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Section 15 C*-Algebras
C -ALGEBRAS
24 7
a
15.
c*-ALGEBRAS
The theory of order-unit Ranach spaces, their states and representations, will now be applied to study an important class of involutive Ranach algebras, the so-called C*-algebras. C*-algebras are the prototype of algehras connected with symmetric Ranach manifolds and reveal an interesting relationship between infinite-dimensional holomorphy and functional analysis.
Since C*-algebras are
also fundamental for the algebraic formulation of quantum mechanics [120] they provide a link for possible applications of infinite-dimensional holomorphy to problems in mathematical physics [144]. 15.1 DEFINITION. algebra over K
.
Suppose 2 is an associative Ranach An involution z + z * of 2 is called a
C*-involution if Iz*zI =
.
2
15.1.1
z E 2 It follows that ( z I L < Iz*I-IzI and hence < (z*I , i.e., Iz1 = I z * I for all z E 2 Therefore
for all 121
IZI
.
every C*-involution is isometric and hence continuous. 2 = Combining this fact with 15.1.1, we get I z z * ( = I z * I for all 15.2
z
E
2
EXAMPLE.
.
.
Let
E
and
F
121
2
be Hilbert spaces over
D E {R,C,H} For any operator z E L ( E , P ) , define the by (zhlk) = (hlz*k) for all adjoint operator z * E L ( F , E )
.
E ExF Then (z*)* = z and (zw)* = w*z* whenever Since F,L) and L is another Hilbert space over D (z*zh k ) = (zhlzk) for all h,k E E , the Cauchy-Schwarz i nequa ity [28; 6.2.11 implies
(hik)
W
E
1
.
SECTION 15
248
Hence z + z* is a C*-involution of the associative unital Ranach algebra Z := L ( E ) over the center K of D , endowed with the operator norm. 15.3 EXAMPLE. Let 2 := g ( S , D ) denote the associative unital Ranach algebra (over the center K of D ) of all bounded D-valued functions on a set S , endowed with the supremum norm. For z E 2 define the adjoint function z* E 2 by z * ( s ) := z ( s ) * , where * denotes the canonical involution of D Since I X * X I = 1 x 1 ~for all x c D , it follows that z + z * is a C*-involution of 2 The closed subalgebras Lm(S,D) , Cm(S,D) and Cu(S,D) of B ( S , D ) , associated with a measure space S , a topological space S and a locally compact space S , respectively, are invariant under the C*-involution.
.
.
15.4 LEMMA. Let 2 be an associative Ranach algebra over K , endowed with a C*-involution. Then z*z = zz* implies n I l/n
I z ( = lim Iz
PROOF.
=
For
I (z*z)
n
N
E
2n-1 2
I
=
,
n+m 15.1.1 implies
2n-2 4 ((z*z) I
=
... =
.
15.4.1
Iz*zI
2"
=
2n+1 121
Hence 1 2 2"
I2-"
=
(z(
.
Since the limit 15.4.1 exists [17; 2.81, it is equal to
.
IzI
O.E.D.
15.5 COROLLARY. Let W be an abelian Ranach algebra over K , endowed with a C*-involution. Let C w be the (locally compact) spectrum space of W Then the Gelfand mapping
.
is an isometric homomorphism onto a closed subalgebra over
K
*
C -ALGEBRAS separating the points of
PROOF. 15.6
Cw
249
.
Apply 15.4 and [17, $171. LEMMA.
Suppose
Z
O.E.D.
is a non-unital associative Ranach
algebra over K I endowed with a C*-involution. Then the canonical involution on the unitization 2 ' := 28K of Z
is
a C*-involution with respect to the Ranach algebra norm I z + s I = sup
.
Izw+swI
IWI
PROOF. F o r x E 2 ' , let Liy : = xy denote left By definition, 1x1 = ILi12 multiplication on 2'
.
L;
operator norm of
x,y
for all
on the ideal
2
of
.
2'
is the
Hence
e E 2' satisfies x = z + s satisfies E 2 If s # 0 , it is a left unit of 2 Since follows that c : = -z/s wc* = (CW*)* = (w*)* = w and c* = cc* = c , c is a unit of 2 This contradiction shows s = 0 Since E
2'
I
and the unit element
.
lel = ILE,I = lid12 = 1 Now suppose 1x1 = 0 Then zw = -sw for all w
.
.
.
lL;lz
<
.
.
and
I Z I
= IL;I
for all
z
E
2
,
*I
it follows that
is a norm on
coincides on 2 with the original norm. z E 2 , we have xz E 2 and hence
lLiZ12 =
(XZl2 =
I(xz)*(xz)l
=
2
.
< Iz*I*Ix*xI*(zI =
IZI
Ix*xI
x
For
E
2' 2'
which and
Iz*(x*x)zl
2
It follows that every x E 2' satisfies 1x1 c Ix*xI < I x * I * I x I , hence 1x1 < Ix*l and therefore ( x I 2 = Ix*xI , showing that the involution on 2' is a C*-involution. In is a complete norm on 2 ' , consider order to show that 1 - 1 the left translation homomorphism L ' : 2 ' + ~ ( 2 ), defined by
2 50
SECTION 1 5
.
x + L' Since L' is isometric, the image L'(Z) is X complete and hence closed in L ( 2 ) Therefore L'(Z') = L'(Z) @ K*idZ is also closed in L(Z) and is therefore complete. This shows that Z' is a Banach algebra.
.
Q.E.D.
LEMMA. Every C*-involution on an associative Ranach L, and R, belong algebra Z over K is -hermitian, i.e., to ui(Z) whenever z* = -2 15.7
.
s unital. Put PROOF. By 15.6, we may assume that 2 w : = exp(z) E 2 Since the involution is continuous, we get
.
w*
=
exp(z*)
=
exp(-z)
exp(z
=
.
Therefore lwI2 = (w*w( = le( < 1 exp(LZ) = Lw E U k ( 2 ) and exp(RZ)
- L-1w.
Therefore Rw E U k ( 2
Q.E.D.
=
15.8 COROLLARY. Every C*-involution on a comp ex associative Banach algebra 2 is hermitian and H ( 2 ) = { x E 2 : x* = x }
.
PROOF. We may assume that Z is unital. By 15.7, every self-adjoint element is hermitian, showing that the involution
is 'hermitian and hence hermitian. hermitian element is self-adjoint.
By 14.29 2, every
Q.E.D.
Note that the identity mapping is a C*- nvolution on , considered as a real Ranach algebra, which is not 'hermitian. Hence 15.8 is not true for K = R . 2 = C
DEFINITION. An associative Ranach algebra 2 over endowed with a C*-involution, is called a C*-algebra if
K
15.9
x = x*
E
2
Cz(x) C R
.
15.9.1
By 15.8, the spectral condition 15.9.1 is redundant for K = c By 15.6 and 14.19, the unitization 2 ' = 2 @ K non-unital C*-algebra Z over K is a C*-algebra.
.
15.10
LEMMA.
Suppose
Z
is a C*-algebra over
,
K
and
of a
W
is
251
C*-ALGEBRAS
a closed *-subalgebra of PROOF. For x = x * E W C \ C,(x) is connected. Cw(x) = C,(x)
.
2
.
Then
W
is a C*-algebra.
, we have
C,(x) c R and hence Therefore [ 1 7 ; 5.141 implies O.E.D.
15.11 EXAMPLE. Let E be a Hilbert space over D E {R,C,E} and consider the associative unital Banach algebra Z := L(E) over the center K of D , endowed with the operator norm and the C*-involution defined in 15.2. In case D = C , 15.8 implies that Z is a complex C*-algebra. In case D = R , Ec : = EORC is a complex Hilbert space and Zc can be identified with L (Ec) Hence x = x * E 2 implies
.
In case D = H , E showing that Z is a real C*-algebra. can be regarded as an even-dimensional complex Hilbert space, denoted by 15.11.1
such that
u
H
carries a conjugation u + and a b z = t L -) : a r b E L (H) } -b a is a purely real closed unital *-subalgebra of L(EC)
-
-
.
au : = au for all a E L(H) and u E H The can be identified with L(EC) complexification 2' above, it follows that 2 is a real C*-algebra.
.
.
Here
As
15.12 EXAMPLE. Let S be a set and consider the associative unital Ranach algebra 2 := B ( S , D ) over the center K of D E {R,CIE} I endowed with the supremum norm and the C*-involution defined in 15.3. In case D = C , 15.8 implies that 2 is a complex C*-algebra. In case D = R , 2 is a real C*-algebra since 2' = S ( S , C ) In case D = H , consider the associative unital complex Ranach algebra A := B ( S , L 2 ( C ) ) of all bounded mappings f : S + L 2 ( C ) , endowed with the supremum norm (with respect to the operator
.
SECTION 1 5
252
norm on L z ( C ) ) and the C*-involution f + f* defined by f*(s) := f ( s ) * for all s E S Then A is a complex C*-algebra by 15.8. and s E S Then a
.
z = t L
Put
-
.
a(s) : = a(s)*
for all
a
E
B(S,C)
b
-)
: a,b E B ( S , C ) } -b a is a purely real closed unital *-subalgebra of A whose complexification Zc can be identified with A Hence Z is a real C*-algebra. qpplying 15.10, it follows that the Ranach *-algebras L m ( S , D ) , C m ( S , D ) and C u ( S , D ) I associated with a measure space S , a topological space S and a locally compact space S , respectively, are C*-algebras over K
.
.
15.13 K , z
E
LEMMA. Suppose W is an abelian C*-algebra over In case K = R , assume in addition that z * = z for all W , Then the Gelfand homomorphism
is an isometric *-isomorphism.
.
PROOF. By 15.9.1, Q is a *-homomorphism into C u ( C w , K ) Hence 15.5 implies that Q is an isometric embedding onto a closed *-subalgebra A of c u ( C w , K ) separating the points of Cw The Stone-Weierstrass Theorem [ 2 8 ; 7.31 implies A = cu(CW,K) O.E.D.
.
15.14 K
.
COROLLARY.
The involution of a C*-algebra
Z
over
is hermitian.
PROOF. For x = x* E Z , consider the closed *-subalgebra W of 2 generated by x By 15.10, W is an abelian C*-algebra over K , Applying 15.13 and [ 1 7 ; 5.141, we get
.
15.15 COROLLARY. An associative Ranach algebra Z over K endowed with a C*-involution, is a C*-algebra if and only if
,
*
C -ALGEBRAS
253
> 0
CZ(Z*Z)
15.15.1
.
In particular, in case for all z E Z C*-involution is positive.
K = C!
,
every
PROOF. The spectral mapping theorem and 15.15.1 imply Conversely, suppose 2 is a Cz(x) C R whenever x* = x C*-algebra. To show 15.15.1, we may assume that 2 is unital. Then the assertion follows from 15.14 and 14.23.
.
Q.E.D.
15.16 COROLLARY. adjoint part X
Let 2 be a C*-algebra over Then the set
.
x+
{ x
:=
of all positive elements in X satisfying
x+= { z
If
x
L
: x
has a unit element C := X + n
X
x }
E
e
G(2) =
x
E
is
{ x
a
{ z*z:
=
,
: C,(X)
K
with self-
> 0 ]
closed convex cone in
2
E
2
}
.
15.16.1
the set E
X : C,(x)
> 0 }
of all strictly positive elements in X is a topologically The order regular open convex cone in X containing e on X coincides with the given norm. unit norm
.
I -Ie
PROOF. In case 2 is unital, the assertion follows from 15.15, 14.16 and 14.24. Now suppose 2 is non-unital and let x E X+ Then x = (y+s) = y 2 + 2sy + s2 , where is self-adjoint. Hence s2 = 0 , i.e., y + s E 2' s = 0 Thus 15.16.1 holds also in the non-unital case.
.
.
Q.E.D.
is a Hilbert space over D E {R,C,E} Let X : = H&(E) denote the self-adjoint part of the unital C*-algebra 2 := L(E) over the center K of D The elements of 15.17
.
EXAMPLE.
.
Suppose
E
S E C T I O N 15
254
are called positive operators on
E
,
and the interior
consists of all strictly positive operators on
E
.
15.18 LEMMA. Suppose 2 is a complex C*-algebra. Then for every x = x* E 2 , the left and right multiplication If operators Lx and Rx are hermitian operators on 2 x is positive, then Lx and Rx are positive.
.
PROOF.
for all
We may assume that
t
E
R
.
Hence
Lx
2
E
is unital.
.
~ a ( 2 )
Ry 15.7,
Since [17; 5.41
2 (Lx ) = C,(x) , it follows that Lx E ~ a + ( z ) if x positive. The analogous statements for Rx follow by C!i
considering the "opposite" C*-algebra of
2
.
implies is O.E.D.
By 15.11 and 15.10, every closed *-subalgebra 2 L ( E ) , for a Hilbert space E over K E {R,C} , is a
of
.
C*-algebra over K Conversely, the Gelfand-Neumark embedding theorem asserts that every C*-algebra 2 over K can be realized as an operator algebra, i.e., as a closed *-subalgebra of L ( E ) for some E
.
Let 2 be a C*-algebra over K , Then there exist a Hilbert space E over K and an isometric *-isomorphism n : 2 + L ( E ) onto a closed *-subalgebra of 15.19
THEOREM.
L(E)
By 15.6, we may assume that 2 is unital. and 14.26, there exist a Hilbert space E over K *-homomorphism n : 2 + L ( E ) such that PROOF.
.
By 15.14 and a
(nzl = ~ z * z I ~ ' =~ I z I for all z E 2 Hence n is isometric and n ( 2 ) is a closed *-suhalgebra of L ( E )
. O.E.D.
C * -ALGEBRAS 15.20
COROLLARY.
C*-algebra PROOF. 15.21
255
The complexification
of a real
2'
is a complex C*-algebra.
2
O.E.D.
Apply 15.19, 15.11, and 15.10. LEMMA,
.
Let
L
be a Hilbert space over
D E {R,C,H} Then a linear operator g E GQ(L) is g*g = id, isometric if and only if g is unitary, i.e., Y
.
In particular, UL(L) = { g
GQ L ) : g*g = id L ]
E
15.21.1
is a real Ranach Lie group in the operator norm topology whose Lie algebra can be identif ed with the closed real subalgebra U ( L ) := {
x
x* + x
gQ(L) :
6
of all skew-adjoint operators on L real Ranach Lie subgroup of GQ(L) PROOF.
Every unitary operator
g
E
.
.
= 0
}
Further,
GQ(L)
UQ(L)
satisfies
is a
lgul
2
= (gulgu) = (g*gulu) = ( u ( u ) = Iu
for all u E L and is therefore isometric. Conversely, every isometric operator g E UL(L) satisfies (gulgu) = ulu) for all u E L
.
Polarizing this identity (in case D # R 1, we get 15.21.1. It follows that Ui(L) is a real algebraic subgroup of degree
.
< 1 in (9,g-l) Now apply 7 . 1 4 . The last assertion follows from the fact that uR(L) is a split real subspace of gn(L)
.
In case
O.E.D.
L = Dn
Uin(D) := UQ(Dn) 15.22
COROLLARY.
is finite-dimensional, we write
and
uQn(D) : = uQ(Dn)
Let
2
.
be a unital C*-algebra over
K
.
Then
.
is the group of all unitary elements in 2 In particular, U(2) is a real Ranach Lie group in the norm topology whose Lie algebra can be identified with the closed real subalgebra
SECTION 15
256
u ( 2 ) :=
{ x
E
of all skew-adjoint elements in 2 real Ranach Lie subgroup of G ( 2 )
g(Z) : x* + x = 0 }
. .
Further,
is a
U(2)
PROOF. By 15.19 and 15.21, an element g E G ( Z ) satisfies lgl = Ig-ll < 1 if and only if g is unitary. Now apply 7.14. The last assertion follows from the fact that u ( 2 ) is a split real subspace of g(Z) O.E.D.
.
By 15.22, exp(u(2)) any unital C*-algebra 2 more is true:
is a neighborhood of
over
K
.
e
U(Z)
for For abelian C*-algebras, E
15.23 LEMMA. L e t W be an abelian unital C*-algebra over K Then the identity component of the Ranach Lie group
.
U(W)
coincides with
exp(u(W))
.
PROOF. The identity component of U ( W ) consists of all elements of the form exp(xl) exp(x2) exp(xn) with n E N and xl, xn E u ( W ) Now the assertion follows O.E.D. from 2.8.2.
...,
...
.
We are now going to prove two fundamental geometric results about unital complex C*-algebras, namely the Russo-Dye Theorem ( 15.24 and 15.25) and the Vidav-Palmer Theorem (15.27). The proof given here makes decisive use of holomorphic mappings and shows how infinite-dimensional holomorphy can be applied to problems in functional analysis. 15.24 THEOREM. Let 2 be a unital complex C*-algebra with open unit ball R and unitary group U ( Z ) Then -B = co U ( Z )
.
.
B C co U ( 2 )
PROOF. It suffices to show that the mapping
is holomorphic for
( c ( < (B)-'
.
.
For
8
E
R
Ry 1.12, it follows that
,
C -ALGEBRAS 6 = g (0) = B
For
b
E
2
,
d5 =
1
257
2n gB(e it )dt
1 0
.
15.24.1
consider the vector field
Then 5.23 implies that
,
$(b) := exp(Xb)(0) = tanh(bb*)1'2 ( b b * )2'1
15.24.2
defines a real-analytic mapping $ : C + 2 for some open neighborhood C of 0 E 2 Since $ ' ( O ) = idz , we may assume that $ : C + D is bianalytic, where D is an open neighborhood of 0 E B For B = $ ( b ) E D , 5.23 implies gB(5) = exp(Xb)(ge) whenever ( 5 1 < 1 Now let f : 2 + 2 be a real-analytic mapping vanishing on the closed real submanifold M := U ( 2 ) of 2 For every z E M , b - z b * z Hence is contained in TZ(M) = { v E 2 : v*z + z * v = 0 }
.
.
.
.
.
xh is tangential to M , and 5.1.1 implies f(gB(5)) = 0 whenever I 5 1 = 1 Applying this to f ( z ) : = e - z z * and f(z) : = e - z * z , we get
.
gs(5)
whenever for all
(51 =
6
E
I3
1
.
E
15.24.3
U(2)
and I s ( is small. By 1.11, this is true Now the assertion follows from 15.24.1. O.E.D.
15.25 PROOF.
COROLLARY.
B
=
co(exp u ( Z ) )
.
By 15.24, it suffices to show that U ( Z ) C co(exp u ( Z ) ) Let E U(2) and 0 < s < 1 Let W be the abelian unital C*-algebra generated by B For ( 5 ) E U(W) Since 151 = 1 , 15.24.3 implies gS B g o ( 3 ) = ge E exp(u(W)) , 15.23 implies ( 5 ) E exp(u(W)) 's8 For s + 1 , the By 15.24.1, we get sB E co(exp u(W)) assertion follows. O.E.D.
.
.
.
.
.
.
The Russo-Dye Theorem, in its stronger form 15.25, will now be applied to obtain a metric characterization of C*-algebras known as the Vidav-Palmer Theorem. Suppose in the
258
SECTION 15
following that Z algebra such that 15.26 x,y
LEMMA. E
H(2)
,
is an associative unital complex Ranach C
~ ( z )
2 =
.
The mapping x + iy + x - iy , for is an algebra involution of Z
.
h 2 E Z has a unique x,y E X Since h and
PROOF. For any h E X := H ( 2 ) , decomposition h2 = x + iy with h2 commute, 14.29.2 implies hx
-
xh = i(yh-hy)
E
.
X A iX = { O }
.
.
Therefore x commutes with h2 and hence with y be a maximal abelian subalgebra of 2 containing x y Then
Let and
.
XZ(h
2
) =
CW(h
2
) =
{ f(x) + if(y) : f
E
Cw }
W
.
Now 14.30 implies f(x) E R , f(y) E R and zZ(h 2 ) = ~ , ( h ) C ~ R Hence f(y) = 0 for all f E C w , showing that C,(y) = Cw(y) = { O } By 14.30, y = 0 It follows that h E X implies h 2 E X and, consequently, xy + yx E X whenever x,y E X Further, i(xy-yx) E X by 14.29.1. Now let a = x+iy , b = h+ik E A Then
.
.
.
.
ab =
Similarly,
.
+ b*a*
=
(x+iy)(h+ik) + (h-ik)(x-iy)
(xh+hx)
-
(yk+ky) + i(yh-hy)
(ab-b*a*)/i
E
X
.
+ i(xk-kx)
Now (ab)* = b*a*
ab = (ab+b*a*)/2 + i(ab-b*a*)/2i
E
.
X
follows from
. O.E.D.
15.27 THEOREM. An associative unital complex Ranach algebra 2 is a C*-algebra if and only if Z = H ( ' 2 ) ' In this case the involution is given by (x+iy)* = x-iy for
.
x,y
E
ff(Z)
.
.
PROOF. Put X := H ( 2 ) Then for every unital complex C*-algebra 2 , X is the self-adjoint part of 2 by 15.8.
c - -ALGEBRAS
259
Hence Z = Xc and the involution is given by (x+iy)* = x-iy C Conversely, suppose Z = X By 14.36 for all x,y E X
.
.
and 15.26, the canonical involution of Z is a continuous algebra involution which is hermitian by 14.30. By 14.27, there exist a complex Hilbert space E and a unital *homomorphism II : 2 + L ( E ) satisfying 14.27.1 and 14.27.2. It follows from 14.36 that II is a homeomorphism onto a unital closed *-subalgebra n ( 2 ) of L ( E ) By 15.10, n ( 2 ) is a C*-algebra. Now assume z E 2 satisfies lnzl = 1 By 15.25, there exists a sequence zn E co(iX) such that Therefore z n + z and hence I z I < 1 = 111zl n(zn) + n ( z ) It follows that since lexp(ix)l = 1 for all x E X NOW assume I z ( < 111zl for I z ( < 111z1 for a 1 z E z z*I < III(z*)\ = ( ( n z ) * l , 14.27.2 implies Since some z
.
.
.
.
.
(IT212 =
z*zI
<
.
<
IZ*I.IZI
I(IIz)*l*(IIzl
= lnzl
2
I
O.E.D.
a contradiction.
*
NOTES. The theory of (complex) C -algebras is by now a major branch of functional analysis, with many applications to operator theory, mathematical physics and the theory of group * representations c29,30,118,22,120 1. Real C -algebras have been considered in connection with "continuous geometries" C141 and occur also in the study of "reversible" Jordan operator algebras C133,134 1. The idea of using Moebius transformations for the proof of the Russo-Dye Theorem 15.24 is due to L. Harris C571, cf. also 117; 5 381. A similar proof applies to the Russo-Dye * Theorem for Jordan C -algebras due to J. Wright and M. Youngson C1571 and to the more general version involving (circular) bounded symmetric domains D with non-empty extremal boundary S C911. More precisely, it is shown in C91; Theorem 5.91 that 6 is the closed convex hull of any connected component of S . The Vidav-Palmer Theorem 15.27 can also be generalized to give a metric characterization * of Jordan C -algebras C159,107,1171.
24 7
a
15.
c*-ALGEBRAS
The theory of order-unit Ranach spaces, their states and representations, will now be applied to study an important class of involutive Ranach algebras, the so-called C*-algebras. C*-algebras are the prototype of algehras connected with symmetric Ranach manifolds and reveal an interesting relationship between infinite-dimensional holomorphy and functional analysis.
Since C*-algebras are
also fundamental for the algebraic formulation of quantum mechanics [120] they provide a link for possible applications of infinite-dimensional holomorphy to problems in mathematical physics [144]. 15.1 DEFINITION. algebra over K
.
Suppose 2 is an associative Ranach An involution z + z * of 2 is called a
C*-involution if Iz*zI =
.
2
15.1.1
z E 2 It follows that ( z I L < Iz*I-IzI and hence < (z*I , i.e., Iz1 = I z * I for all z E 2 Therefore
for all 121
IZI
.
every C*-involution is isometric and hence continuous. 2 = Combining this fact with 15.1.1, we get I z z * ( = I z * I for all 15.2
z
E
2
EXAMPLE.
.
.
Let
E
and
F
121
2
be Hilbert spaces over
D E {R,C,H} For any operator z E L ( E , P ) , define the by (zhlk) = (hlz*k) for all adjoint operator z * E L ( F , E )
.
E ExF Then (z*)* = z and (zw)* = w*z* whenever Since F,L) and L is another Hilbert space over D (z*zh k ) = (zhlzk) for all h,k E E , the Cauchy-Schwarz i nequa ity [28; 6.2.11 implies
(hik)
W
E
1
.
SECTION 15
248
Hence z + z* is a C*-involution of the associative unital Ranach algebra Z := L ( E ) over the center K of D , endowed with the operator norm. 15.3 EXAMPLE. Let 2 := g ( S , D ) denote the associative unital Ranach algebra (over the center K of D ) of all bounded D-valued functions on a set S , endowed with the supremum norm. For z E 2 define the adjoint function z* E 2 by z * ( s ) := z ( s ) * , where * denotes the canonical involution of D Since I X * X I = 1 x 1 ~for all x c D , it follows that z + z * is a C*-involution of 2 The closed subalgebras Lm(S,D) , Cm(S,D) and Cu(S,D) of B ( S , D ) , associated with a measure space S , a topological space S and a locally compact space S , respectively, are invariant under the C*-involution.
.
.
15.4 LEMMA. Let 2 be an associative Ranach algebra over K , endowed with a C*-involution. Then z*z = zz* implies n I l/n
I z ( = lim Iz
PROOF.
=
For
I (z*z)
n
N
E
2n-1 2
I
=
,
n+m 15.1.1 implies
2n-2 4 ((z*z) I
=
... =
.
15.4.1
Iz*zI
2"
=
2n+1 121
Hence 1 2 2"
I2-"
=
(z(
.
Since the limit 15.4.1 exists [17; 2.81, it is equal to
.
IzI
O.E.D.
15.5 COROLLARY. Let W be an abelian Ranach algebra over K , endowed with a C*-involution. Let C w be the (locally compact) spectrum space of W Then the Gelfand mapping
.
is an isometric homomorphism onto a closed subalgebra over
K
*
C -ALGEBRAS separating the points of
PROOF. 15.6
Cw
249
.
Apply 15.4 and [17, $171. LEMMA.
Suppose
Z
O.E.D.
is a non-unital associative Ranach
algebra over K I endowed with a C*-involution. Then the canonical involution on the unitization 2 ' := 28K of Z
is
a C*-involution with respect to the Ranach algebra norm I z + s I = sup
.
Izw+swI
IWI
PROOF. F o r x E 2 ' , let Liy : = xy denote left By definition, 1x1 = ILi12 multiplication on 2'
.
L;
operator norm of
x,y
for all
on the ideal
2
of
.
2'
is the
Hence
e E 2' satisfies x = z + s satisfies E 2 If s # 0 , it is a left unit of 2 Since follows that c : = -z/s wc* = (CW*)* = (w*)* = w and c* = cc* = c , c is a unit of 2 This contradiction shows s = 0 Since E
2'
I
and the unit element
.
lel = ILE,I = lid12 = 1 Now suppose 1x1 = 0 Then zw = -sw for all w
.
.
.
lL;lz
<
.
.
and
I Z I
= IL;I
for all
z
E
2
,
*I
it follows that
is a norm on
coincides on 2 with the original norm. z E 2 , we have xz E 2 and hence
lLiZ12 =
(XZl2 =
I(xz)*(xz)l
=
2
.
< Iz*I*Ix*xI*(zI =
IZI
Ix*xI
x
For
E
2' 2'
which and
Iz*(x*x)zl
2
It follows that every x E 2' satisfies 1x1 c Ix*xI < I x * I * I x I , hence 1x1 < Ix*l and therefore ( x I 2 = Ix*xI , showing that the involution on 2' is a C*-involution. In is a complete norm on 2 ' , consider order to show that 1 - 1 the left translation homomorphism L ' : 2 ' + ~ ( 2 ), defined by
2 50
SECTION 1 5
.
x + L' Since L' is isometric, the image L'(Z) is X complete and hence closed in L ( 2 ) Therefore L'(Z') = L'(Z) @ K*idZ is also closed in L(Z) and is therefore complete. This shows that Z' is a Banach algebra.
.
Q.E.D.
LEMMA. Every C*-involution on an associative Ranach L, and R, belong algebra Z over K is -hermitian, i.e., to ui(Z) whenever z* = -2 15.7
.
s unital. Put PROOF. By 15.6, we may assume that 2 w : = exp(z) E 2 Since the involution is continuous, we get
.
w*
=
exp(z*)
=
exp(-z)
exp(z
=
.
Therefore lwI2 = (w*w( = le( < 1 exp(LZ) = Lw E U k ( 2 ) and exp(RZ)
- L-1w.
Therefore Rw E U k ( 2
Q.E.D.
=
15.8 COROLLARY. Every C*-involution on a comp ex associative Banach algebra 2 is hermitian and H ( 2 ) = { x E 2 : x* = x }
.
PROOF. We may assume that Z is unital. By 15.7, every self-adjoint element is hermitian, showing that the involution
is 'hermitian and hence hermitian. hermitian element is self-adjoint.
By 14.29 2, every
Q.E.D.
Note that the identity mapping is a C*- nvolution on , considered as a real Ranach algebra, which is not 'hermitian. Hence 15.8 is not true for K = R . 2 = C
DEFINITION. An associative Ranach algebra 2 over endowed with a C*-involution, is called a C*-algebra if
K
15.9
x = x*
E
2
Cz(x) C R
.
15.9.1
By 15.8, the spectral condition 15.9.1 is redundant for K = c By 15.6 and 14.19, the unitization 2 ' = 2 @ K non-unital C*-algebra Z over K is a C*-algebra.
.
15.10
LEMMA.
Suppose
Z
is a C*-algebra over
,
K
and
of a
W
is
251
C*-ALGEBRAS
a closed *-subalgebra of PROOF. For x = x * E W C \ C,(x) is connected. Cw(x) = C,(x)
.
2
.
Then
W
is a C*-algebra.
, we have
C,(x) c R and hence Therefore [ 1 7 ; 5.141 implies O.E.D.
15.11 EXAMPLE. Let E be a Hilbert space over D E {R,C,E} and consider the associative unital Banach algebra Z := L(E) over the center K of D , endowed with the operator norm and the C*-involution defined in 15.2. In case D = C , 15.8 implies that Z is a complex C*-algebra. In case D = R , Ec : = EORC is a complex Hilbert space and Zc can be identified with L (Ec) Hence x = x * E 2 implies
.
In case D = H , E showing that Z is a real C*-algebra. can be regarded as an even-dimensional complex Hilbert space, denoted by 15.11.1
such that
u
H
carries a conjugation u + and a b z = t L -) : a r b E L (H) } -b a is a purely real closed unital *-subalgebra of L(EC)
-
-
.
au : = au for all a E L(H) and u E H The can be identified with L(EC) complexification 2' above, it follows that 2 is a real C*-algebra.
.
.
Here
As
15.12 EXAMPLE. Let S be a set and consider the associative unital Ranach algebra 2 := B ( S , D ) over the center K of D E {R,CIE} I endowed with the supremum norm and the C*-involution defined in 15.3. In case D = C , 15.8 implies that 2 is a complex C*-algebra. In case D = R , 2 is a real C*-algebra since 2' = S ( S , C ) In case D = H , consider the associative unital complex Ranach algebra A := B ( S , L 2 ( C ) ) of all bounded mappings f : S + L 2 ( C ) , endowed with the supremum norm (with respect to the operator
.
SECTION 1 5
252
norm on L z ( C ) ) and the C*-involution f + f* defined by f*(s) := f ( s ) * for all s E S Then A is a complex C*-algebra by 15.8. and s E S Then a
.
z = t L
Put
-
.
a(s) : = a(s)*
for all
a
E
B(S,C)
b
-)
: a,b E B ( S , C ) } -b a is a purely real closed unital *-subalgebra of A whose complexification Zc can be identified with A Hence Z is a real C*-algebra. qpplying 15.10, it follows that the Ranach *-algebras L m ( S , D ) , C m ( S , D ) and C u ( S , D ) I associated with a measure space S , a topological space S and a locally compact space S , respectively, are C*-algebras over K
.
.
15.13 K , z
E
LEMMA. Suppose W is an abelian C*-algebra over In case K = R , assume in addition that z * = z for all W , Then the Gelfand homomorphism
is an isometric *-isomorphism.
.
PROOF. By 15.9.1, Q is a *-homomorphism into C u ( C w , K ) Hence 15.5 implies that Q is an isometric embedding onto a closed *-subalgebra A of c u ( C w , K ) separating the points of Cw The Stone-Weierstrass Theorem [ 2 8 ; 7.31 implies A = cu(CW,K) O.E.D.
.
15.14 K
.
COROLLARY.
The involution of a C*-algebra
Z
over
is hermitian.
PROOF. For x = x* E Z , consider the closed *-subalgebra W of 2 generated by x By 15.10, W is an abelian C*-algebra over K , Applying 15.13 and [ 1 7 ; 5.141, we get
.
15.15 COROLLARY. An associative Ranach algebra Z over K endowed with a C*-involution, is a C*-algebra if and only if
,
*
C -ALGEBRAS
253
> 0
CZ(Z*Z)
15.15.1
.
In particular, in case for all z E Z C*-involution is positive.
K = C!
,
every
PROOF. The spectral mapping theorem and 15.15.1 imply Conversely, suppose 2 is a Cz(x) C R whenever x* = x C*-algebra. To show 15.15.1, we may assume that 2 is unital. Then the assertion follows from 15.14 and 14.23.
.
Q.E.D.
15.16 COROLLARY. adjoint part X
Let 2 be a C*-algebra over Then the set
.
x+
{ x
:=
of all positive elements in X satisfying
x+= { z
If
x
L
: x
has a unit element C := X + n
X
x }
E
e
G(2) =
x
E
is
{ x
a
{ z*z:
=
,
: C,(X)
K
with self-
> 0 ]
closed convex cone in
2
E
2
}
.
15.16.1
the set E
X : C,(x)
> 0 }
of all strictly positive elements in X is a topologically The order regular open convex cone in X containing e on X coincides with the given norm. unit norm
.
I -Ie
PROOF. In case 2 is unital, the assertion follows from 15.15, 14.16 and 14.24. Now suppose 2 is non-unital and let x E X+ Then x = (y+s) = y 2 + 2sy + s2 , where is self-adjoint. Hence s2 = 0 , i.e., y + s E 2' s = 0 Thus 15.16.1 holds also in the non-unital case.
.
.
Q.E.D.
is a Hilbert space over D E {R,C,E} Let X : = H&(E) denote the self-adjoint part of the unital C*-algebra 2 := L(E) over the center K of D The elements of 15.17
.
EXAMPLE.
.
Suppose
E
S E C T I O N 15
254
are called positive operators on
E
,
and the interior
consists of all strictly positive operators on
E
.
15.18 LEMMA. Suppose 2 is a complex C*-algebra. Then for every x = x* E 2 , the left and right multiplication If operators Lx and Rx are hermitian operators on 2 x is positive, then Lx and Rx are positive.
.
PROOF.
for all
We may assume that
t
E
R
.
Hence
Lx
2
E
is unital.
.
~ a ( 2 )
Ry 15.7,
Since [17; 5.41
2 (Lx ) = C,(x) , it follows that Lx E ~ a + ( z ) if x positive. The analogous statements for Rx follow by C!i
considering the "opposite" C*-algebra of
2
.
implies is O.E.D.
By 15.11 and 15.10, every closed *-subalgebra 2 L ( E ) , for a Hilbert space E over K E {R,C} , is a
of
.
C*-algebra over K Conversely, the Gelfand-Neumark embedding theorem asserts that every C*-algebra 2 over K can be realized as an operator algebra, i.e., as a closed *-subalgebra of L ( E ) for some E
.
Let 2 be a C*-algebra over K , Then there exist a Hilbert space E over K and an isometric *-isomorphism n : 2 + L ( E ) onto a closed *-subalgebra of 15.19
THEOREM.
L(E)
By 15.6, we may assume that 2 is unital. and 14.26, there exist a Hilbert space E over K *-homomorphism n : 2 + L ( E ) such that PROOF.
.
By 15.14 and a
(nzl = ~ z * z I ~ ' =~ I z I for all z E 2 Hence n is isometric and n ( 2 ) is a closed *-suhalgebra of L ( E )
. O.E.D.
C * -ALGEBRAS 15.20
COROLLARY.
C*-algebra PROOF. 15.21
255
The complexification
of a real
2'
is a complex C*-algebra.
2
O.E.D.
Apply 15.19, 15.11, and 15.10. LEMMA,
.
Let
L
be a Hilbert space over
D E {R,C,H} Then a linear operator g E GQ(L) is g*g = id, isometric if and only if g is unitary, i.e., Y
.
In particular, UL(L) = { g
GQ L ) : g*g = id L ]
E
15.21.1
is a real Ranach Lie group in the operator norm topology whose Lie algebra can be identif ed with the closed real subalgebra U ( L ) := {
x
x* + x
gQ(L) :
6
of all skew-adjoint operators on L real Ranach Lie subgroup of GQ(L) PROOF.
Every unitary operator
g
E
.
.
= 0
}
Further,
GQ(L)
UQ(L)
satisfies
is a
lgul
2
= (gulgu) = (g*gulu) = ( u ( u ) = Iu
for all u E L and is therefore isometric. Conversely, every isometric operator g E UL(L) satisfies (gulgu) = ulu) for all u E L
.
Polarizing this identity (in case D # R 1, we get 15.21.1. It follows that Ui(L) is a real algebraic subgroup of degree
.
< 1 in (9,g-l) Now apply 7 . 1 4 . The last assertion follows from the fact that uR(L) is a split real subspace of gn(L)
.
In case
O.E.D.
L = Dn
Uin(D) := UQ(Dn) 15.22
COROLLARY.
is finite-dimensional, we write
and
uQn(D) : = uQ(Dn)
Let
2
.
be a unital C*-algebra over
K
.
Then
.
is the group of all unitary elements in 2 In particular, U(2) is a real Ranach Lie group in the norm topology whose Lie algebra can be identified with the closed real subalgebra
SECTION 15
256
u ( 2 ) :=
{ x
E
of all skew-adjoint elements in 2 real Ranach Lie subgroup of G ( 2 )
g(Z) : x* + x = 0 }
. .
Further,
is a
U(2)
PROOF. By 15.19 and 15.21, an element g E G ( Z ) satisfies lgl = Ig-ll < 1 if and only if g is unitary. Now apply 7.14. The last assertion follows from the fact that u ( 2 ) is a split real subspace of g(Z) O.E.D.
.
By 15.22, exp(u(2)) any unital C*-algebra 2 more is true:
is a neighborhood of
over
K
.
e
U(Z)
for For abelian C*-algebras, E
15.23 LEMMA. L e t W be an abelian unital C*-algebra over K Then the identity component of the Ranach Lie group
.
U(W)
coincides with
exp(u(W))
.
PROOF. The identity component of U ( W ) consists of all elements of the form exp(xl) exp(x2) exp(xn) with n E N and xl, xn E u ( W ) Now the assertion follows O.E.D. from 2.8.2.
...,
...
.
We are now going to prove two fundamental geometric results about unital complex C*-algebras, namely the Russo-Dye Theorem ( 15.24 and 15.25) and the Vidav-Palmer Theorem (15.27). The proof given here makes decisive use of holomorphic mappings and shows how infinite-dimensional holomorphy can be applied to problems in functional analysis. 15.24 THEOREM. Let 2 be a unital complex C*-algebra with open unit ball R and unitary group U ( Z ) Then -B = co U ( Z )
.
.
B C co U ( 2 )
PROOF. It suffices to show that the mapping
is holomorphic for
( c ( < (B)-'
.
.
For
8
E
R
Ry 1.12, it follows that
,
C -ALGEBRAS 6 = g (0) = B
For
b
E
2
,
d5 =
1
257
2n gB(e it )dt
1 0
.
15.24.1
consider the vector field
Then 5.23 implies that
,
$(b) := exp(Xb)(0) = tanh(bb*)1'2 ( b b * )2'1
15.24.2
defines a real-analytic mapping $ : C + 2 for some open neighborhood C of 0 E 2 Since $ ' ( O ) = idz , we may assume that $ : C + D is bianalytic, where D is an open neighborhood of 0 E B For B = $ ( b ) E D , 5.23 implies gB(5) = exp(Xb)(ge) whenever ( 5 1 < 1 Now let f : 2 + 2 be a real-analytic mapping vanishing on the closed real submanifold M := U ( 2 ) of 2 For every z E M , b - z b * z Hence is contained in TZ(M) = { v E 2 : v*z + z * v = 0 }
.
.
.
.
.
xh is tangential to M , and 5.1.1 implies f(gB(5)) = 0 whenever I 5 1 = 1 Applying this to f ( z ) : = e - z z * and f(z) : = e - z * z , we get
.
gs(5)
whenever for all
(51 =
6
E
I3
1
.
E
15.24.3
U(2)
and I s ( is small. By 1.11, this is true Now the assertion follows from 15.24.1. O.E.D.
15.25 PROOF.
COROLLARY.
B
=
co(exp u ( Z ) )
.
By 15.24, it suffices to show that U ( Z ) C co(exp u ( Z ) ) Let E U(2) and 0 < s < 1 Let W be the abelian unital C*-algebra generated by B For ( 5 ) E U(W) Since 151 = 1 , 15.24.3 implies gS B g o ( 3 ) = ge E exp(u(W)) , 15.23 implies ( 5 ) E exp(u(W)) 's8 For s + 1 , the By 15.24.1, we get sB E co(exp u(W)) assertion follows. O.E.D.
.
.
.
.
.
.
The Russo-Dye Theorem, in its stronger form 15.25, will now be applied to obtain a metric characterization of C*-algebras known as the Vidav-Palmer Theorem. Suppose in the
258
SECTION 15
following that Z algebra such that 15.26 x,y
LEMMA. E
H(2)
,
is an associative unital complex Ranach C
~ ( z )
2 =
.
The mapping x + iy + x - iy , for is an algebra involution of Z
.
h 2 E Z has a unique x,y E X Since h and
PROOF. For any h E X := H ( 2 ) , decomposition h2 = x + iy with h2 commute, 14.29.2 implies hx
-
xh = i(yh-hy)
E
.
X A iX = { O }
.
.
Therefore x commutes with h2 and hence with y be a maximal abelian subalgebra of 2 containing x y Then
Let and
.
XZ(h
2
) =
CW(h
2
) =
{ f(x) + if(y) : f
E
Cw }
W
.
Now 14.30 implies f(x) E R , f(y) E R and zZ(h 2 ) = ~ , ( h ) C ~ R Hence f(y) = 0 for all f E C w , showing that C,(y) = Cw(y) = { O } By 14.30, y = 0 It follows that h E X implies h 2 E X and, consequently, xy + yx E X whenever x,y E X Further, i(xy-yx) E X by 14.29.1. Now let a = x+iy , b = h+ik E A Then
.
.
.
.
ab =
Similarly,
.
+ b*a*
=
(x+iy)(h+ik) + (h-ik)(x-iy)
(xh+hx)
-
(yk+ky) + i(yh-hy)
(ab-b*a*)/i
E
X
.
+ i(xk-kx)
Now (ab)* = b*a*
ab = (ab+b*a*)/2 + i(ab-b*a*)/2i
E
.
X
follows from
. O.E.D.
15.27 THEOREM. An associative unital complex Ranach algebra 2 is a C*-algebra if and only if Z = H ( ' 2 ) ' In this case the involution is given by (x+iy)* = x-iy for
.
x,y
E
ff(Z)
.
.
PROOF. Put X := H ( 2 ) Then for every unital complex C*-algebra 2 , X is the self-adjoint part of 2 by 15.8.
c - -ALGEBRAS
259
Hence Z = Xc and the involution is given by (x+iy)* = x-iy C Conversely, suppose Z = X By 14.36 for all x,y E X
.
.
and 15.26, the canonical involution of Z is a continuous algebra involution which is hermitian by 14.30. By 14.27, there exist a complex Hilbert space E and a unital *homomorphism II : 2 + L ( E ) satisfying 14.27.1 and 14.27.2. It follows from 14.36 that II is a homeomorphism onto a unital closed *-subalgebra n ( 2 ) of L ( E ) By 15.10, n ( 2 ) is a C*-algebra. Now assume z E 2 satisfies lnzl = 1 By 15.25, there exists a sequence zn E co(iX) such that Therefore z n + z and hence I z I < 1 = 111zl n(zn) + n ( z ) It follows that since lexp(ix)l = 1 for all x E X NOW assume I z ( < 111zl for I z ( < 111z1 for a 1 z E z z*I < III(z*)\ = ( ( n z ) * l , 14.27.2 implies Since some z
.
.
.
.
.
(IT212 =
z*zI
<
.
<
IZ*I.IZI
I(IIz)*l*(IIzl
= lnzl
2
I
O.E.D.
a contradiction.
*
NOTES. The theory of (complex) C -algebras is by now a major branch of functional analysis, with many applications to operator theory, mathematical physics and the theory of group * representations c29,30,118,22,120 1. Real C -algebras have been considered in connection with "continuous geometries" C141 and occur also in the study of "reversible" Jordan operator algebras C133,134 1. The idea of using Moebius transformations for the proof of the Russo-Dye Theorem 15.24 is due to L. Harris C571, cf. also 117; 5 381. A similar proof applies to the Russo-Dye * Theorem for Jordan C -algebras due to J. Wright and M. Youngson C1571 and to the more general version involving (circular) bounded symmetric domains D with non-empty extremal boundary S C911. More precisely, it is shown in C91; Theorem 5.91 that 6 is the closed convex hull of any connected component of S . The Vidav-Palmer Theorem 15.27 can also be generalized to give a metric characterization * of Jordan C -algebras C159,107,1171.