’s, a linear

contains

mappingan oversetof eachBe 8.

T : E—> F is bounded if it maps

If (FE, 8), (F,%,)boundedaresets of FE

into bounded sets in F..If E has a locally convex space structure, the set ofbounded, absolutely convex subsets of E defines a bornology on E -it is called
the von Neumann bornology.

Let A be an algebra over €. An involution on 4 is an antilinear mappingx => x* from A into itself so that (xy)* = y*x* and (x*)¥ = x for eachx,y € A. An algebra with involution is called a *-algebra. A *-algebra A with unit
e is symmetric if (e + x* x) is invertible for each xe E. If X is a set, the space§ (X) of all'mappings from X into & has a natural *-algebra structure (given bypointwise algebraic operations and complex conjugation as involution). We will
be concerned with subalgebras of §(X), closed under complex conjugation,
and will always regard them as *-algebras with the induced structure.
A C*-algebra is a@@) (4,11*-algebra A together with a norm so that1]) 1s a Banach algebra;(i) || x*x]||= || x||* for each x €A.
We shall only be intereseted in commutative C*-algebras with a unit and so shalluse the name “C*-algebra” in this restricted sense. If X is a compact space, then
C(X), the space of continuous functions in & (X) is a C*-algebra with the supre-

mum norm
xl->sup {|x(t)|:te X}
If Ais a C*-algebra, the spectrum M(A) of A is defined to be the set of linear mul-

tiplicative functions f on 4 which map e into 1. M(4) is a o(A4’, A)-compact
subset of A” and we regard it as a compact space with the induced topology. If
x € A, then the mappingx:

fil>f(x)Proceedings of the Symposium on Functional Analysis

77

from M(A) into ¥ is continuous, i.e. is an element of C(M(A)). We have thus constructed a mapping XI>x from A into C(M(A4)).
can be summarised in the theorem :The theory of C*-algebras

The Gelfand-Neumark

isomorphism

transform

is a C*-algebrafrom

A ontoC(M(4)).
A C*-algebra is said to be a W*-algebra if, as a Banach space, it is the dualof a Banach space. Let M be a locally compact space which is the topologicalsum of a family of compact sets, p a positive Radon measure on M. Then theLebesgue spaces L!(M; pn) and L*=(M; p) are Banach spaces and L®(M; pn) can be
identified with the dual of LY(M; 1). L=(M; wn) has a natural C*-algebra structureand so is a W*-algebra. The representation theory for W*-algebras is as follows:
Let A be a W*-algebra. Then A is isomorphic {as a C*-algebra)
L*(M; pn) (M, pn as above).
§ 1. Extended

to a W*-algebra

C*-algebras.

1.1. Definition : Let (4,, |] ||) be a C*-algebra and suppose that 4, is embedded
as a subalgebra of an algebra 4 with involution so that(i) e is a unit for 4 ;(ii) for each xe 4, (e+x*x)
lie in A4,.

is invertible and (e4+x*x)™

and x(e+4x*x)™!

Then we call A an extended C*-algebra over A, .

We note that this implies that every element of A4 is expressible in the formyz~! where y, z€ A, i.e. 4 is a quotient algebra over 4, and so 4 is commutative.
We define a barnology 8 on A as follows: Let B(4,) denote the unit ball of(4g, Il ||). Then a subset B of 4 isin B if there is a ze 4, (z invertible in A)

so that BC B(4y) 27.
1.2. Proposition: (i) (4, ||||) is a complete CBS;(ii) x|1> x* is a bounded mapping ;(iii) multiplication is a jointly bounded mapping.
Proof. (i) It is clear from the above remark that 4 = u B. Also the mapping’x1-> xz"!

is -linear and

maps

B(4,)

Beep

onto

B;: = B(4,) z'

andso the latter

is a ball. The same mapping defines an isomorphism between (4,, ||the normed space Ag, and so B, is even a Banach ball.
(#7) Involution maps B(A4y)z™" onto B(A4y) (z*)~.
(iii) Multiplication

maps

B(4,) z,™! X B(4y) z,”!

into

B(A4y) (z, 2)! .

||) and

78

COOPER J. B.; "Extended C* -Algebras and W* -Algebras”

1.3. Examples: A. Let X be a completely regular topological space and let 4, be
the C*-algebra C*(X) of bounded, continuous, complex-valued functions on X

with the supremum norm. If A is the algebra C(X) of all continuous, complexvalued functions on X (with complex conjugation as involution), then 1({) and(ii) are obviously satisfied and so we can regard C(X) as an extended C*-algebra
over C*(X). The bornology of C(X) defined above is easily seen to be the bornology with a basis {B,: xe C(X)} whereB,:={yeCX):|y|l =< |x|}
If X 1s locally compact and countable at infinity, then this is the von Neumannbornology associated with the structure of compact convergence on X.
B. Let A be a commutative

algebra "of unbounded

operators

in a Hilbert

space H. That is, 4 is a set of closed, densely defined partial linear operators in
H so that(i)

ifSeA,

thenS*e 4;(i)

if S,Te A, then S+

T and ST are densely defined and closeable andtheir closures lie in A (we write $+ T and S.T for their closures);(Fi) if S,Te A, then ST=T.5;(iv) Ie A and if Te A, then ([+ T*T)™
(v)

ANL(H)isandT(/4 T*T)™'

are in 4;

norm-closed.

Then A is an extended C*-algebra over

AN L(H).

C. Let M be a locally compact space, i a positive Radon measure on M.
We consider the C*-algebra L=(M ;p) of equivalence classes of‘complex-valued
functions which are bounded (on the complement of a locally p-negligeable set).

Let S(M;p) denote the *-algebra of equivalence classes of p-measurable
complex-valued functions. Then 1(i) and (ii) are satisfied and we can regard
S(M ;p) as an extended

C*-algebra

over

L*(M ;p).

The

bornology

definedon S(M ;p) has as basis the sets {B,:x€S (M ;p.)} whereB,:={yeSM;p):|y| < |x| (locally p—a.e.)}
D. Let X be a compact space. We consider pairs (Y, 4) where(i) Yis an open dense subspace of X;(if) Ayis a symmetric *-subalgebra of C(Y) so thatNow we consider a family &Ay Nn C*(Y)= C(X):of open, dense subsets of X which containsX and isclosed under the formation of finite intersections and suppose that for each Ye &,we are given an algebra Ay as above. In addition, we suppose that if ¥Y, Ze &,YSZ, then Az < Ay. Then the system {dy : Ye &} with the natural injectionProceedings of the Symposium on Functional Analysis

79

mappings form an inductive system of *-algebras. Their union is an extended
C*-algebra over C(X)-we denote it by Cp (X). The bornology of Cg (X) has as
basis the sets {By,: Ye,

xe Ay}

where.

By,x:={yedy:|y|<|x|onY}.
(Note that example 4 corresponds to the special case where & is the pair {X, 8X}

and Ay is C(X)).
1.4. Definition. Let A (resp. B) be an extended C*-algebra over A, (resp. By).
Then an (extended C*-) morphism from A into B is a unit-preserving *-algebramorphism from 4 into B which maps 4, into By . If the morphism is bijective
and maps 4, onto B,, it is called an (extended C*-) isomorphism.
1.5. Proposition:bounded.

Let

A, Bbeasind,© a morphism from A into B. Then ® is

Proof. Since ® maps 4, into B, , it maps the unit ball B(4,) of 4, into the unit
ball B(B,) of B,. Hence it maps B(A4y)z™! into B(B,) {®(z)}"?! (z invertiblein A).
1.6. Proposition. Let A be an extended C*-algebra over Ay, x the spectrum M(A,)
of Ay.

Then

there is a family &of dense,

open

subsetsof X with associatedalgebras {Ay:Y € &} so that A is isomorphic tv Cg (X).
Proof. Let M denote the multiplicative semigroup in 4, consisting of thoseelements which are invertible in 4. If ze M, define
Yz:= {fe M(4):f(2) = 0)
Then Yz is open in X. It is dense. For suppose that X\ Yz contains an open
set. Then we can find a positive function in C(X) which vanishes on ¥z andwe can suppose that this function is the image under the Gelfand-Neumark
transform (for 4;) of an element x of 4,. Then xz = 0 and so z is a divisor ofzero in A, and so also in 4 which is impossible (z is invertible in A).

Let &¥:={Yz:zeM}.

Then

YN

Yz, =Yzz

and

so &

is closed underfinite intersections. If ze M, put
Mz:

=

{z,e

M

: Sz,

=

Sz}

and define 4, to be the set of those elements of 4 which can be written in theform xy™ (xe A,, ye Mz). Az is a *-subalgebra of 4 and the Gelfand-Neumark

transform x —morphismfrom Azx

from4, onto

C(X)

¢an

be

extended

to aonto analgebraof functions in ¥Y:= Y,*-algebra

by mapping

xy? onto (x|y)/(y|y). We denote this algbra by Ay. Ay is symmetric since Az

80|

1s. WeCOOPER

show

that,J.B. ; "Extended C¥* -Algebras and W* Algebras”

Ay N C*(Y)= C(X), i.e. that each bounded function in Ay

can be extended continuously to X. Such a function is the image x under theextended Gelfand-Neumark transform of an element x of A. Then the function{(e + x*x)71}"" is bounded away from zero on Y and so on X. Hence (e+x*x)"

is bounded on X and the product{x(e-+ x*x)7'}""(e + x*x)” is the required

extension.
1.7. Corollary: Let A be an extended C*-algebra over A,. Then B(Ay) is the largest idempotent, self-adjoint, absolutely convex bounded subset of A.
1.8. Corollary: If A (resp. B) is an extended C*-algebra over A(resp. B) and ®
is a bounded unit-preserving *-algebra morphism from A into B, then © is a morphism.
1.9. Corollary: Let A be an extended C*-algebra over Ay. Then A is isomorphic
to the algebra of continuous functions on a completely regular space (cf. 3A) if thefollowing conditions hold:@) Y:=N

SL is a dense subset of X: = M(Ay)and X is the Stone-Cech

compactification of Y ;(ii) if xe A does not vanish on Y, then x is invertible in A.
1.10. Definition: A linear form
f(x*x) = 0 for each xe A.

f on

1.11. Proposition:bounded,linear functional(ii) if Nn(i) A positive

anextendedC*-algebra

is positive if

on an extended C*-algebra isFis dense in M(Ay), then the set of positive linear forms oni A separates A.

1.12. Proposition: Let A be an extended C*-algebra.
linear form f on A is bounded.

Then every multiplicative

Proof. The restriction of f to 4, is bounded in absolute value by one on B(4,).

Hence if ye B(4,).z7!

then [f(»)| < 1/2).

1.13. Remark: We mention some categorical constructions in the category ofextended C*-algebras. If 4 is an extended C*-algebra over A4,, a subobject of A
is a *-subalgebra B of A containing e so that(i) B [) 4o is closed in 4, ;(i) (e+x*x)"! is in B for each xe B.
Then B is an extended C*-algebra over BN4,.

Proceedingsof the Symposium on Functional Analysis

81

Now let I be a self-adjoint ideal in 4 so that IN A4,is closed in 4; (note that
this is equivalent to the closedness of 7 in the bornology of 4). Then A/I (with its
natural structure as a *-algebra)

is an extended

C*-algebra overA4y/INA4, andthe resulting bornology is the quotient bornology.
If {A':ieI}

A:= I1 A!

has

is a familya naturalof extended

C*-algebras,

structure of a *-algebra

ier

then

with a unit.

theproduct

Consider the}subset B:= |] B(d%). Then this is the unit ball of a C*-algebra AoCA (4, isiel
the C*-algebra product of theC*-algebras 4) and A is an extended C*-algebra

over A, . Similarly, the unrestricted projective limit- of a projective
extended C*-algebras is an extended C*-algebra.

system of

1.14. Example:

Let {jo : A. > A4,, a < B, a, Bel}be a projéctive system

of C*-algebras

and let A be its projective limit (in the

category of sets). Then 4

has

the

structureof acommutative*-algebra

with a unit.

Let 4,

be theC*-algebra projective limit of the system. Then 4, < 4 and A is an extended
C*-algebra over A. The bornology of A is precisely the projective limit of thehorm bornologies on 4, and so is the von Neumann bornology of the projective
locally convex structure on A.

1.15. Example: Let H be a Hilbert space, 57 a family of closed subspaces withthe following properties :.
.
(i) or, ordered by inclusion,

is a directed set;(i) {J K is dense in H.
xX
Suppose given a family {4x: Ke H} where, Ax is a commutative C*-algebraof operators on K so that if K<K; then K is invariant under Ax, and Ag
coincides with the set of restrictions of the operators ofAx,

to K. Then if wewrite jk, x for the natural mapping from Ag, onto Ag,{jx,.x : Ax,—> Ax, K<K, , K, K, € 24}
is a projective system. The C*-algebra projective limit of this system can be
identified with a C*-algebra 4, of operators on H and the projective limit
described in 14 can be identified with an algebra of unbounded operators in H.

1.16. Remark:

We comment

briefly on the relation

betweenthe aboveandthe results of Allan. If 4 is commutative Cp-*algebra (see the papers of Allan
quoted below

forthe definition), then the subalgebra 4, of bounded

elements

defined by Allan is a C*-algebra and 4 is an extended C*-algebra over 4,.
The bornology defined in 1 is.finer than the von ‘Neumann bornology of A.

82

COOPER,J. B.; "Extended C* -Algebras and W* -Algebras”§ 2. Extended JW *-algebras.
2.1. Definition: Let 4, 4, be extended C*-algebras over 4,. We say that 4, 1s
greater than A (written A, > A) if there is an injective morphism ® from A into

A, extending the identity on A,.
« &A, —/> 4

\°

NW

A

2.2. Proposition: Let A and A; be extended C*-algebras over A, and let inv (Ay)(resp. inv; (4y)) denotethe setof "elements

of

A,

which

are

invertibleinA(resp. in Ay). Then A; > A if and only if inv (4) <inv, (4).
Proof. The necessity of the condition is clear. Suppose that it is fulfilled. Then

® can be defined by mapping xz(xe 4, zeinv(4,)—z™'

denotes the inversein A) into xz7! (inverse in A4,).
2.3. Definition:

Let 4,A, is an extended
ordering.

be a W*-algebra.

C*-algebra

over 4,Thenan extended

W*-algebra

which

is maximal

under

over

the above

24. Lemma: Let A be an extended C*-algebra over a W*-algebra Ay. Then A isan extended W*-algebra over A, if every non zero divisor in A, is invertible in A.

2.5. Example: Let M be a locally compact space which is the direct sum of compact spaces and let i be a positive Radon measure on M. Then L*(M;p) is a
W*-algebra and S(M ;p) is an extended W*-algebra over L>(M ;p). (For x
is a non zero' divisor in L*(M ; p) if and only if p{te M: x(t) = 0} = 0 andthen x is invertible in S(M ; p)).
2.6. Proposition: Let A be an extended W*-algebra over Ay.
pair (M ; 1) as in 5 so that A is isomorphic to S(M ; 1).
Proof. By

the representation

theoremfor

W*-algebras,

Then there is a4, is isomorphic

to aspace L®(M ; p). Then, up to isomorphism, S(M ; pn) and A are extended W*-algebras over L*(M ; pu). But it follows immediately from 2 that there is at mostone

extendedW#*-algebra over a given

W*-algebraandso A andS(M; p)are isomorphic.

2.7. Remark:Let 4 be an extended W*-algebra over 4,. We give A an ordering

as follows :x = 0 if and only if x = y*y for some ye A.

Proceedings of the Symposium on Functional Analysis

83

Then A, with this ordering, is a Dedekind complete vector lattice (this follows
immediately from 6).
2.8. Definition: Let 4 be an extended

W*-algebra over 4,. "We consider thebasis

B, : = {B(4,)z7! : zeinv (4,)}

for the bornology 8. If Be 8, , then Ap is isomorphic as a Banach space to 4,
and so ‘we cdr transfer the

Mackeytopology#(4,, F) to a-topology -on

Ag(F denotes the predual of 4;). We say that a net (x, ) in A converges boundedultrastrongly to x, in A if there is a Be ®, so that x, — x, in this topology

on Ag. This is equivalent to the existence of a zeinv (4,) so that {x, z} isbounded in 4g and x, z = x, zn

7(4,, F).

279. Example:

W*-algebraboundedIn theultrastrongly

exténded
if andonly if there

S(M; nu) (seeis a strictly5) thenpositivex, — x,measurable

function z'so that {x,z} is bounded and x, z—>'x,z in L'(K; px) for eachcompact subset K of M (ux is the restriction of p to K).
2.10.- Definition:

Let 4 be an extended

W*-algebra over 4,. An element p of

A is said to be a projection if it is selfadjoint and idempotent (then pe A,
and is a projection in A;). If x is an element of A, there exists a smallest
projection p so that px = x. We call this the support of x. (If A is the extended
W*-algebra S(M ;u), then the projections are precisely the functions of the form
xn - the characteristic function of the measurable subset N of M. If xe S(M ; un),the support of x is yn whereN:={te M : x(t)0}).

2.11. Notation: Let 4 be an extended W*-algebra over 4,, x a self-adjoint element of A. Then for each Ae 2’, we denote by e()\) the support of (Ae— x)*(: = sup {(Ae— x), 0}).
2.12. Proposition: Let x be a self-adjoint element of an extended W*-algebra A.
Then@OfFr<spinX, eM) < e(p);
(@)

lim
AS—o0

i)

e(A)=0,lime(M) =e(in o(4,, F));A~®

if Mt Min XH, then e(M,)—> e(\)

in o(A4,, F);(iv) x = / A de (\) in the sense that the natural Riemann-Stieltjes
A
this integral converge to x bounded ultrastrongly,

Proof. This follows from the representation theorem 6.

sums for84

COOPER, J. B, ; "Extended C* -Algebras and W* -Algebras”

BIBLIOGRAPHICAL REMARKSThe theory of C*-algebras and WW *-algebras is presented in
S. SAKAI,

C*-algebras andWW *-algebras (Berlin, 1971).

See also
J. DIXMIER, Les

C*-algébres et leurs représentations (Paris, 1964).

J. DIXMIER, Les algébres d’opérateurs dans I’espace Hilbertien (Paris, 1969).
The Lebesgue spdces used above are defined in

N. BOURBAK], Eléments de Mathématique. Livre VI, Intégration. (Paris, 1965).
Allan’s papers are as follows:
G.R. ALLAN, A spectral theory for locally convex algebras, Proc. London Math. Soc.
15 (3) : 399-421 (1965).
G.R. ALLAN, On a tlass of locally convex algebras, Proc. London Math.
91-114 (1967).

Soc. 17(3):Mathematisches Institut
Hochschule Linz
Linz - Auhof

Osterreich