TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 180, June 1973
P COMMUTATIVEBANACH^ALGEBRASt1), (2)
BY
WAYNE TILLER
ABSTRACT.
Let
A be a complex
*-algebra.
If / is a positive
A, let /, = ¡x £ A: f{x * x) = 0| be the corresponding
where
the
intersection
P-commutative
and examples
commutative.
the
extend
results
2
if and only
1. Introduction.
speaking,
of this
called
erties
Banach
The structure
which
bras,
some
prove
discuss
several
-algebra
-algebras
tative
we shall
theorems
A
Banach
functional
-algebra
Presented
and, in revised
-algebras
are
on
If
A is a pure
the
spectral
radius
positive
I. Murphy
on A implies
a nonzero
January
properly
for P-commutative
-al-
the class
the notation
P-commutative
ideals,
and then
Banach
on A is automatically
for commutative
[5] gave
show
identity,
a different
that
Banach
proof
of a multiplicative
linear
17, 1972; received
which
if A is a P-commu-
then the existence
the existence
and
alge-
and quotients,
functional
result
prop-
algebras.
if A is a P-commutative
We also
positive
theory.
Banach
contains
In §2 we introduce
the same
with an apptoximate
noncommutative
In §3 we define
that
is, generally
many of the well-known
true
subalgebras,
involution.
to the Society,
them
*
-algebras
which
see,
remain
is as follows.
[9] proved
on A; and in fact,
Among
of noncommutative
-algebras,
as we shall
of the involution.
functional
Banach
a new class
= A, then every
continuity
P-
A is continuous.
then
than the corresponding
In §4 we show
with continuous
zero positive
of commutative
concerning
examples.
are
*-algebras
functional
If A is symmetric,
use throughout.
N. Varopoulos
did not require
positive
Banach
Banach
satisfying
continuous.
a nonzero
is P-
which
Banach
on
A is called
*-algebra
seminorm.
of the paper
terminology
then
-algebras;
of commutative
Then
*-algebras.
functional
is to study
P-commutative
of commutative
Banach
positive
understood
paper
A.
*-algebras
every
The theory
much better
The purpose
on
commutative
for P-commutative
if it is multiplicative.
algebra
Every
of noncommutative
for commutative
then
identity,
in A is a continuous
gebras,
-A,
on
left ideal of A. Set P = I I /,,
functionals
x, y £ A.
are obtained
known
If A
an approximate
state
all positive
are given
Many results
following:
A has
over
if xy —yx £ P for all
commutative
which
is
functional
functional
of a non-
linear
is a pure
by the editors
February
state
22,
if
1972
form, July 24, 1972.
AMS(MOS)subject classifications (1970). Primary 46H05, 46J99; Secondary 46K99.
Key words and phrases.
Banach
*-algebra,
positive
multiplicative
linear functional,
symmetric
*-algebra.
(1)
Texas
(2)
functional,
*-representation,
The contents
of this paper form a portion of the author's
doctoral
dissertation
Christian
University
under the direction
of Professor
Robert
S. Doran.
This
research
was
partially
supported
by the National
Science
at
Foundation.
Copyright © 1973, TVmericanMathematical Society
327
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328
WAYNETILLER
and only
which
if it is multiplicative.
are symmetric.
algebra
is a continuous
tion of symmetry
-algebra
vix
-algebra
to have
by one.
A by adjoining
and
norm one,
in [7]).
ideal
R ).
iated
"MLF".
If / is a positive
of A; that
The words
g (x
v(A
/
denote
on A, we set
"pure"
and
obtained
from
radical
of A,
of the kernels
the
the radical
functional
x) < f(x
will
(R is called
linear
be called
a con-
be two-sided
is, the intersection
we let
is,
the Jacobson
spaces
algebras.
identities
the algebra
"multiplicative
/ on A will
g on A satisfying
radius
x—> x ; that
will
by A
(or / .) denote
convenience
we write
functional
Banach
linear
algebras
identities
of A on Hilbert
larly,
A positive
/
complex
In normed
we denote
We let
-representations
For typographical
in such an
the assump-
if its spectral
an involution
two.
and approximate
-algebra,
R (or R .) the reducing
that
an arbitrary
associative
with
of period
an identity.
all irreducible
only
an algebra
If A is a
show
that
is P-commutative
consider
anti-automorphism
be assumed
bounded
mean
will
we show
-algebras
radius
x.
We shall
we will
Banach
that the spectral
An example
Finally
identity
for all
2. Preliminaries.
jugate-linear
seminorm.
be dropped.
x) < (x)
P-commutative
among other things,
algebra
cannot
with an approximate
satisfies
By a
In §5 we study
We show,
functional"
of
-radical
of A
(simi-
will be abbrev-
I, = \x e A: f (x x) = 0\.
if for every
x) for all
x £ A, it follows
-algebra.
A continuous
positive
that
functional
g = Kf for some
0< kK 1.
Now let
A will
A denote
be called
trum of an element
remark
that
tor
exists
r) £ II such
continuous,
that
show
/
that
(x) = f (y
then
functional
book
and let
[7] will
not require
of J. Ford's
immediate
P = I I /,,
o . (x))
the spectral
radius
every
x e A.
root
lemma
We
/ on
H and a cyclic
that
vec-
/ is Hermitian,
any results
of local
we use
continuity
of
[3].
and some
the intersection
of x.
functional
In particular,
the assumption
square
/ on
the spec-
to be representable
space
we remark
consequences,
where
positive
/ on A is said
Finally,
functional
(or
77 of A on a Hilbert
f (R) = 0.
positive
by ct(x)
(or v . (x))
f (x) = (?t(x) r¡ \ rf) for every
by virtue
examples.
is taken
over
Let
A be
all positive
on A.
Proposition
Prool.
We denote
identity,
-representation
3. The definition,
-algebra,
= 1.
[8]; a positive
a
and satisfies
the involution
functionals
||/||
x in A, and by vix)
from C. Rickart's
a
if
if A has an approximate
A is representable
if there
a Banach
a "state"
Since
3.1.
P is a two-sided
P is the intersection
P is a right
xy)
ideal
for each
ideal.
of a family
If / is a positive
x £ A.
Then
in A.
/
of left
functional
is a positive
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ideals,
it suffices
on A and
functional
to
y 6 A, set
on A, and if
P-COMMUTATIVE BANACH *-ALGEBRAS
x £ P, we have
right ideal.
The
/[(xy)
x
xy) = /
(x
x) = 0.
Hence
xy £ P and
P is a
Q.E.D.
following
Definition
every
(xy)] = fiy
329
definition
3.2.
was
A -algebra
suggested
to the author
A is said
by Professor
to be P-commutative
R. S. Doran.
if xy-yx
£ P for
x, y £ A.
Proposition
3.3.
Lez" A be a P-commutative
-algebra.
Then
A
is P-commu-
tative.
Proof.
-(y,
It is clear that
a) (x, A) = ixy-yx,
P-commutative.
3.4.
-homomorphism
Proof.
Let
<7j(x)e
-cb(y)
Let
since
A is P-commutative.
A be a P-commutative
of A onto
B.
Then
/ be an arbitrary
tive functional
thus,
0) £ P .
Then (x, A)(y, a)
Therefore,
A
is
Q.E.D.
Proposition
a
PA C P . Now let (x, A), (y, a) £ A£.
P_.
Hence,
It follows
positive
functional
if 0 (x), cbiy)
on B.
0 = if ° (b) ix
are arbitrary
e <tj(P .) C P„.
from Proposition
commutative
B a
-algebra,
and
d)
B is P-commutative.
on A ; so if x e P A, then
cbix) = 0(xy-yx)
-algebra,
Therefore,
3.4 that
Then
f ° cb is a posi-
x) = f[cb(x)
elements
cb (x)], and
of B, then
q>ix)cb(y)
B is P-commutative.
if / is a
-ideal
in A, then
Q.E.D.
A/1
is a P-
-algebra.
For the remainder
of this
paper,
unless
stated
otherwise,
A will
denote
a
a.
Banach
-algebra.
Since
for every
Banach
-algebra
A, it is true
(see [7, 4.6.8,
p. 226]), it is true that
P-commutative,
then
and reduced
identity,
and
(i.e.,
then
A/R
is commutative.
R = iOi), then
every
positive
A is P-commutative
need
an approximate
then
that
a closed
not be P-commutative.
tion to illustrate
Proposition
functional
this
fact.
3.5.
However,
}e j.
/ representable
that if A is
if A is P-commutative
Now if A has
on A is representable,
if A/R
an approximate
and hence,
P = R
is commutative.
-subalgebra
of a P-commutative
We will give
we have
Lez: A be a P-commutative
identity
R = I 1/,,
In particular,
A is commutative.
if and only
It is not surprising
-algebra
that
P C R. So we see immediately
// B is a closed
an example
the following
symmetric
later
Banach
in this
sec-
result.
Banach
-subalgebra
-algebra
of A containing
with
¡e
B is P-commutative.
Proof.
notation
ducing
o
We adjoin
ßj
ideals
in this
identities
proposition
R e of A e and
to A and
B to obtain
only so that
R, \ of B, \J).
trie [7, 4.7.9, p. 233], and ß j is a closed
A
and
we may distinguish
Now
B,
between
A e is P-commutative
-subalgebra
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(we use the
the re-
and symme'
of A . Since both A and
¡,
330
WAYNE TILLER
B possess
approximate
identities,
all positive
functionals
are representable,
and
it follows that PA = RA = Re and Pß = Rß = R v So by [7, 4.7.19, p. 237], we
have that
Pß = R{ = By O Rg = ßJ n P^ = ß n P^; it follows that
mutative.
ß is P-com-
Q.E.D.
A related
result
Proposition
is given
3.6.
1 is a closed
Let
-ideal
next.
A be a P-commutative
of A having
symmetric
an approximate
Banach
identity,
then
-algebra.
If
I is P-commuta-
tive.
Proof.
it follows
We have that
P.
C R . = / . and P. = P. = /,.
that /, = / O JA.
/ H P . is obvious;
We now give
mutative,
hence
some
3.7.
by giving
We next
Example
norm
||-||
ments
3.8.
Let
norm
Let
/ be a positive
set
a = (x, 0).
arbitrary,
a £ P.
functional
-algebra
having
use
algebra
need
-subalgebra
tation
of A
has
no nonzero
We next
it is reduced.
which implies
noncommutative
Furnish
Banach
We now show
A
is
that
element
a) = 0; hence
(x, y)eP
an identity
e, then
with
ele-
algebraic
= (y , x ).
Then
A is P-commutative.
element
in A , and
a £ I, and since
of A of the form
that
algebra
We denote
(x, y)
x be an arbitrary
/(zz
|.
A with pointwise
, and involution
on A, let
every
if A
on A, choose
Example
0) £ A = P
A = ¡(x, y): x, y e A
+ |y||
in A, it follows
3.8,
of A. Then |/(«)|2
tive
Let
a = 0 implies
Similarly,
(i.e.,
(0, y) is in
for every
/ was
P.
x, y £ A ; i.e.,
But,
A =
A is P-commutative.
In Example
functional
a
-algebra
is semisimple.
be a semisimple,
x—> x .
and semisimple.
P is an ideal
Therefore,
which
(x, y) or a, b, etc.
Then
Banach
is noncommutative.
||(x, y)|| = ||x||
A is noncommutative
P.
A
of A by either
operations,
since
A
an example
and involution
of noncomr
P e = P e = A. Now if (x, A), (y,
a.) £
J
a) (x, A) = ixy-yx,
Obviously,
construct
a few examples
radical
/J A. = P A. = A).' Then, ' for A e , we have that
P-commutative.
P, C
-algebras.
A be a noncommutative
A , then (x, A) (y, a)--(y,
/ is an ideal,
The inclusion
Q.E.D.
We begin
Banach
Let
Since
P, 3 / C\ PA.
/ is P-commutative.
examples.
Prcommutative
Example
Thus, we obtain
Indeed,
a £ A, and let
3.8 to show
that
let
of a Banach
/ be a positive
1 = (e, e) denote
the identity
= 0 and so f = 0 on A.
a closed
-subalgebra
of a P-commuta-
not be P-commutative.
Let
Now let
of A.
functionals.
= |/(1 -a)\2 < f(l)f(a*a)
Example
3.,9.
positive
an arbitrary
A is an example
A
be as in 3.8 above
ß = í(x, x): x e A j.
We show
on some Hilbert
that
space
but with the added
Then
ß is a closed
B is not P-commutative.
H.
Let
assumption
noncommutative
n be a -represen-
Then the map 77R: B —• B(H)
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that
defined
by
P-COMMUTATIVE BANACH *-ALGEBRAS
77g [(x, x)) = 77(x) is a
(x, x)eker(/7R).
-representation
Since
commutative,
A'
it follows
Example
3.10.
with continuous
Let
A is noncommutative
B is symmetric
ments
of the form
x, y £ A, i.e.
identity
e.
We show
P,
z + r, where
every
z £ Z, r£RR,
P„
is non-
by Z and
P„.
A is P-commutative.
= RA = A C\ PR = PR.
(z x + r x) (z, + r2)-U2
in A and since
that
-algebra
that
Now
+ r 21 ^z l + rl^ = rlr2~r2rl
element
of A is the limit
it follows
that
xy-yx
e
of ele-
£ R . for every
A is P-commutative.
We next
shows
Then
P . is closed
B is non-
Banach
of B generated
and e £ A, we have that
z2 + r 2£ A.
Since
since
Z, and assume
-subalgebra
and has
if and only if
Therefore,
symmetric
e, and center
A be the closed
Since
+ t.,
identity
x £ ker(7r)
be P-commutative.
B be any noncommutative,
Then
P ..
ß is reduced.
B cannot
involution,
commutative.
let z
is reduced,
that
Let
of B on H and
331
give
that,
an example
in general,
showing
nothing
that
P is not always
can be said
about
equal
to P.
the relationship
It also
between
/ and
P.
2\a
Example
3.11.
\/n\
(it will
< »
the usual
tion
2a
Let
algebraic
A be the set
be understood
operations,
zn—>2â~
z".
7Z
that
the
power
sum runs
norm defined
Then
72
of all fotmal
by
from
\\2a
A is a commutative,
series,
2 a z",
n = 1 to oo).
zn\\ = 2 \a
radical
\/n\,
Banach
Then
/ is clearly
there
implies
linear,
exist
/ is positive
series
since
in A for which
a
f[2ä~
/
-algebra,
&
zn) i2a
'
so
zn) = «y
zn)] -- \a A
0, it follows
A
that
I,/
> 0.
A which
P / A.
A. Multiplicative
A is a Banach
of pairs
and
power
Give
and involu-
/ = P = A. We show that P / A. Consider f: A — C defined by f(la
Since
satisfying
linear
-algebra,
of elements
Theorem.
dominate
from
Assume
A.
A
a continuous
functionals
we denote
and continuity
by A
I. Murphy
the set
[5] proved
= A and let every
nonzero
positive
of positive
of all finite
the following
nonzero
functional.
If
of products
result:
positive
Then
functionals.
sums
every
functional
on A
positive
functional
on A is continuous.
then
As a corollary,
Murphy
every
functional
positive
commutative
being
algebras.
in showing
Theorem
then every
Prool.
since
every
(x
if A is commutative
x) = /
follows
iu
functional
/ be a nonzero
Murphy's
and satisfies
We extend
this
essentially,
A2 = A,
result
the main
to Pdeviation
u).
// A is a P-commutative
positive
Let
/
that
on A is continuous.
The proof
that
4.1.
proved
Banach
-algebra
satisfying
A
= A,
on A is continuous.
positive
x in A can be written
functional
on A.
in the form x = 2"_
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Then
, X.x .x.
/ is Hermitian
(see
[5]).
For
WAYNE TILLER
332
every
u £ A, let /
3.1.
Then /
every
u £ A, then
/(A3)
= 0 which
0, and since
lemma
that
be the positive
is continuous
|/(z/*x*y)|2
is false
/
= |a|
then
provides
(x
x) = / iu
/
is well known
defined
by
functional
< fiu *x *xu)fiy
since
A^ = A.
in the proof of Proposition
v £ A
u) for every
that the linear
x £ A.
space
(x + I, \y + IA = fiy
that
such
together
/,
that
that
root
in A.
product
It
(• | •)
Furthermore,
zzx + /, = xzz + /, for every
4
Now we show
is a left ideal
space.
/
square
u.
with the inner
x), is a pre-Hilbert
it follows
Ford's
v v = e-u
that
zz, x, y £ A; hence,
u £ A such
\\u u\\ < 1.
that
Recall
A/1
Now if fu= 0 fot
*y) = 0 for every
So we may choose
/ , we may assume
an element
A is P-commutative,
defined
on A (see [7, 4.5.4, (ii), p. 215]).
since
x e A.
Therefore,
we have
/ (x x) = fiu
x xu) = ixu + I Axu + I A
= (zzx + IftAux + IA = fix
Now for every
x £ A,
if-f)
(x*x)
= f(x*x)-fu(x*x)
= fix*x)-fxiu*u)
= f[x*íe-u*u)x]
since
/ is positive
an application
on A and
of Murphy's
continuous.
The following
lemma
Lemma
4.2.
Let
the identity
= f[ívx)*ívx)]
> 0
in A . So / dominates
we see that
It is well
every
positive
/ , and by
functional
on A is
on H.
Let
Then
p. 28] for the case
of the proof
involution.
-algebra
We therefore
with approximate
of A on a Hilbert
Then
A
that it
omit the proof.
identity
space
lim 77(ea) = 1, where
when
in [2] shows
je
H, and let
the limit
¡.
Let
1 de-
is in the
on B (H).
known that
4.3.
\ea\.
with arbitrary
-representation
topology
[2, 2.2.10,
modification
A be a Banach
is a pure state
Theorem
known
A slight
operator
operator
identity
is well
algebras
v be a nondegenerate
functional
A is an ideal
theorem,
involution.
is true for Banach
note
= fíx*v*vx)
Q.E.D.
has isometric
strong
u ux) = f x iu u).
on a commutative
Banach
-algebra
a nonzero
positive
if and only if it is multiplicative.
A be a P-commutative
a nonzero
positive
functional
Banach
-algebra
with approximate
f on A is a pure state
if and
only if it is multiplicative.
Proof.
f(R)=
Clearly
that
Let
/ be a pure state
0 and so we may define
/
is linear
11/11< ||/
||.
and positive,
Now positive
on A.
a function
Since
and it is easy
scalar
A has an approximate
identity,
/ ': A/R —> C by / ' (x + P) = fix).
multiples
to check
that
of pure
positive
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it is also
pure and
functionals
are
333
P-COMMUTATIVE BANACH "-ALGEBRAS
pure
positive
A/'is
functionals;
a pure
state
multiplicative
able,
on A/R.
which
and
space
f(x)/,
||/||
n £ H.
since
the first
Now assume
define
/
that
state.
77 is a
a.
that
A<
/
limf(ea)
positive
[7, 4.4.12,
relative
MLF on A.
If we choose
which
implies
MLF
Again
on A.
then,
f(R) = 0, so we
on A/R
x £ A such
limf(ej
and,
that
irreducible
||/||
then
of A must
the set of scalar
If we furnish
case,
M with the
M is a locally
M is compact.
functions
statej
the set of irreducible
(f £ M).
compact
If we define
on M which
x—> x is a norm decreasing
But
we set
is just
M is precisely
as in the commutative
> 1.
Q.E.D.
-representation
image
hence,
f (x) /= 0, then
= 1; hence
identity,
of each
R = flker(/)
Cn(zVI) (the continuous
then
suchthat
\\r]\\2 < ||/|| =
on A\ = ¡/: / is a pure
of every
the commutant
If A has an identity,
x—> x of A into
= fix),
the image
Therefore,
-topology,
space.
positive
p. 211], we see that
of A.
weak
Hausdorff
by x(f)
Since
and since
-representations
xeA
(ea) which implies
and multiplicative
and has an approximate
M = \f : j is a nonzero
operators
= ||771| ;
A= 1 and so / is multiplicative
is positive
/ is pure.
If A is P-commutative
be commutative,
is
on the
1. But lim/(ea)=
11/11
S 11/'II= 1 implies u/11= 1, and therefore / is a state.
(M may be empty).
A/
lima/(ea)
Now if we choose
1 since
that
Now / is represent-
-representation
4.2, we see
= (A/) (x) lim(A/)
1/A>
= 1; thus
Then
= fix)
it follows
on A.
that
of the theorem.
Therefore,
l'im/(xea)
A, 0 < A < 1, such
is commutative,
rj), where
/ is a nonzero
as before.
is a pure
f(x)=
half
a scalar
by Lemma
= lim(A/)(xea)
1/A = lim/(ea)
proves
A/R
< 1 for every
1 or lim/(ea)=
1, and hence
This
exists
A/ is multiplicative
Then
||ej|
0, then (A/)(x)
lim(A/)(ea)=
that
f(x) = ivix)r\\
H and
||z7||2<
there
Since
implies
so we may write
Hilbert
hence,
vanish
-homomorphism.
the map
at infinity)
It is injective
if and only if A is reduced.
5. Symmetric,
commutative
bras
have
already
-algebras
5.1.
Let
in A is primitive
Proof.
in A.. Then
Banach
which
been obtained
Proposition
ideal
P-commutative
Banach
The "if"
since
(see
A be symmetric
implies
modular in A.
l/R
is maximal
section
we study
for such
and P-commutative.
P-
alge-
Now let
P = /, and therefore,
is a primitive
modular
which
Then a two-sided
modular.
part is true for any algebra.
A is symmetric
In this
Two results
3.5, 3.6).
if and only if it is maximal
map of A onto A/R, then d(l) = l/R
mutative
-algebras.
are symmetric.
/ be a primitive
PC/.
ideal in A/R.
But A/R
implies
/ = d " 1 ÍI/R)
Banach
*-algebra,
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then
com-
is maximal
Q.E.D.
It is well known that if A is a commutative
ideal
If d is the quotient
A is
334
WAYNETILLER
symmetric
if and only if every
Theorem
5.2.
is Hermitian
// A is symmetric
(hence
Proof.
Let
MLF on A is Hermitian.
and P-commutative,
then
every
MLF on A
positive).
/ be a MLF on A.
Then
R = ] Cket(f)
since
ker(/)
tive ideal in A. If we define
/ ' : A/R —>C by / ' (x + P) = fix),
Hermitian
A/R
MLF on A/R
since
is commutative.
is a primi-
then / ' is a
Therefore,
/ is Hermitian.
Q.E.D.
The
converse
P-commutative
of Theorem
algebras
Example
5.3.
e, involution
Let
semisimple
tions,
Banach
on D, then
A be a noncommutative
-algebra
(x, y, z)
Example
with identity
on D.
thus
+ z
Let
z) which
cannot
We remark
modular
between
two-sided
ideals
implies
R = J; therefore
as used
for the analogous
Theorem
5.4.
PC/
Let
/.-norm.
But since
Then
in D.
algebra
D is just
Thus,
let
and so
A/I
MLF's.
of A.
Set
</>R is
and commuta-
It remains
to show
that
y = - (x~
)
and
z) = (0, 0¡ e
D is not symmetric.
then
there
is a one-to-
on A and the set of maximal
/ be such
an ideal.
is commutative.
result
now applies
and
P-commutative.
A be symmetric
opera-
the pro-
no nonzero
B is symmetric
and symmetric,
Indeed,
commutative
symmetric,
Now if cb is a MLF
A has
element
the set of all MLF's
of A.
identity
and involution
D pointwise
cb is Hermitian.
if A is P-commutative
one correspondence
by e also)
(x, y, z) = (e + y x, e + x y, e + z
be invertible
that
having
by <7j[(x, y, z)] = </> (z), where
x be an invertible
(e, e, e) + (x, y, z)
-algebra
Give
x, y £ A since
tive,
Then
nonsymmetric
ß be a commutative,
(denoted
4>: D—> C, defined
D is not symmetric.
exist
D is P-commutative.
a MLF on B, are the only MLF's
z £ B.
Let
= (y , x , z ), and
3.8; therefore,
</>d's are Hermitian;
is, there
Banach
MLF's.
4>[ix, y, 0)] = 0 for every
the functionals
all the
that
MLF is Hermitian.
D = i(x, y, z): x, y £ A, z £ B\.
involution
of B with
Thus,
every
x—» x , and no nonzero
z —>z . Now set
duct
5.2 is not true;
on which
Then A symmetric
The
(see
same
argument
[6, p. 192]).
Then
R - ¡x £ A:
vix) = 0|.
Proof.
x £ A, vix)
v (x) and
the theorem
implies
Since
= 0.
is clear).
5.5.
Then
ker(/)
that
x e P implies
is P-commutative
= 0; hence
x e (I
Theorem 4.3).
Lemma
A
R . = P , we may assume
vAiR(x)
Therefore,
P = /, it follows
Since
/
that
(usual
0= /'(x
A has an identity.
definition)
+ P) = fix)
(f £ M) implies
v(x)
and symmetric,
= 0,
Conversely,
and since
Let
f £ M (if M = 0.
is a MLF on A/R.
since
A/R
But
vix)
is commutative.
x £ R (see the paragraph
following
Q.E.D.
Suppose
A is symmetric,
P-commutative,
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and has
let
vA (x) =
an identity.
= 0
P-COMMUTATIVE BANACH *-ALGEBRAS
Then
for every
Proof.
x in A, a(x)
= {fix):
The containment
4.3 and [4, Theorem
(x + P) = fixfa
A, it follows
the lemma follows.
Q.E.D.
Theorem
5.6.
Let
< vix)v(y)
Proof.,
< (sup
over all
give
Example
assume
that
Let
Then
Theorem
x and
without
for every
x, y £
5.5, v(xy)
the supremums
=
ate taken
+ sup|/(y)|
=
algebra
seminorm
in a
the assumption
of symmetry,
note that
extended
P.
This
Example
that
-algebra
algebra
5.3).
We may
A of Example
It is easy
vA [(x, y)iy,
to check
x)\ = vA [(xy,
yx)]
with an approximate
identity.
If
= il Ker(/),
is proved
where
situation-
Now let
since
is over all
= fiyx)
for every
and every
positive
x £ A, |/(x)|
x, y £ A; A is therefore
with an iden-
RA = P
/ be an arbitrary
for every
3] fixy)
the intersection
in [6, p. 259] for algebras
< /'
x, y e A.
functional
(e) vix
x)
Hence
P-commutative.
xy-
Q.E.D.
BIBLIOGRAPHY
1. J. W. Baker
and J. S. Pym,
A remark
on continuous
bilinear
mappings,
Proc.
Edinburgh Math. Soc. 17 (1971), 245-248.
2'. J. Dixmier,
Les
C -algebres
et leurs
représentations,
2ieme
ed.,
Cahiers
Scienti-
fiques, fase. 29, Gauthier-Villars, Paris, 1969. MR 39 #7442.
3.
J. W. M. Ford,
42(1967), 521-522.
A square
=
A is P-commutative.
to A . Then
= RA for every
with an identity
^y¡ *^-
to the present
and by [l, Corollary
-algebra
P-commutative
(see
It follows
is representable.
/ ' its extension
Banach
v„ (xy) > v„ (x) vR (y).
Now form the
x £ A, then
on A.
functional
< / (e)i/(x),
by Lemma
where
is a continuous
that,
A be a Banach
for every
We first
yx e MKer(/)
Then
+ y)| < sup|/(x)|
y satisfying
va ^x- y^va
Let
functionals
on A and
v(-)
A is not symmetric
5.8.
It is easily
positive
Then
viy),
ß be a noncommutative
vb ^=
x) < vix)
tity.
oA/fR(x + R) Co(x),
-algebra.
to show
v„ (y) = v„ (x).
B.
Proof.
f £ M. Since
seminorm.
oA L(x, y)] = oß (x) U oß iy).
positive
Since
P-commutative.
= vix)
Banach
an example
5.7.
vb ^xy^> v b ^
vix
for some
+ R).
an identity.
5.6 we see that
two elements
that
3.8 using
and
v(x + y) = sup|/(x
not be an algebra
and having
r\£ oA/R(x
(sup|/(y)|)
P-commutative
need
A= fix)
from Theorem
Q.E.D.
We next
v(-)
A has
|/(x)|)
From Theorem
symmetric,
that
immediately
v(x + y) < v (x) + viy).
/ in M. Similarly,
vix) + viy).
that
A be symmetric
and
We may assume
sup|/(xy)|
f £ M\ follows
Now suppose
/
A, v(xy)
f £ M\.
o~(x) C \fíx):
4.10].
335
root lemma
for Banach
*-algebras,
MR 35 #5950.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
J. London
Math.
Soc.
336
WAYNE TILLER
4.
J. W. M. Ford,
Dissertation,
Subalgebras
Newcastle
5. I. S. Murphy,
upon
of Banach
Tyne,
Continuity
algebras
generated
by semigroups,
Ph.D.
1966.
of positive
linear
functionals
on Banach
-algebras,
Bull.
London Math. Soc. 1 (1969), 171- 173. MR 40 #3320.
6.
M. A. Naimark,
Noordhoff,
7.
C. E. Rickart,
Van Nostrand,
9.
General
Princeton,
8. S. Shirali,
145-150.
Normed
rings,
''Nauka",
Moscow,
1968;
English
theory
of Banach
algebras,
University
transi.,
Wolters-
1970.
N.J.,
I960.
Representability
Series
in Higher Math.,
MR 22 #5903.
of positive
functionals,
J. London Math. Soc. (2) 3 (1971),
MR 43 #2517.
N. Th.
Varopoulos,
Continuité
des formes
linéaires
positives
sur une algebre
de
Banach avec involution, C. R. Acad. Sei. Paris 258 (1964), 1121- 1124. MR 28 #4387.
DEPARTMENT OF MATHEMATICS, TEXAS CHRISTIAN UNIVERSITY, FORT WORTH, TEXAS
76109
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address:
Bocholt
Marienschule,
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