- Email: [email protected]
Operator theory in the C∗-algebra framework
Vol. 31 (1992)
REPORTS ON MATHEMATICAL
OPERATOR
No. 3
PHYSICS
THEORY IN THE C*-ALGEBRA
FRAMEWORK
S. L. WORONOWICZand K. NAPI~RKOWSKI Department Faculty
of Physics,
of Mathematical
Warsaw
University,
Methods Hoia
in Physics,
74, 00-682 Warszawa,
Poland
(Received April 4, 1992)
Properties of operators affiliated with a C*-algebra are studied. A functional calculus of normal elements is constructed. Representations of locally compact groups in a C*-algebra are considered. Generalizations of Stone and Nelson theorems are investigated. It is shown that the tensor product of affiliated elements is affiliated with the tensor product of corresponding algebras.
0. Introduction
Let H be a Hilbert space and CB(H) be the algebra of all compact operators acting on H. It was pointed out in [17] that the classical theory of unbounded closed operators acting in H [8, 9, 31 is in a sense related to CB(H). It seems to be interesting to replace in this context CB(H) by any non-unital G-algebra. A step in this direction is done in the present paper. We shall deal with the following topics: the functional calculus of normal elements (Section l), the representation theory of Lie groups including the Stone theorem (Section 2, 3 and 4) and the extensions of symmetric elements (Section 5). Section 6 contains elementary results related to tensor products. The perturbation theory (in the spirit of T. Kato) is not covered in this paper. The elementary results in this direction are contained the previous paper of the first author (cf. [17, Examples 1, 2 and 3 pp. 412-4131). To fix the notation we remind the basic definitions and results [17]. Let A be a C*-algebra and T be a linear mapping acting on A defined on a dense linear domain D(T). The adjoint mapping T* is introduced by the following equivalence (X,Y E A) * for any a E D(T) x E D(T*) (0.1) and y = T’x ( ) ( (Tu)*x = a*y > ’ Clearly T* is a closed linear mapping. Even in T is bounded there is no guarantee that D(T*) is dense in A. However, if T is bounded, so is T’. Indeed, approximating y by a E D(T) we obtain Y*Y = P’Y)*G 11~11~ L IITYII lbll I IITII llvll II41 and llvll 5 IITII 1141. Let B(A) denote the Banach algebra of all bounded linear mappings acting on A defined on the whole A (D(a) = A for all a E B(A)) and let M(A)
= {u E B(A) [3531
: u* E B(A)}.
S. L. WORONOWICZ
354
and K. NAPIbRKOWSKI
Then M(A) is a norm-closed subalgebra of B(A), a* E M(A) for any a E M(A) and M(A) endowed with the *-operation is a unital C*-algebra. It coincides with the multiplier algebra of A (see e.g. [12]). Assuming that elements of A act on A by left multiplication we include A L-) AI(A). A is an ideal in M(A). The natural topology on Al’(A) is the topology of almost uniform convergence. We say that a net (a,) of elements of M(A) converges (almost uniformly) to 0 if /]a,~]] --f 0 and IlaLz]] + 0 for any 5 E A. For example if (e,) is an approximate unity for A ([5]) then e, converges to I almost uniformly. Using this fact one can easily show that A is dense in AI(A). The almost uniform topology remembers the position of A in M(A): A =
17:E AI(A) : 1
To prove this approximate unity hand e, E A and Remembering
lim ]]unxlI = 0 for any net (a,) of elements of Al(A) converging almost uniformly to 0 1
relation it is sufficient to consider a, = I - e,, where (e,) is the for A. Assuming l]ua~]] -+ 0 we get e,z + z in norm. On the other A is an ideal in Al(A). Therefore, e,x E A and x E A (A is complete!) that T* is bounded for any bounded T we get
PROPOSITION 0.1. Let A be a C*-algebra in A. Then T E Al(A).
and T E B(A).
Assume
that D(T*)
ti dense
Let A be a C*-algebra and T be a linear mapping acting on A having a dense domain D(T). We say that T is affiliated with A and write TVA if and only if there exists z E M(A) such that l]zl] 5 1, D(T) = JI-z”zA and Tdmu = zu for all a E A. It is known that z is determined by T. We call it z-transform of T and denote by zr. If TVA then T is closed. The set of all bounded elements affiliated with A coincides with Al(A). According to Theorem 1.4 of [17], if TVA then T*vA and zT* = z;.
(0.2)
Let A and B be C*-algebras. The set of all “-algebra homomorphisms ‘p from A into AI(B) such that (p(A)B is dense in B will be denoted by Mor(A, B). If TVA and cp E Mor(A, B) then there exists unique (p(T)vB such that (p(D(T))B is a core of v(T) and ~Q)cp(a)b
= ‘p(Ta)b
for any a E D(T) and b E B (see [17, Theorem 1.21). One can check that p(T) E M(B) for any T E Al(A). The resulting mapping cp : Ai + M(B) is a continuous unital “-algebra homomorphism (both multiplier algebras are considered with the topology of almost uniform convergence). In the general case (TVA), we have z,(T) = (p(zT). Remark: If DO is a core of T then p(Do)B is a core of p(T). This follows immediately from the definition. One can easily show (cf. [17, Theorem 1.41) that cp(T*) = q(T)*. The affiliation relation n was introduced by Baaj [4]. It was rediscovered in [17], where its properties were investigated in detail. The set of morphisms Mor(A, B) was introduced and investigated in [15] and [14]. In the following we shall need the following version of polar decomposition (cf. [17, Proposition 0.21).
OPERATORTHEORYIN THE C”-ALGEBRAFRAMEWORK
355
PROPOSITION0.2. Let c E M(A), where A is a C’-algebra. Assume that CA (c*A resp.) is a dense subset of eA (f A resp.), where e, f are selfadjoint projections belonging to M(A). Then there exists unique u E M(A) such that UU* = e and u*u = f.
c= u&G,
Proof Using [5, Lemma 2.9.41 one can easily show that c*cA = c*A and c*cA = &%A. Therefore, &%A is a dense subset of fA and the set of elements of the form a = &z
+ (I - f)y,
(0.3)
where 2, y E A is dense in A. Let us notice that u*a = z*z*cz = y*(I - f)y > (cs)*cz. Therefore, there exists a unique bounded linear mapping u : A + A such that ua = cz for any a of the form (0.3). In the similar way one can show that the set of elements of the form b = Jcctz’
+ (I - e)y’,
(0.4) where z’, y’ E A is dense in A and construct a unique bounded linear mapping u’ : A + A such that u’b = C*X’for any b of the form (0.4). Let us notice that (uu)‘b = 2*c*&zc Therefore, u’ = u*, hence u E M(A). requirements. 1. Functional
= X*&%*X’
= u*(u’b).
One can easily check that u satisfies all the Q.E.D.
calculus of normal elements
An element T affiliated with a C*-algebra A is called normal if D(T) = D(T*) and (Tu)*(Tu) = (T*u)*(T*a) for any a E D(T). This is equivalent to z$zr = zrz$. T is called selfadjoint if T = T*. We remind the description of the universal normal element given in [17]. Let _4be a closed subset of C and (4 be the element affiliated with C,(A) introduced by the formula
w(&I)
= T.
Let T be a normal element affiliated with a c-algebra tinuous function defined on A = Sp T. Then frtC,(A) in Theorem 1.1 we set
A and f be a complex valued conan d using the notation introduced
f(T) = VT(~). (1.1) Clearly f(T)vA. f(T) is a normal element (f is normal and due to [17, Theorem 1.21 any image of a normal element is normal). If f(x) = 1 for all A E C then f = It_(c)) and f(T) = CpT(Lc_(C) = LA. (For any C*-algebra A, by IA we denote the unity of M(A).) Similarly, if f(x) = X for all X E C then f = C and f(T) = (PT(C) = T. In general the definition (1.1) gives the meaning to many algebraic and analytical expressions containing T. For example, (I + T*T)sin(2T) = @(f), wh ere f is the continuous function on C
356
S. L. WORONOWICZ
and K. NAPIbRKOWSKI
such that f(x) = (1 + XX)sin(2X) for all X E C. Similarly, T(I + T*T)-l/* = (or where f(x) = X(1 + %A)-‘/’ for any X E C. One can easily verify that in this case f = ZC. Therefore, ZT = T(I + T*T)-l’*
(1.2)
for any normal TVA. The reader easily examines how the functional calculus introduced by (1.1) works in particular examples. If A is a C*-algebra of bounded operators acting in a Hilbert space (examples 3 and 4 in [17]) then f(T)
=
s f(XP(A)> SPT
where dE(.) is the spectral measure associated with the normal operator T: T =
s XdE(X). SPT
Considering A = C,(A), where A is a locally compact topological space, one can see that any T&‘,(A) is normal, SpT = T(A) and f(T)
= f o T.
(1.3)
Let A, B be C*-algebras and @ E Mor(A,B). is a normal element affiliated with B. Moreover, = ~&D(T)((). Therefore,
P*(T) = @ 0 VT
=
and
@(‘PT(.f))
=
@(f(T))
and
f(W’))
for
=
any
Then for any normal TVA, G(T) (@ o (PT)(<) = @(VT(<)) = Q(T) f
E
C(C)
we
get
.f(@(T))
=
‘P@(T)(f)
@U-(T)),
(14
Let A be a C*-algebra; ft, f2 E C(C) and TVA. Assume that T is normal. Replacing in (1.4) A, B, @, T and f by C,(C), A, VT, f2 and fr resp. and remembering that ft(f~) = fr o f2 (cf. (1.3) we obtain f1U2GY)
= (.fi 0 f2)Gf’h
(1.5)
Clearly (cf. Section 0) f(T)*
= f(T),
0.6)
where by definition f(x) = f(x) for any X E SpT. If f E Cbounded(SpT) then f(T) E M(A). Remembering we see that (A& +
~z.fi)(T)
(5
. f2)V)
= Xh(T) = h(T)
that PT E Mor(C,(SpT),
+ X2f2Q’)
. f2V)
A)
(1.7) (1.8)
for any fr , f2 E Choundrd(SpT) and XI, X2 E C. Moreover, if f E Cboun&d(SpT) and (fiL)n=1,2.... is a sequence of bounded continuous functions with a common bound such that f,, --+ f uniformly on any compact subset of SpT then endowing M(A) with the
OPERATOR THEORY IN THE C*-ALGEBRA
FRAMEWORK
357
almost uniform topology we have fn(T) + f(T).
(1.9)
Due to relations (1.5)-(1.8) the notation introduced by (1.1) is selfconsistent. over, if f(A) is a continuous bounded function such that Xf(X) is bounded then
f(T)z Kf(T)a)
More-
E D(T),
(1.10)
= (Tf(T))a:
(1.11)
for any 2 E A. The simple proof is left to the reader. Let A be a C*-algebra and let TVA. We say that T is skew-adjoint (skew-symmetric resp.) if T* = -T (T* > (-T) resp.). Like in the Hilbert space operator theory we have LEMMA
Proof
S c (-S*)
1.2. Skew-adjoint elements have no proper skew-symmetric extensions. Let A be a C*-algebra, T, SvA, T* = -T, c (-T*) = T and S = T.
s* > (-S)
and T c S. Then Q.E.D.
Clearly, any skew-adjoint element T is normal. ‘Its spectrum is contained in iR. Indeed, denoting by C the element affiliated with Coo(C) introduced by the formula [(A) = X for all X E C, we have: (PT(ZC) = zr = -2; = -(or. Therefore, VT vanishes on the closed ideal in C,,,,,,,(C) generated by the function zc + Zc, which consists of all bounded continuous functions vanishing on iR. It means (cf. [17, formula (1.20)] that Sp T c iR. Let T be a skew-adjoint element affiliated with a C*-algebra A. Taking into account the localization of Sp T described above and using (1.5)-( 1.9) we obtain (ctT)* = e---tT = (&y-l, (1.12) &Tet27-
=
e(tl
+t,v
(1.13)
>
fimoetT = 1, etT - I
%
-(I
t
(1.14)
+ T*T)- l/2 = T(I + T*T)-1/2
for any t, tl, t2 E R. The above limits are understood in the sense of almost uniform convergence. Let us notice that (I + T*T)-‘/2 = (I - .z$z~)-~/~. Therefore, for any a E D(T) there exists 2 E A such that a = (I + T*T)-1/2z and using the last of the above formulae we get etT - I norm- lim ---a=Ta (1.15) t+o
t
for all a E D(T). 2. Infinitesimal representations of Lie groups
Let G be a locally compact group and A be a C*-algebra. u : G + M(A) is a unitary representation of G in A if
We say that a mapping
1. u(g) is unitary for any g E G. 2. u(gigz) = u(gi)u(sz) for any gi,g2 E G. 3. For any a E A the mapping G 3 g - u(g)a E A is norm continuous.
358
S.L. WORONOWICZandK.NAPI~RKOWSKl
Let us remind that elements of nf(A) are continuous linear operators on A (they act by left multiplication). Therefore, we may use all the concepts and results of the general theory of linear actions of locally compact groups on Banach spaces [l]. In particular, in the Lie group case, we may consider infinitesimal representations of G. From now till the end of this Section we assume that G is an N-dimensional Lie group. Let g be tHe Lie algebra and & be the enveloping algebra of G. By definition g consists of all right-invariant real vector fields on G. Consequently, elements of E are differential operators on G commuting with the right shifts. & is equipped with an antilinear and antimultiplicative involution I 3 A1 H AP E E such that X+ = -X for all X E g. Let u be a unitary representation of G in a C*-algebra A. The corresponding infinitesimal representation will be denoted by dl~. Operators &(A[) (where M E I) are defined on the invariant dense domain D”(U)
=
a E A:
1
the mapping G 3 g H u(g)u E A is of C30-class in the sense . of norm topology in A I
For any Af E E and a E D”(u) &(A@
= nrz1(g)&]9,e.
Clearly, D”(u) is invariant under all u(h), h E G. Replacing in the above formula a by u(h)u and remembering that A1 commutes with right translations we get du(M)zl(h)u = AIu(g)ul,=h. This formula may be rewritten in the following way: = Ilfu(h)u.
du(Al)u(h)u
(2.1)
Let a, b E D”(U) and X E g. Applying X to the constant function (u(g)u)*u(g)b and using the Leibniz rule we get
(du(X)u)*b
= u*b
= -u*(du(X)b).
Therefore (du(Af)u)*b = u*du(M+)b for any a, b E Dm(u) and A1 E 1. Using this formula one can easily show that for all A1 E & operators du(A1) are closable. In what follows the closure of du(M) will be denoted by u(Af). AFFILIATION PROBLEM.
unitary
representation
Characterize
all elements A1 E I such that u(i’bl)~A for any
u of G in a C*-algebra A.
The following theorem provides a simple necessary condition THEOREM 2.1. Let u be a unitary representation of G in a C*-algebra A and M E E. Assume that f = 0 is the only bounded C”-solution to the differential equation M+Mf = -f_ Then u(Ar), z~(Al+)nA and u(Af)* = u(hf+).
Proof
By virtue
of [17, Proposition a - du(Af
K
2.21 it is sufficient
+)b
dzl(hf)u + b >
: a, b E IF(u)
to show that the set
OPERATOR THEORY IN THE C*-ALGEBRA
FRAMEWORK
359
is dense in A @ A. Let w and w’ be continuous linear functionals on A such that w(a - du(M+)b) + w’(du(M)a
+ b) = 0
for all a,b E DO”(u). We have to show that w = w’ = 0. Let c E D”(U). (2.2) u(g)c and --CIU(M)U(g)c instead of a and b we get f + M+Mf
(2.2) Inserting in
= 0,
on G. Therefore, w(u(g)c) = 0, where f(g) = w(u(g) c) 1s . a bounded C”-function w(c) = 0 and (D”( ‘u.) is dense in A) w = 0. Now, inserting in (2.2) a = 0 we get w’(b) = 0 for all b E IF(u) and w’ = 0. Q.E.D. It is known that all elements 111 E g satisfy the assumption of Theorem 2.1. The same is true for the Nelson operator A = CX+Xi, where (Xi)+t,~,...,~ is a basis in g. Elements u(X~) (i = 1,2, . . . , N) are called infinitesimal generators of ‘1~.They are skew-adjoint and affiliated with A. Element USA is selfadjoint and positive. THEOREM 2.2. Let A, B be (Y-algebras, ‘p E Mor(B, A), u be a unitary representation of G in B and u(g) = cp(v(g)) for all g E G. Then 1. u is a unitary representation of G in A. 2. ZfM E E and v(M)qB then USA and
u(M) = cp(v(M)). Proof Statement 1 is trivial. We prove Statement 2. By definition cp(D”(v))A core for cp(w(M)) and for any b E D”(v) and a E A we have:
is a
cp(v(M))cp(b)a = cp(v(M)b)a = cp(Mv(g)bl,=,)a = (Mp(v(g)b)l,=,)a = (Mu(g)cp(b)lg=,)a = u(M)cp(b)a. Therefore, it is sufficient to show that cp(D”(w))A is a core for U(M). Let c E 050(u) and E > 0. Then u(h)c and zl(M)u(h)c (= Mu(h)c by virtue of (2.1)) depend continuously on h and there exists a neighbourhood U of e such that (2.3) Ilu(M)u(h)c - u(M)cII
< ;
(2.4)
for any h E U. Let f be a C”-function on G with a compact support contained in U such that f(h) 2 0 for all h E G and Jo f(h)dh = 1 (dh denotes the left-invariant Haar measure on G). We set %, =
jG
f(h)@)&
72, = 1 f(h)v(h)dh. G
S. L.
360 Clearly R,, E Al(A),
WORONOWICZ
and K. NAPI6RKOWSKI
and R, = cp(R,). Moreover,
R,, E M(B)
for all a E A. R,,, and R,, are regularizing operators: R,A c IF(u) and R,B c D”(V) (cf. the concept of the G&ding domain [6]). Let a E A. Using (2.1) and integrating by parts we get u(N)R,,a = u(N) J f(h)u(h)adh = J f(h)(&fu(h)u)dh = J (M+f)(h)u(h)udh. Therefore,
Il4W%4l
I QIMI,
(2.6)
where Q = J I(Al+f)(h)ldh. Let 8 = i .min(l, Q-l).
We know (cf. Section 0) that (p(B)A is dense in A. Therefore,
there exist b, , bz, . . . , bN E B and a],U2,...,aN
E A such that
N
IL
cp(bh)uk - c
L=l
II
5 6’.
Inserting in (2.5) and (2.6) a = C p(bk)uk - c we get N
IL dR,,h)a~ Il4W5 &‘Wda~
-R,c
k=l
-
II
(2.7)
I ;,
u(Af,RucII L 5.
(2.8)
k=1
On the other hand, due to (2.3) and (2.4) we have
lI%LC- cl1I ;>
(2.9)
I~u(AI)K!,c - u(M)cll 5 ;.
(2.10)
Combining (2.7) with (2.9) and (2.8) with (2.10) we get (2.11)
llc’ - cl1 5 c, Ilu(AI)c’ - u(AI)cll 5 E,
(2.12)
where c’ = C cp(R,,bk)uk. This way we showed that for any c E D”(u) and any E > 0 there exists c’ E p(D”(v))A such that the estimates (2.11) and (2.12) hold. It means that Q.E.D. cp(lF(w))A is a core for u(A1). The remaining part of this Section is mainly devoted to the case G = R. 2.3. Let u be a representation of R in a C’-algebra generator of u. Then
THEOREM
finitesimal
A and TVA be the in-
1. Tu = lim
t-0
zl(t)a - a t
.
(2.13)
OPERATORTHEORYINTHEC*-ALGEBRAFRAMEWORK
361
In particular, a E D(T) if and only if the limit on the right-hand side of (2.13) exists in the sense of the norm topology on A. 2. T is skew-adjoint (T* = -T) and using the functional calculus of normal elements we have (2.14)
u(t) = etT for all t E R
Proof: Ad 1. Let T be the operator introduced by (2.13). One can easily see that T d d is a closed extension of du z . Therefore, T II u z . On the other hand, due to the ( > ( > Leibniz rule (Ta)*b + a*Tb = 0 for all a, b E D(T). It means that T is skew-symmetric d and using Lemma 1.2 we obtain T = u z .
( > Ad 2. By virtue of (1.12)-(1.15) the mapping R 3 t H etT E M(A) is a unitary representation of R in A and its infinitesimal generator c0ntains.T. Using Lemma 1.2 and remembering that any representation is uniquely determined we get (2.14). Q.E.D. COROLLARY 2.4. Let u be a representation of G in a Cm-algebra A and let Y,, Y2, . . . , YN be infinitesimal generators of u. Then
1
n
D(x)=
{atA:
i=1,2,...,N
the mapping G 3 g I-+ u(g)a E A is of Cl-class
2. For any tl, t2,. . . , tN E R u( exp (CtiXi))
= eCtaYi,
(2.15)
where exp : g -+ G is the exponential mapping of the theory of Lie groups and the right-hand side is understood in the sense of functional calculus of normal (skew-adjoint) elements (cf Section 1). 3. Elements affiliated with C*(G)
Let G be a locally compact group. It is well-known that unitary representations of G are in l-l correspondence with the representations of the algebra C*(G). By Theorem 2.2, investigating the affiliation problem formulated in Section 2 it is sufficient to consider the universal representation of G in C*(G). We refer to [12] for the definition and basic properties of b(G). By definition Ll(G) c C*(G) : C*(G) is the completion of Ll(G) with respect to a C*-norm. Moreover, the space of all finite (complex valued) measures on G is contained in M(C’(G)). Identifying any element g E G with the probability measure concentrated at g we define the embedding G c, M(C*(G)). One can easily show that this embedding G 3 g I-+ U(g) E M(C*(G)) is a unitary representation of G in C*(G). It is called the universal representation for the following: THEOREM 3.1. Let u be a representation of G in a (?-algebra % E Mor(C*(G), A) such that u(g) = ;ii(U(g)) for all g E G.
A. Then there exists unique
362
S. L.WORONOWICZand
K.NApI6RKOWSKI
Conversely, if U E Mor(C*(G), A) then (cf. Theorem define a representation u of G. Using Theorem 3.1 one can easily prove
3.2. There exisrs unique Q, E Mor(C*(G),
PROPOSITION
@(U(s))
2.2.1) setting
C*(G)
u(g) = e(U(g))
@ C*(G))
we
such that
= U(g) @ D(g).
3.3.Ler p : G, + Gx be a homomorphism of locally compact groups. Then there exists unique u E Mor(C*(Gi), C*(G?)) such that Cp(Ul(g)) = Uz((p(g)) for all g E G1 (U? denotes the universal representation of G,, i = 1,2). PROPOSITION
The algebra
C*(G)
may be considered
as the algebra
of all “continuous
vanishing
at
infinity functions” on a quantum space G. The comultiplication @ introduced in Proposition 3.2 defines a group structure on G. G endowed with this structure is the Pontryagin dual of G (cf. [15]). It turns out that the pair (C*(G), @) determines the group G uniquely: Using the Tatsuuma duality [2] one can show that G (or more precisely U(G)) coincides with {u E AI(C*(G)) : u # 0 and @j(u) = ‘u @ v}. Moreover, restricting to G the topology of almost uniform convergence on Al(C*(G)) we get the original topology of G. Indeed, the mapping G 3 g H U(g) E AI(C*(G)) is . continuous (U is a representation of G in C*(G)). To prove the continuity of the reverse mapping it is sufficient to consider the left regular representation L of G: one can easily show that L Bn * L.k in the strong ( operator topology ) for any sequence
(go) of elements
w(g,
+933)
of G and any g% E G.
A deeper analysis (see [16]) shows that the Pontryagin dual of G coincides with G. Let g be the Lie algebra of G, (Xi, X2,. . . , Assume now that G is a Lie group. Xnr=
LX,..Xl] =
c
f&xj
The enveloping algebra of g will be denoted by E. In particular, A = C X+X, E E. In what follows we identify G with its U-image. AI E E will be identified with the corresponding closed unbounded C*(G). It means that we shall omit the symbol U writing g and nf IJ(Af) resp. We also set DX = D”(U). We already know (cf. Section 2, remarks after Theorem 2.1) that and A are affiliated with C*(G). X, are skew-adjoint; A is selfadjoint fore, SpA c [0, M[. Let b,, = (I + A)-‘. By virtue of [ll, f(x)
Theorem
3.11 b,, E C*(G) (see also [13]). Inserting
the Nelson operator Consequently, each operator acting on instead of U(g) and Xi (i = 1,2, . , N) and positive. There(3.1) in (1.10) T = A1i2,
= (1 + X’)-‘1’ and .r = hX” we obtain ho E D(A’/‘). Like in the Hilbert space operator theory one can show that any core of A is a core of A’/‘. Using now the obvious estimate l]Xla]] 5 ~]A’/‘Q]] for all a E D” we see that
OPERATOR THEORY IN THE C*-ALGEBRA
FRAMEWORK
363
D(A1/2) > D(Xi). In particular 6s E D(Xi) for all i = 1,2,. . . , N. Let bi = Xibo. Then bo, bt, . . . , biv E C*(G). Clearly, bo, bl, . . . . bN coincide with the generators introduced in Lemma 2 of [13]. They satisfy the following Szymanski relations: b;, = bo, bib0 = -bobk, bo(l-bo)=eb;bj, j=l b’,bl - b;bl, = &&bobi, j=l b; + bk = 5
c&(b;bi
+ bfbj)
i,j=l
for all Ic,1 = 1,2, . . . , IV. If G is connected and simply connected then C*(G) coincides with the (non-unital) C*-algebra generated by abstract elements be, bl, . . . , bN satisfying the above relations [13]. We shall use the following C*-version of Lemma 2 of [13]. PROPOSITION 3.4. Let A be a C*-algebra and b& b’,, . . . , bX be elements of M(A) sutisfiing the Szymanski relations. Assume that bbA is dense in A and that G is connected and simp& connected. Then there exists unique cp E Mor(C*(G), A) such that b(i = v(bj) for all j = 0, 1, . . . ) N.
Proof We may assume that A c B(H), where H is a Hilbert space and the embedding is non-degenerate. Then bb, b’,, . . . , bh E B(H) and using Lemma 2 of [13] we get bl, = cp(bj), where cp is a representation of C’(G) in H. Remembering that b, bl, . . . , bN generate C*(G) we see that (p(C*(G)) c M(A). Moreover, (p(C*(G))A I cp(b~)A = bbA. Q.E.D. Therefore, (p(C*(G))A is dense in A and cp E Mor(C*(G), A). 4. The integrahility theorem of E. Nelson
Let Y be a linear operator acting on a C*-algebra A. W say that Y is skew-symmetric if (Ya)*b = -a*(Yb) for all a, b E D(Y). In this section we assume that G is a connected, simply connected Lie group. We shall use the notation introduced in Sections 2 and 3. The following theorem is the P-version of Theorem 5 and Corollary 9.1 of [lo]. THEOREM 4.1. Let Y, (i = 1,2,. . . , N) be closed skew-symmetric operators acting on a P-algebra A and D be a dense linear subset of A such that for all a E D and i,j = 1,2,..., N we have 1. a E D(Y) and Yiu E D(Yj),
2. YiYjU - YjYiU = Cf=“=, c:jYka. Assume that the closure of I&
Y$l D is a selfadjoint element afiliated with A.
364
S. L.
WORONOWICZ
and K. NAPIbRKOWSKI
Then YzvA (i = 1,2,. . . , N) and there exists unique representation ZL(X~) = Yi (i = 1,2,. . . , N).
u of G such that
Proof: The uniqueness of ‘u.follows from the genera1 theory of Banach space representations of Lie group. We have to show the existence. Let 2 be the closure of - Cr=“=,Y:In and C = I+ i. Then D is a core of C, C* = QA and C 2 I. It implies that CD(C) = A. The inverse of C will be denoted by bb. Clearly bb E hi(A), 0 5 b; 5 I and Z&A = D(C). Let i E (1,2, . . . , N}. For any a E D we have 5 ~(u,u)*(y,u)
(Y,u)‘(~a)
= a*(C - I)a 5
I*.
k=l
The last estimate follows from the equality C - I = (C/2)2 - (C/2 - 1)2. Therefore, IlKall < ;llCall for all a E D. Remembering that D is a core of C and that Yi is closed we get D(C) c D(Y,) and llYzull 5 ~\jCull for all a E D(C). It shows that the operator bi = Yzbbis defined on the whole A and /lb:/] < 4. Therefore b: E B(A). Let z E D(K). Then for any z E A we have (biz)*x = (xbbz)*s = -(bbz)*&z = z*(-b$z). It shows that 3: E D((bi)*), D(Yi) c D((bi)*) and by virtue of Proposition 0.1 we obtain b’,E M(A) for i = 1,2,. . . , N. By simple computations one can check that bk, b’,, . . . , bh satisfy the Szymanski relations. Therefore (cf. Proposition 3.4) there exists unique cp E Mor(C(G), A) such that cp(bk) = b’, for Ic = 1,2,. . , N. Let i E {1,2,. . . , N}. For any a E D(C) we have Y,u = Y,b&‘u = b:Cu = @,)Cu = ‘p(X&@l
= cp(X,bo)Cu
= cp(X&(bo)Cu
= cp(X&.
According to (3.1), &Z’*(g) = D(A) > Dm. Therefore, b&*(G) is a core of Xi and consequently (cf. Section 0, Remark before Proposition 0.2) cp&)A = bbA = D(C) is a core of cp(Xi). Taking into account the above computations and remembering that x is closed we conclude that cp(Xi) c Y,. By virtue of Lemma 1.2, p(Xi) = Yi. In particular Y,qA.
For any g E G c Ar(C*(G)) we set u(g) = cp(g). By virtue of Theorem 2.2, u is a Q.E.D. unitary representation of G in A and u(X~) = cp(Xi) = Y, for i = 1,2,. . . , N. 5. Selfadjoint
extensions
of symmetric
elements
Let A be a C*-algebra and TVA. We say that T is symmetric if T c T*. It means that y*Ta: = (Ts)*z
(5.1)
for any Z, y E D(T). Let z be the z-transform of TVA. Then 2 = d-a, TX = zu, y = v’mb and Ty = zb, where a, b are elements of A. Inserting these data into (5.1) we get d-d
= z*JK-Z.
(5.2)
365
OPERATORTHEORYINTHEC*-ALGEBRAFRAMEWORK
Conversely, (5.2) immediately implies (5.1). Like in the theory on the Hilbert space level the problem of extensions of symmetric elements can be solved by the Cayley transform method (see e.g. [9]). Let T be a symmetric element affiliated with a C*-algebra and z be the z-transform of T. We set CT = ‘w-w;,
(5.3)
where wk = z f id-. Due to (5.2) w+ and w_ are isometries: w;w* = I. Therefore, CT is a partial isometry: +cT = w+wr and CT+ = w-w? are projections (note that an element c of a C*-algebra is a partial isometry if and only if cc*c = c). We call CT the Cayley transform of T. To justify this terminology one can verify that CT(T
+ il)a: =
(T - iI)a:
for any x E D(T). Let b E A. Then
$CT-
I)c$b =
and S(C~ - I)c$b E D(T).
$w_w; -
I)w+w:b
= dmw”b
Let us notice that each element of D(T) is of this form.
Indeed, for any a E A we have a = wYb, where b = w-u. With the same notations we have T ;(c~ - I)c;b = zw:b = ;(w_ (’ >
+ w+)w:b = ;(cTc;
+ c;)b = ;(cT + I)c$b.
This way we showed that ( x;zDg))
there exists b E +A such that @ ( x = i(cT - I)b and y = (CT + I)b >
Therefore, T is uniquely determined is dense in A.
(5.4)
by its Cayley transform CT. Moreover, (CT - I)c&A
PROPOSITION 5.1.Let c be an element of M(A) such that CC*C= c (i.e. c is a partial isometry). Assume that (c - I)? A is dense in A. Then c is the Cay@ transform of a symmetric TVA.
Proof: (I-c*)cc*A = (c-I)c*A is, by assumption, dense in A. Therefore, (I-C*)A is dense in A and ~(1 - C*)A is dense in CA = cc*A (because cc*A c CA = cc*cA c cc*A). Applying Proposition 0.2 to the operator ~(1 - c*) we get c(1 - c’) = w- IQ-
c*)l,
where V?W_ = I, w-w? = cc* and Ic(l - c*)I=J(I - c)c*c(l- c*)=Jc*c - c - c* + cc*. Let w+ = c*w_. Then w:w+ = w~cc*w_ = wlw-wZw_ = I, w+w; = C*V_V:C = C*CC*C = C*Cand v-v:
= w-w’c = cc*c = c.
S. L. WORONOWICZ
366
and K. NAPIbRKOWSKI
We shall apply Proposition 0.2 to the operator (w+ - v-)*. In this case e = f = I. Indeed, (v+ - v_)A 1 (v+ - v_)w?A = (c* - cc*)A = (I - c)c*A is, by the assumption, dense in A and f = I. Furthermore, (v+ - v_)*A = w;v+(v+ - w_)*A = w;(c*c - c*)A = wTc*(I-c)A. We know that (I-c)c*A is dense in A, so is (I-c)A, whence (w+-v-)*A is dense in wl;c*A = w*cc*A > wrcc*cA = w:cA = wfA > vfv+A = A and e = I. By virtue of Proposition 0.2 (w+ - w-)* = rs, where r, s E M(A), r is unitary, s > 0 and w+ - w_ = sr*. Let W+ = iv+r, w- = iv-r. By computation
we get
w:w+ = I,
w”w-
= I,
w+w+* = c*c,
w-w:
= cc+,
(5.5)
f (w+ -w_)>O, c =
(5.6)
w-w;.
Let
(5.7)
z = i(w+ + w-). Then z*z+
2i w+-w_ -% ( and taking into account (5.6) we get
))
* ;( 20+-w_) (
=I >
$( w+ - w-) = J_.
(5.8)
Combining (5.7) and (5.8) we get w* = zfiJI_z’z.
(5.9)
Since JI-z*zA
= i(w+ - w_)rA
= (w+ - w_)A
is dense in A, z is the z-transform of an element T affiliated with A. T is symmetric. Indeed, combining (5.5) and (5.9) we get (5.2). Clearly, c is the Cayley transform of T. Q.E.D. Now we are able to prove THEOREM
z-transfom
5.2. Let T be a symmetric element afiliated with a C’-algebra of T and E +, E- be elements of M(A) introduced by E’
= z*z * (zdm
- d-z*)
- zz*.
A, z be the (5.10)
OPERATOR
THEORY
IN THE C*-ALGEBRA
FRAMEWORK
367
Then 1. Ei and E- are orthogonal projections: E** = E* = (E*)2. Moreover, E’ # I # E-. 2. T is selfadjoint if and only if E’ = 0. 3. The set of ail selfadjoint extensions of T is in one to one correspondence with the set of all elements v E M(A) such that v*v = Ef and vu* = E-. 4. The set of all symmetric extensions of T is in one to one correspondence with the set of all partial isometrics v E M(A) such that v*v < E’ and vu* 5 E-. Proof Ad 1. Using the relations derived in the proof of Proposition 5.1 one can easily verify that E+ = I - c;cT, E- = I-c& where CT is the Cayley transform of T and the statements follow (CT is a partial isometry). Ad 2. If T is selfadjoint then z* = z and (cf. (5.10)) E* = 0. Conversely, if E’ = 0 Therefore, z* commutes with z*z and then z*z = zz* and zdm = d-z*. zdm = z*vm. Remembering that D(T) = dr-z*zA is dense in A we get z = z* and the selfadjointness of T follows (cf. (0.2)). It follows immediately from the above considerations that a symmetric element is selfadjoint if and only if its Cayley transform is unitary. Ad 4. Let v E M(A) be a partial isometry such that v*v < E+ and vu* I E-. Then v*v < I- C+T and vu* < I- cTc& Therefore, (UC;)*(+)
= cTv*2)c; 5 cT(I - c;cT)c;
(V*cr)*(V*c~) =
CGVV*C*
5
C>(I
-
CTC&)CT
= 0, =
0
and vc; = V*cT = 0.
(5.11)
Using these relations one can easily prove that c = CT = v is a partial isometry. Indeed, c*c = I - (E+ - v*v), CC*= I-(E-
-VU*)
are orthogonal projections. Let b E c;A. Then b = c&a (where a E A), vb = vc;a = 0 (cf. (5.11)) and cb = CTb. Moreover, b = c$?rc$a = (c; + ZI*)CTC$U= c*cTc$a and b E c*A. Therefore, (c - I)c*A > (CT - I)c$A is dense in A, c is the Cayley transform of a symmetric SQA (see Proposition 5.1) and using (5.4) we get T c S. Conversely, assume that 5’ is a symmetric extension of T. By virtue of (5.4) for any b E CGA there exists b’ E c:A such that i(cT - I)b = i(cs - I)b’, (CT + I)b = (cs + I)b’.
368
S. L.WORONOWICZandK.NAPI6RKOWSKI
Solving these equations we get b’ = b and csb’ = crb. It shows that c c;A
(5.12)
c&& = c&.
(5.13)
c;A
and We know that C$SC$ = c& hence czcsa = a for any a E c;A. By virtue of (5.12) the same relation holds for a E +A. In particular c~csc>cr = C&CT.Therefore, c+r I c&s - c$cs - c$-CT is an orthogonal projection. Let 21= cs(c$,$
- c>cr).
Then V*V= c$s - c$cr. It shows that w is a partial isometry and V*V< I - c1;c~ = E+. Using (5.13) one can verify that WV*= csc; - crc$ 5 E-. To end the proof of Statement 4 we notice that by virtue of (5.13) v =
c,c;c, -
c&c,
= cs -
CT.
Ad 3. According to the remark at the end of the proof of Statement 2 the extension and only if cs is unitary. Clearly, this is the case if and only if V*V and vu* = I - c,c;. = E-. Q.E.D.
s is selfadjoint if = I - c$cT = E+
COROLLARY 5.3. Let A be a C*-algebra such that M(A) 0 and I. Then every symmetric TVA is selfadjoint,
Remembering
contains no projections
except
that E* = I - w *wg we get the following enhancement
COROLLARY 5.4.Let A be a C-algebra such that each isometty unitary. Then each symmetric TVA is selfadjoint.
belonging to M(A)
Let TVA. We say that T is positive if x*Tx > 0 for any x E D(T). z-transform of T. One can easily check that T is positive if and only if v-z
is
Let z be the
2 0.
In particular any positive element is symmetric. PROPOSITION 5.5.Every positive element T ashated joint extension. Proof: Let z be the z-transform
with a C*-algebra admits a selfad-
of T. For any X E C such that IX] = 1, %J+ 2 0 we set WA= z + A&-=-Z.
Then wl;w~ = I + (2%x)v’mz 2 I. It shows that Sp(w;wx) c [l, cc#. Therefore, (cf. [7, Problem 691) Sp(wxwi) c (0) u [l, m[. Let f be a continuous function on [0, m[ such that f(0) = 1 and f(t) = 0 for t 2 1. We set E(X) = f(wxw;,). Then E(X) is an orthogonal projection belonging to M(A); E(A) depends continuously The existence of a partial isometry 2) such that ‘U*V= E+ on X and E(fi) = E*.
369
OPERATORTHEORYINTHEC*-ALGEBRAFRAMEWORK
and vu* = EC*-algebras.
follows now from the following lemma well known in the K-theory
of
LEMMA 5.6.Let e, f be orthogonalprojections belonging to M(A) such that lie - fll < 1. Then there exists u E M(A) such that u*u = e and uu* = f. Proof Apply Proposition
0.2 to the element c = ef.
Q.E.D.
Remark: At the moment we are not able to show that any positive element admits a positive selfadjoint extension. Let T be a positive element affiliated with C*-algebra A. We endow D(T) with the norm
lll4ll = dIW(T + Wll,
2
E WY.
Let E)(T) be the completion of D(T) with respect to this norm, DF = E)(T) n D(T*) and TF = T*[D~. TF is called the Friedrichs extension of T. PROBLEM: Is TF affiliated with A? It seems that the answer is negative in general. However, if A is balanced (i.e. any left multiplier is a multiplier: L&I(A) = M(A)) then TFVA for any positive TVA. In this case TF is a positive selfadjoint extension of T. 6. Tensor product of affiliated elements
In this section we shall prove the following THEOREM 6.1.Let Ai be a C*-algebra and TivAi (i = 1,2). Then there exists unique Ti 8 TzrtA1 @ A2 such that D(Tl) glatg D(T2) is a core of Tl @ T2 and (TI @
T2)(w
@ a2)
=
Tlal
8 T2a2
(6.1)
for any al E D(Tl) and ~42E D(T2). Moreover, (TI @IT2)* = T; @ T;.
(6.2)
We shall use the following: PROPOSITION 6.2.Let C be a unital C*-algebra; J c C be a closed ideal; bl, b2, . . . , bN be Hermitian, mutually commuting elements of C, A c RN be the joint spectrum of (bI, b2, . . . , bN) and f, g E C(A). Assume that (f(A)
= 0 *
(s(X) = 0)
(6.3)
for any A E A. Then g(bl, b2, . . . , bN)J is contained in the closure of f (bl, b2, . . . , bN) J. Proof: Due to (6.3) the closed ideal of C(_4) generated by f contains g (cf. [5, Lemma 2.9.41). It means that for any E > 0 there exists h, E C(n) such that
19(X)- f(xMx)l for all X E A. Let z E J.
I 6
Then gE = h,(bl, b2,. . . , bN)z E J, using the above inequality we get Q.E.D. f (bl, b2,. . . , bN)yell 5 e and the statement follows.
lldh, b, . . . >blv)s -
370
S. L. WORONOWICZ
and K. NAPIbRKOWSKI
We shall use Proposition 6.2 in the following situation: C = M(Al @A*); J = Al @AZ; bl = z;z, @zI and b, = I @ z;z~, where zi and z2 are z-transforms of Ti and T2 resp. Then A c [0, 112 and the functions f(xi, X2) = (1 - X1)(1 - X2) + X1X2 and g(Xi, X2) = ,,/(l - X,)(1 - XZ) satisfy the relation (6.3). Indeed, the only points, where f vanishes, are (1,0) and (0,l). Notice that in this case g(bl, b&J = (dECJm) (Al CCA?) 3 D(Tl) @n/qD(Tz) is dense in Al 8 AZ. Therefore, we have LEMMA
Replacing LEMMA
6.3. The range of (I - z;z,) @ (I - ~$22) + z;zI @ ~33 is dense in Al @ AZ. Tl and T2 by T; and T; resp. we get 6.4. The range of (I - ZIZ;) @ (I - z2z;) + zlz; c%z2z; is dense in Al ~3AZ.
Proof if Theorem 6.1: We shall use Theorem
2.3 of [17]. Let
CL=JI-Z1ZT@d_, b = c = z1 @ x2, d=dw@,/‘-,
Q=
(;I
;:).
Clearly a, b, c, d E iU(Al @JAZ). By direct computation one many check that ab = cd. Next a(Al cz A2) contains a(Al @‘alg A2) = D(T;) glalg D(T,*) and d(Al @3AZ) contains d(Al @lalSA2) = D(T,) Q,D(Tz). Both these sets are dense in Al @ AZ. Finally 0 a2 + bb*
QQ* =
and using Lemmas 6.3 and 6.4 we see that the range of Q is dense in (Al EIA~)CB(AI@A2). By virtue of Theorem 2.3 of [17] there exists an element TI $3T2qA1 @ A2 such that d(Al @ AZ) is a core of TL @ T2 and (T, @ Tz)dx = bx
(6.4)
that Al galg A2 is dense in Al ~3 A2 we see that for all x E Al ECAZ. Remembering if al E D(TI) and d(Ai @alg A2) = D(Tl) @al9 D(T2) is a core of TI ~3 T2. Moreover, a2 E D(T2) then al = ,/-x1, a2 = d-x:2 and inserting in (6.4) x = xi 8 x2 we get (6.1). Let x, y E A, @ AZ. By virtue of (0.1) x E D((Tl @ T2)*) and y = (TI @ T~)*x if and only if ((TI @ T2)c)*x = x*y
(6.5)
for all c E D(T, 8 T2). We may assume that c is of the form c = dc’, where c’ E Al 8 A2 (d(A, C$AZ) is a core of TLc$T2). Then (TL ~3T2)c = bc’ and relation (6.5) is equivalent to b*x = d*y.
(6.6)
the other hand, replacing Tl and T2 and T; and T,* resp. (and consequently a by d and c by b*) and using Statement 2 of Theorem 2.3 of [17] we see that (6.6) is equivalent Q.E.D. to x E D(T; @ T;) and y = (T;” ~3T;)x. This way formula (6.2) is proved. On
OPERATOR THEORY IN THE C*-ALGEBRA
FRAMEWORK
371
REFERENCES (11 [21
[31 141 I51 [61 171 [81 [91 [lOI 1111 [121 [131 [141 [151
[161 [171
0. Bratteli: Derivations, Dissipations and Group Actions on (Y-algebras, Springer Lecture Notes in Mathematics 1229 (1986). N. Tatsuuma: Duality theorem for locally compact groups and some related topics, Algebres d’operateurs et leurs applications en physique mathematique (Proc. Colloq. Marseille, 1977), pp. 387-408, Colloques Internat. CNRS. 274, CNRS, Paris 1979. N.I. Achiezer and I.M. Glazman: Theory of Linear Operators in Hilbert Spaces, Nauka, Moskva 1966. S. Baaj and P. Jungl: C. R. Acad. Sci. Parts, S&e I, 296 (1983) 875-878, see also S. Baaj: Multiplicateur non born&, These 3eme Cycle, Universite Paris VI, I1 Decembre 1980. J. Dixmier: Les C*-algebres et leur representations, Gauthier-Villars, Paris 1967. L. Girding: Proc. Nat. Acad. Sci. USA 33 (1947) 331-332. P.R. Halmos: A Hilbert Space Problem Book, Springer-Verlag, Berlin-Heidelberg-New York, 1974. T. Kato: Perturbation Theory for Linear Operators, Springer-Verlag, Berlin-Heidelberg-New York 1966. K. Maurin: Methods of Hilbert Spaces, Warszawa 1967. E. Nelson: Ann. of Math. 70 (1959), 572-615. E. Nelson and W.F. Stinespring: Amer. J. Math. 81 (1959) 547-560. G.K. Pedersen: C*-Algebras and their Automorphism Groups, Academic Press, London-New York-San Francisco 1979. W. Szymadski: Proc. Amer. Math. Sot. 107, No. 2 (1989) 353-359. J.M. Vallin: Proc. Lond. Math. Sot. (3) 50 (1985), No. 1. S.L. Woronowicz: “Pseudospaces, pseudogroups and Pontryagin duality”, Proceedings of the International Conference on Mathematical Physics, Lausanne 1979, Lecture Notes in Physics, Vol. 116, Springer, BerlinHeidelberg-New York. S.L. Woronowicz: Duality of Locally Compact Groups, CPT CNRS Marseille, June 1983, preprint CPT83ip.1519. S.L. Woronowicz: Commun. Math. Phys. 136 (1991) 399-432.
REPORTS ON MATHEMATICAL
OPERATOR
No. 3
PHYSICS
THEORY IN THE C*-ALGEBRA
FRAMEWORK
S. L. WORONOWICZand K. NAPI~RKOWSKI Department Faculty
of Physics,
of Mathematical
Warsaw
University,
Methods Hoia
in Physics,
74, 00-682 Warszawa,
Poland
(Received April 4, 1992)
Properties of operators affiliated with a C*-algebra are studied. A functional calculus of normal elements is constructed. Representations of locally compact groups in a C*-algebra are considered. Generalizations of Stone and Nelson theorems are investigated. It is shown that the tensor product of affiliated elements is affiliated with the tensor product of corresponding algebras.
0. Introduction
Let H be a Hilbert space and CB(H) be the algebra of all compact operators acting on H. It was pointed out in [17] that the classical theory of unbounded closed operators acting in H [8, 9, 31 is in a sense related to CB(H). It seems to be interesting to replace in this context CB(H) by any non-unital G-algebra. A step in this direction is done in the present paper. We shall deal with the following topics: the functional calculus of normal elements (Section l), the representation theory of Lie groups including the Stone theorem (Section 2, 3 and 4) and the extensions of symmetric elements (Section 5). Section 6 contains elementary results related to tensor products. The perturbation theory (in the spirit of T. Kato) is not covered in this paper. The elementary results in this direction are contained the previous paper of the first author (cf. [17, Examples 1, 2 and 3 pp. 412-4131). To fix the notation we remind the basic definitions and results [17]. Let A be a C*-algebra and T be a linear mapping acting on A defined on a dense linear domain D(T). The adjoint mapping T* is introduced by the following equivalence (X,Y E A) * for any a E D(T) x E D(T*) (0.1) and y = T’x ( ) ( (Tu)*x = a*y > ’ Clearly T* is a closed linear mapping. Even in T is bounded there is no guarantee that D(T*) is dense in A. However, if T is bounded, so is T’. Indeed, approximating y by a E D(T) we obtain Y*Y = P’Y)*G 11~11~ L IITYII lbll I IITII llvll II41 and llvll 5 IITII 1141. Let B(A) denote the Banach algebra of all bounded linear mappings acting on A defined on the whole A (D(a) = A for all a E B(A)) and let M(A)
= {u E B(A) [3531
: u* E B(A)}.
S. L. WORONOWICZ
354
and K. NAPIbRKOWSKI
Then M(A) is a norm-closed subalgebra of B(A), a* E M(A) for any a E M(A) and M(A) endowed with the *-operation is a unital C*-algebra. It coincides with the multiplier algebra of A (see e.g. [12]). Assuming that elements of A act on A by left multiplication we include A L-) AI(A). A is an ideal in M(A). The natural topology on Al’(A) is the topology of almost uniform convergence. We say that a net (a,) of elements of M(A) converges (almost uniformly) to 0 if /]a,~]] --f 0 and IlaLz]] + 0 for any 5 E A. For example if (e,) is an approximate unity for A ([5]) then e, converges to I almost uniformly. Using this fact one can easily show that A is dense in AI(A). The almost uniform topology remembers the position of A in M(A): A =
17:E AI(A) : 1
To prove this approximate unity hand e, E A and Remembering
lim ]]unxlI = 0 for any net (a,) of elements of Al(A) converging almost uniformly to 0 1
relation it is sufficient to consider a, = I - e,, where (e,) is the for A. Assuming l]ua~]] -+ 0 we get e,z + z in norm. On the other A is an ideal in Al(A). Therefore, e,x E A and x E A (A is complete!) that T* is bounded for any bounded T we get
PROPOSITION 0.1. Let A be a C*-algebra in A. Then T E Al(A).
and T E B(A).
Assume
that D(T*)
ti dense
Let A be a C*-algebra and T be a linear mapping acting on A having a dense domain D(T). We say that T is affiliated with A and write TVA if and only if there exists z E M(A) such that l]zl] 5 1, D(T) = JI-z”zA and Tdmu = zu for all a E A. It is known that z is determined by T. We call it z-transform of T and denote by zr. If TVA then T is closed. The set of all bounded elements affiliated with A coincides with Al(A). According to Theorem 1.4 of [17], if TVA then T*vA and zT* = z;.
(0.2)
Let A and B be C*-algebras. The set of all “-algebra homomorphisms ‘p from A into AI(B) such that (p(A)B is dense in B will be denoted by Mor(A, B). If TVA and cp E Mor(A, B) then there exists unique (p(T)vB such that (p(D(T))B is a core of v(T) and ~Q)cp(a)b
= ‘p(Ta)b
for any a E D(T) and b E B (see [17, Theorem 1.21). One can check that p(T) E M(B) for any T E Al(A). The resulting mapping cp : Ai + M(B) is a continuous unital “-algebra homomorphism (both multiplier algebras are considered with the topology of almost uniform convergence). In the general case (TVA), we have z,(T) = (p(zT). Remark: If DO is a core of T then p(Do)B is a core of p(T). This follows immediately from the definition. One can easily show (cf. [17, Theorem 1.41) that cp(T*) = q(T)*. The affiliation relation n was introduced by Baaj [4]. It was rediscovered in [17], where its properties were investigated in detail. The set of morphisms Mor(A, B) was introduced and investigated in [15] and [14]. In the following we shall need the following version of polar decomposition (cf. [17, Proposition 0.21).
OPERATORTHEORYIN THE C”-ALGEBRAFRAMEWORK
355
PROPOSITION0.2. Let c E M(A), where A is a C’-algebra. Assume that CA (c*A resp.) is a dense subset of eA (f A resp.), where e, f are selfadjoint projections belonging to M(A). Then there exists unique u E M(A) such that UU* = e and u*u = f.
c= u&G,
Proof Using [5, Lemma 2.9.41 one can easily show that c*cA = c*A and c*cA = &%A. Therefore, &%A is a dense subset of fA and the set of elements of the form a = &z
+ (I - f)y,
(0.3)
where 2, y E A is dense in A. Let us notice that u*a = z*z*cz = y*(I - f)y > (cs)*cz. Therefore, there exists a unique bounded linear mapping u : A + A such that ua = cz for any a of the form (0.3). In the similar way one can show that the set of elements of the form b = Jcctz’
+ (I - e)y’,
(0.4) where z’, y’ E A is dense in A and construct a unique bounded linear mapping u’ : A + A such that u’b = C*X’for any b of the form (0.4). Let us notice that (uu)‘b = 2*c*&zc Therefore, u’ = u*, hence u E M(A). requirements. 1. Functional
= X*&%*X’
= u*(u’b).
One can easily check that u satisfies all the Q.E.D.
calculus of normal elements
An element T affiliated with a C*-algebra A is called normal if D(T) = D(T*) and (Tu)*(Tu) = (T*u)*(T*a) for any a E D(T). This is equivalent to z$zr = zrz$. T is called selfadjoint if T = T*. We remind the description of the universal normal element given in [17]. Let _4be a closed subset of C and (4 be the element affiliated with C,(A) introduced by the formula
w(&I)
= T.
Let T be a normal element affiliated with a c-algebra tinuous function defined on A = Sp T. Then frtC,(A) in Theorem 1.1 we set
A and f be a complex valued conan d using the notation introduced
f(T) = VT(~). (1.1) Clearly f(T)vA. f(T) is a normal element (f is normal and due to [17, Theorem 1.21 any image of a normal element is normal). If f(x) = 1 for all A E C then f = It_(c)) and f(T) = CpT(Lc_(C) = LA. (For any C*-algebra A, by IA we denote the unity of M(A).) Similarly, if f(x) = X for all X E C then f = C and f(T) = (PT(C) = T. In general the definition (1.1) gives the meaning to many algebraic and analytical expressions containing T. For example, (I + T*T)sin(2T) = @(f), wh ere f is the continuous function on C
356
S. L. WORONOWICZ
and K. NAPIbRKOWSKI
such that f(x) = (1 + XX)sin(2X) for all X E C. Similarly, T(I + T*T)-l/* = (or where f(x) = X(1 + %A)-‘/’ for any X E C. One can easily verify that in this case f = ZC. Therefore, ZT = T(I + T*T)-l’*
(1.2)
for any normal TVA. The reader easily examines how the functional calculus introduced by (1.1) works in particular examples. If A is a C*-algebra of bounded operators acting in a Hilbert space (examples 3 and 4 in [17]) then f(T)
=
s f(XP(A)> SPT
where dE(.) is the spectral measure associated with the normal operator T: T =
s XdE(X). SPT
Considering A = C,(A), where A is a locally compact topological space, one can see that any T&‘,(A) is normal, SpT = T(A) and f(T)
= f o T.
(1.3)
Let A, B be C*-algebras and @ E Mor(A,B). is a normal element affiliated with B. Moreover, = ~&D(T)((). Therefore,
P*(T) = @ 0 VT
=
and
@(‘PT(.f))
=
@(f(T))
and
f(W’))
for
=
any
Then for any normal TVA, G(T) (@ o (PT)(<) = @(VT(<)) = Q(T) f
E
C(C)
we
get
.f(@(T))
=
‘P@(T)(f)
@U-(T)),
(14
Let A be a C*-algebra; ft, f2 E C(C) and TVA. Assume that T is normal. Replacing in (1.4) A, B, @, T and f by C,(C), A, VT, f2 and fr resp. and remembering that ft(f~) = fr o f2 (cf. (1.3) we obtain f1U2GY)
= (.fi 0 f2)Gf’h
(1.5)
Clearly (cf. Section 0) f(T)*
= f(T),
0.6)
where by definition f(x) = f(x) for any X E SpT. If f E Cbounded(SpT) then f(T) E M(A). Remembering we see that (A& +
~z.fi)(T)
(5
. f2)V)
= Xh(T) = h(T)
that PT E Mor(C,(SpT),
+ X2f2Q’)
. f2V)
A)
(1.7) (1.8)
for any fr , f2 E Choundrd(SpT) and XI, X2 E C. Moreover, if f E Cboun&d(SpT) and (fiL)n=1,2.... is a sequence of bounded continuous functions with a common bound such that f,, --+ f uniformly on any compact subset of SpT then endowing M(A) with the
OPERATOR THEORY IN THE C*-ALGEBRA
FRAMEWORK
357
almost uniform topology we have fn(T) + f(T).
(1.9)
Due to relations (1.5)-(1.8) the notation introduced by (1.1) is selfconsistent. over, if f(A) is a continuous bounded function such that Xf(X) is bounded then
f(T)z Kf(T)a)
More-
E D(T),
(1.10)
= (Tf(T))a:
(1.11)
for any 2 E A. The simple proof is left to the reader. Let A be a C*-algebra and let TVA. We say that T is skew-adjoint (skew-symmetric resp.) if T* = -T (T* > (-T) resp.). Like in the Hilbert space operator theory we have LEMMA
Proof
S c (-S*)
1.2. Skew-adjoint elements have no proper skew-symmetric extensions. Let A be a C*-algebra, T, SvA, T* = -T, c (-T*) = T and S = T.
s* > (-S)
and T c S. Then Q.E.D.
Clearly, any skew-adjoint element T is normal. ‘Its spectrum is contained in iR. Indeed, denoting by C the element affiliated with Coo(C) introduced by the formula [(A) = X for all X E C, we have: (PT(ZC) = zr = -2; = -(or. Therefore, VT vanishes on the closed ideal in C,,,,,,,(C) generated by the function zc + Zc, which consists of all bounded continuous functions vanishing on iR. It means (cf. [17, formula (1.20)] that Sp T c iR. Let T be a skew-adjoint element affiliated with a C*-algebra A. Taking into account the localization of Sp T described above and using (1.5)-( 1.9) we obtain (ctT)* = e---tT = (&y-l, (1.12) &Tet27-
=
e(tl
+t,v
(1.13)
>
fimoetT = 1, etT - I
%
-(I
t
(1.14)
+ T*T)- l/2 = T(I + T*T)-1/2
for any t, tl, t2 E R. The above limits are understood in the sense of almost uniform convergence. Let us notice that (I + T*T)-‘/2 = (I - .z$z~)-~/~. Therefore, for any a E D(T) there exists 2 E A such that a = (I + T*T)-1/2z and using the last of the above formulae we get etT - I norm- lim ---a=Ta (1.15) t+o
t
for all a E D(T). 2. Infinitesimal representations of Lie groups
Let G be a locally compact group and A be a C*-algebra. u : G + M(A) is a unitary representation of G in A if
We say that a mapping
1. u(g) is unitary for any g E G. 2. u(gigz) = u(gi)u(sz) for any gi,g2 E G. 3. For any a E A the mapping G 3 g - u(g)a E A is norm continuous.
358
S.L. WORONOWICZandK.NAPI~RKOWSKl
Let us remind that elements of nf(A) are continuous linear operators on A (they act by left multiplication). Therefore, we may use all the concepts and results of the general theory of linear actions of locally compact groups on Banach spaces [l]. In particular, in the Lie group case, we may consider infinitesimal representations of G. From now till the end of this Section we assume that G is an N-dimensional Lie group. Let g be tHe Lie algebra and & be the enveloping algebra of G. By definition g consists of all right-invariant real vector fields on G. Consequently, elements of E are differential operators on G commuting with the right shifts. & is equipped with an antilinear and antimultiplicative involution I 3 A1 H AP E E such that X+ = -X for all X E g. Let u be a unitary representation of G in a C*-algebra A. The corresponding infinitesimal representation will be denoted by dl~. Operators &(A[) (where M E I) are defined on the invariant dense domain D”(U)
=
a E A:
1
the mapping G 3 g H u(g)u E A is of C30-class in the sense . of norm topology in A I
For any Af E E and a E D”(u) &(A@
= nrz1(g)&]9,e.
Clearly, D”(u) is invariant under all u(h), h E G. Replacing in the above formula a by u(h)u and remembering that A1 commutes with right translations we get du(M)zl(h)u = AIu(g)ul,=h. This formula may be rewritten in the following way: = Ilfu(h)u.
du(Al)u(h)u
(2.1)
Let a, b E D”(U) and X E g. Applying X to the constant function (u(g)u)*u(g)b and using the Leibniz rule we get
(du(X)u)*b
= u*b
= -u*(du(X)b).
Therefore (du(Af)u)*b = u*du(M+)b for any a, b E Dm(u) and A1 E 1. Using this formula one can easily show that for all A1 E & operators du(A1) are closable. In what follows the closure of du(M) will be denoted by u(Af). AFFILIATION PROBLEM.
unitary
representation
Characterize
all elements A1 E I such that u(i’bl)~A for any
u of G in a C*-algebra A.
The following theorem provides a simple necessary condition THEOREM 2.1. Let u be a unitary representation of G in a C*-algebra A and M E E. Assume that f = 0 is the only bounded C”-solution to the differential equation M+Mf = -f_ Then u(Ar), z~(Al+)nA and u(Af)* = u(hf+).
Proof
By virtue
of [17, Proposition a - du(Af
K
2.21 it is sufficient
+)b
dzl(hf)u + b >
: a, b E IF(u)
to show that the set
OPERATOR THEORY IN THE C*-ALGEBRA
FRAMEWORK
359
is dense in A @ A. Let w and w’ be continuous linear functionals on A such that w(a - du(M+)b) + w’(du(M)a
+ b) = 0
for all a,b E DO”(u). We have to show that w = w’ = 0. Let c E D”(U). (2.2) u(g)c and --CIU(M)U(g)c instead of a and b we get f + M+Mf
(2.2) Inserting in
= 0,
on G. Therefore, w(u(g)c) = 0, where f(g) = w(u(g) c) 1s . a bounded C”-function w(c) = 0 and (D”( ‘u.) is dense in A) w = 0. Now, inserting in (2.2) a = 0 we get w’(b) = 0 for all b E IF(u) and w’ = 0. Q.E.D. It is known that all elements 111 E g satisfy the assumption of Theorem 2.1. The same is true for the Nelson operator A = CX+Xi, where (Xi)+t,~,...,~ is a basis in g. Elements u(X~) (i = 1,2, . . . , N) are called infinitesimal generators of ‘1~.They are skew-adjoint and affiliated with A. Element USA is selfadjoint and positive. THEOREM 2.2. Let A, B be (Y-algebras, ‘p E Mor(B, A), u be a unitary representation of G in B and u(g) = cp(v(g)) for all g E G. Then 1. u is a unitary representation of G in A. 2. ZfM E E and v(M)qB then USA and
u(M) = cp(v(M)). Proof Statement 1 is trivial. We prove Statement 2. By definition cp(D”(v))A core for cp(w(M)) and for any b E D”(v) and a E A we have:
is a
cp(v(M))cp(b)a = cp(v(M)b)a = cp(Mv(g)bl,=,)a = (Mp(v(g)b)l,=,)a = (Mu(g)cp(b)lg=,)a = u(M)cp(b)a. Therefore, it is sufficient to show that cp(D”(w))A is a core for U(M). Let c E 050(u) and E > 0. Then u(h)c and zl(M)u(h)c (= Mu(h)c by virtue of (2.1)) depend continuously on h and there exists a neighbourhood U of e such that (2.3) Ilu(M)u(h)c - u(M)cII
< ;
(2.4)
for any h E U. Let f be a C”-function on G with a compact support contained in U such that f(h) 2 0 for all h E G and Jo f(h)dh = 1 (dh denotes the left-invariant Haar measure on G). We set %, =
jG
f(h)@)&
72, = 1 f(h)v(h)dh. G
S. L.
360 Clearly R,, E Al(A),
WORONOWICZ
and K. NAPI6RKOWSKI
and R, = cp(R,). Moreover,
R,, E M(B)
for all a E A. R,,, and R,, are regularizing operators: R,A c IF(u) and R,B c D”(V) (cf. the concept of the G&ding domain [6]). Let a E A. Using (2.1) and integrating by parts we get u(N)R,,a = u(N) J f(h)u(h)adh = J f(h)(&fu(h)u)dh = J (M+f)(h)u(h)udh. Therefore,
Il4W%4l
I QIMI,
(2.6)
where Q = J I(Al+f)(h)ldh. Let 8 = i .min(l, Q-l).
We know (cf. Section 0) that (p(B)A is dense in A. Therefore,
there exist b, , bz, . . . , bN E B and a],U2,...,aN
E A such that
N
IL
cp(bh)uk - c
L=l
II
5 6’.
Inserting in (2.5) and (2.6) a = C p(bk)uk - c we get N
IL dR,,h)a~ Il4W5 &‘Wda~
-R,c
k=l
-
II
(2.7)
I ;,
u(Af,RucII L 5.
(2.8)
k=1
On the other hand, due to (2.3) and (2.4) we have
lI%LC- cl1I ;>
(2.9)
I~u(AI)K!,c - u(M)cll 5 ;.
(2.10)
Combining (2.7) with (2.9) and (2.8) with (2.10) we get (2.11)
llc’ - cl1 5 c, Ilu(AI)c’ - u(AI)cll 5 E,
(2.12)
where c’ = C cp(R,,bk)uk. This way we showed that for any c E D”(u) and any E > 0 there exists c’ E p(D”(v))A such that the estimates (2.11) and (2.12) hold. It means that Q.E.D. cp(lF(w))A is a core for u(A1). The remaining part of this Section is mainly devoted to the case G = R. 2.3. Let u be a representation of R in a C’-algebra generator of u. Then
THEOREM
finitesimal
A and TVA be the in-
1. Tu = lim
t-0
zl(t)a - a t
.
(2.13)
OPERATORTHEORYINTHEC*-ALGEBRAFRAMEWORK
361
In particular, a E D(T) if and only if the limit on the right-hand side of (2.13) exists in the sense of the norm topology on A. 2. T is skew-adjoint (T* = -T) and using the functional calculus of normal elements we have (2.14)
u(t) = etT for all t E R
Proof: Ad 1. Let T be the operator introduced by (2.13). One can easily see that T d d is a closed extension of du z . Therefore, T II u z . On the other hand, due to the ( > ( > Leibniz rule (Ta)*b + a*Tb = 0 for all a, b E D(T). It means that T is skew-symmetric d and using Lemma 1.2 we obtain T = u z .
( > Ad 2. By virtue of (1.12)-(1.15) the mapping R 3 t H etT E M(A) is a unitary representation of R in A and its infinitesimal generator c0ntains.T. Using Lemma 1.2 and remembering that any representation is uniquely determined we get (2.14). Q.E.D. COROLLARY 2.4. Let u be a representation of G in a Cm-algebra A and let Y,, Y2, . . . , YN be infinitesimal generators of u. Then
1
n
D(x)=
{atA:
i=1,2,...,N
the mapping G 3 g I-+ u(g)a E A is of Cl-class
2. For any tl, t2,. . . , tN E R u( exp (CtiXi))
= eCtaYi,
(2.15)
where exp : g -+ G is the exponential mapping of the theory of Lie groups and the right-hand side is understood in the sense of functional calculus of normal (skew-adjoint) elements (cf Section 1). 3. Elements affiliated with C*(G)
Let G be a locally compact group. It is well-known that unitary representations of G are in l-l correspondence with the representations of the algebra C*(G). By Theorem 2.2, investigating the affiliation problem formulated in Section 2 it is sufficient to consider the universal representation of G in C*(G). We refer to [12] for the definition and basic properties of b(G). By definition Ll(G) c C*(G) : C*(G) is the completion of Ll(G) with respect to a C*-norm. Moreover, the space of all finite (complex valued) measures on G is contained in M(C’(G)). Identifying any element g E G with the probability measure concentrated at g we define the embedding G c, M(C*(G)). One can easily show that this embedding G 3 g I-+ U(g) E M(C*(G)) is a unitary representation of G in C*(G). It is called the universal representation for the following: THEOREM 3.1. Let u be a representation of G in a (?-algebra % E Mor(C*(G), A) such that u(g) = ;ii(U(g)) for all g E G.
A. Then there exists unique
362
S. L.WORONOWICZand
K.NApI6RKOWSKI
Conversely, if U E Mor(C*(G), A) then (cf. Theorem define a representation u of G. Using Theorem 3.1 one can easily prove
3.2. There exisrs unique Q, E Mor(C*(G),
PROPOSITION
@(U(s))
2.2.1) setting
C*(G)
u(g) = e(U(g))
@ C*(G))
we
such that
= U(g) @ D(g).
3.3.Ler p : G, + Gx be a homomorphism of locally compact groups. Then there exists unique u E Mor(C*(Gi), C*(G?)) such that Cp(Ul(g)) = Uz((p(g)) for all g E G1 (U? denotes the universal representation of G,, i = 1,2). PROPOSITION
The algebra
C*(G)
may be considered
as the algebra
of all “continuous
vanishing
at
infinity functions” on a quantum space G. The comultiplication @ introduced in Proposition 3.2 defines a group structure on G. G endowed with this structure is the Pontryagin dual of G (cf. [15]). It turns out that the pair (C*(G), @) determines the group G uniquely: Using the Tatsuuma duality [2] one can show that G (or more precisely U(G)) coincides with {u E AI(C*(G)) : u # 0 and @j(u) = ‘u @ v}. Moreover, restricting to G the topology of almost uniform convergence on Al(C*(G)) we get the original topology of G. Indeed, the mapping G 3 g H U(g) E AI(C*(G)) is . continuous (U is a representation of G in C*(G)). To prove the continuity of the reverse mapping it is sufficient to consider the left regular representation L of G: one can easily show that L Bn * L.k in the strong ( operator topology ) for any sequence
(go) of elements
w(g,
+933)
of G and any g% E G.
A deeper analysis (see [16]) shows that the Pontryagin dual of G coincides with G. Let g be the Lie algebra of G, (Xi, X2,. . . , Assume now that G is a Lie group. Xnr=
LX,..Xl] =
c
f&xj
The enveloping algebra of g will be denoted by E. In particular, A = C X+X, E E. In what follows we identify G with its U-image. AI E E will be identified with the corresponding closed unbounded C*(G). It means that we shall omit the symbol U writing g and nf IJ(Af) resp. We also set DX = D”(U). We already know (cf. Section 2, remarks after Theorem 2.1) that and A are affiliated with C*(G). X, are skew-adjoint; A is selfadjoint fore, SpA c [0, M[. Let b,, = (I + A)-‘. By virtue of [ll, f(x)
Theorem
3.11 b,, E C*(G) (see also [13]). Inserting
the Nelson operator Consequently, each operator acting on instead of U(g) and Xi (i = 1,2, . , N) and positive. There(3.1) in (1.10) T = A1i2,
= (1 + X’)-‘1’ and .r = hX” we obtain ho E D(A’/‘). Like in the Hilbert space operator theory one can show that any core of A is a core of A’/‘. Using now the obvious estimate l]Xla]] 5 ~]A’/‘Q]] for all a E D” we see that
OPERATOR THEORY IN THE C*-ALGEBRA
FRAMEWORK
363
D(A1/2) > D(Xi). In particular 6s E D(Xi) for all i = 1,2,. . . , N. Let bi = Xibo. Then bo, bt, . . . , biv E C*(G). Clearly, bo, bl, . . . . bN coincide with the generators introduced in Lemma 2 of [13]. They satisfy the following Szymanski relations: b;, = bo, bib0 = -bobk, bo(l-bo)=eb;bj, j=l b’,bl - b;bl, = &&bobi, j=l b; + bk = 5
c&(b;bi
+ bfbj)
i,j=l
for all Ic,1 = 1,2, . . . , IV. If G is connected and simply connected then C*(G) coincides with the (non-unital) C*-algebra generated by abstract elements be, bl, . . . , bN satisfying the above relations [13]. We shall use the following C*-version of Lemma 2 of [13]. PROPOSITION 3.4. Let A be a C*-algebra and b& b’,, . . . , bX be elements of M(A) sutisfiing the Szymanski relations. Assume that bbA is dense in A and that G is connected and simp& connected. Then there exists unique cp E Mor(C*(G), A) such that b(i = v(bj) for all j = 0, 1, . . . ) N.
Proof We may assume that A c B(H), where H is a Hilbert space and the embedding is non-degenerate. Then bb, b’,, . . . , bh E B(H) and using Lemma 2 of [13] we get bl, = cp(bj), where cp is a representation of C’(G) in H. Remembering that b, bl, . . . , bN generate C*(G) we see that (p(C*(G)) c M(A). Moreover, (p(C*(G))A I cp(b~)A = bbA. Q.E.D. Therefore, (p(C*(G))A is dense in A and cp E Mor(C*(G), A). 4. The integrahility theorem of E. Nelson
Let Y be a linear operator acting on a C*-algebra A. W say that Y is skew-symmetric if (Ya)*b = -a*(Yb) for all a, b E D(Y). In this section we assume that G is a connected, simply connected Lie group. We shall use the notation introduced in Sections 2 and 3. The following theorem is the P-version of Theorem 5 and Corollary 9.1 of [lo]. THEOREM 4.1. Let Y, (i = 1,2,. . . , N) be closed skew-symmetric operators acting on a P-algebra A and D be a dense linear subset of A such that for all a E D and i,j = 1,2,..., N we have 1. a E D(Y) and Yiu E D(Yj),
2. YiYjU - YjYiU = Cf=“=, c:jYka. Assume that the closure of I&
Y$l D is a selfadjoint element afiliated with A.
364
S. L.
WORONOWICZ
and K. NAPIbRKOWSKI
Then YzvA (i = 1,2,. . . , N) and there exists unique representation ZL(X~) = Yi (i = 1,2,. . . , N).
u of G such that
Proof: The uniqueness of ‘u.follows from the genera1 theory of Banach space representations of Lie group. We have to show the existence. Let 2 be the closure of - Cr=“=,Y:In and C = I+ i. Then D is a core of C, C* = QA and C 2 I. It implies that CD(C) = A. The inverse of C will be denoted by bb. Clearly bb E hi(A), 0 5 b; 5 I and Z&A = D(C). Let i E (1,2, . . . , N}. For any a E D we have 5 ~(u,u)*(y,u)
(Y,u)‘(~a)
= a*(C - I)a 5
I*.
k=l
The last estimate follows from the equality C - I = (C/2)2 - (C/2 - 1)2. Therefore, IlKall < ;llCall for all a E D. Remembering that D is a core of C and that Yi is closed we get D(C) c D(Y,) and llYzull 5 ~\jCull for all a E D(C). It shows that the operator bi = Yzbbis defined on the whole A and /lb:/] < 4. Therefore b: E B(A). Let z E D(K). Then for any z E A we have (biz)*x = (xbbz)*s = -(bbz)*&z = z*(-b$z). It shows that 3: E D((bi)*), D(Yi) c D((bi)*) and by virtue of Proposition 0.1 we obtain b’,E M(A) for i = 1,2,. . . , N. By simple computations one can check that bk, b’,, . . . , bh satisfy the Szymanski relations. Therefore (cf. Proposition 3.4) there exists unique cp E Mor(C(G), A) such that cp(bk) = b’, for Ic = 1,2,. . , N. Let i E {1,2,. . . , N}. For any a E D(C) we have Y,u = Y,b&‘u = b:Cu = @,)Cu = ‘p(X&@l
= cp(X,bo)Cu
= cp(X&(bo)Cu
= cp(X&.
According to (3.1), &Z’*(g) = D(A) > Dm. Therefore, b&*(G) is a core of Xi and consequently (cf. Section 0, Remark before Proposition 0.2) cp&)A = bbA = D(C) is a core of cp(Xi). Taking into account the above computations and remembering that x is closed we conclude that cp(Xi) c Y,. By virtue of Lemma 1.2, p(Xi) = Yi. In particular Y,qA.
For any g E G c Ar(C*(G)) we set u(g) = cp(g). By virtue of Theorem 2.2, u is a Q.E.D. unitary representation of G in A and u(X~) = cp(Xi) = Y, for i = 1,2,. . . , N. 5. Selfadjoint
extensions
of symmetric
elements
Let A be a C*-algebra and TVA. We say that T is symmetric if T c T*. It means that y*Ta: = (Ts)*z
(5.1)
for any Z, y E D(T). Let z be the z-transform of TVA. Then 2 = d-a, TX = zu, y = v’mb and Ty = zb, where a, b are elements of A. Inserting these data into (5.1) we get d-d
= z*JK-Z.
(5.2)
365
OPERATORTHEORYINTHEC*-ALGEBRAFRAMEWORK
Conversely, (5.2) immediately implies (5.1). Like in the theory on the Hilbert space level the problem of extensions of symmetric elements can be solved by the Cayley transform method (see e.g. [9]). Let T be a symmetric element affiliated with a C*-algebra and z be the z-transform of T. We set CT = ‘w-w;,
(5.3)
where wk = z f id-. Due to (5.2) w+ and w_ are isometries: w;w* = I. Therefore, CT is a partial isometry: +cT = w+wr and CT+ = w-w? are projections (note that an element c of a C*-algebra is a partial isometry if and only if cc*c = c). We call CT the Cayley transform of T. To justify this terminology one can verify that CT(T
+ il)a: =
(T - iI)a:
for any x E D(T). Let b E A. Then
$CT-
I)c$b =
and S(C~ - I)c$b E D(T).
$w_w; -
I)w+w:b
= dmw”b
Let us notice that each element of D(T) is of this form.
Indeed, for any a E A we have a = wYb, where b = w-u. With the same notations we have T ;(c~ - I)c;b = zw:b = ;(w_ (’ >
+ w+)w:b = ;(cTc;
+ c;)b = ;(cT + I)c$b.
This way we showed that ( x;zDg))
there exists b E +A such that @ ( x = i(cT - I)b and y = (CT + I)b >
Therefore, T is uniquely determined is dense in A.
(5.4)
by its Cayley transform CT. Moreover, (CT - I)c&A
PROPOSITION 5.1.Let c be an element of M(A) such that CC*C= c (i.e. c is a partial isometry). Assume that (c - I)? A is dense in A. Then c is the Cay@ transform of a symmetric TVA.
Proof: (I-c*)cc*A = (c-I)c*A is, by assumption, dense in A. Therefore, (I-C*)A is dense in A and ~(1 - C*)A is dense in CA = cc*A (because cc*A c CA = cc*cA c cc*A). Applying Proposition 0.2 to the operator ~(1 - c*) we get c(1 - c’) = w- IQ-
c*)l,
where V?W_ = I, w-w? = cc* and Ic(l - c*)I=J(I - c)c*c(l- c*)=Jc*c - c - c* + cc*. Let w+ = c*w_. Then w:w+ = w~cc*w_ = wlw-wZw_ = I, w+w; = C*V_V:C = C*CC*C = C*Cand v-v:
= w-w’c = cc*c = c.
S. L. WORONOWICZ
366
and K. NAPIbRKOWSKI
We shall apply Proposition 0.2 to the operator (w+ - v-)*. In this case e = f = I. Indeed, (v+ - v_)A 1 (v+ - v_)w?A = (c* - cc*)A = (I - c)c*A is, by the assumption, dense in A and f = I. Furthermore, (v+ - v_)*A = w;v+(v+ - w_)*A = w;(c*c - c*)A = wTc*(I-c)A. We know that (I-c)c*A is dense in A, so is (I-c)A, whence (w+-v-)*A is dense in wl;c*A = w*cc*A > wrcc*cA = w:cA = wfA > vfv+A = A and e = I. By virtue of Proposition 0.2 (w+ - w-)* = rs, where r, s E M(A), r is unitary, s > 0 and w+ - w_ = sr*. Let W+ = iv+r, w- = iv-r. By computation
we get
w:w+ = I,
w”w-
= I,
w+w+* = c*c,
w-w:
= cc+,
(5.5)
f (w+ -w_)>O, c =
(5.6)
w-w;.
Let
(5.7)
z = i(w+ + w-). Then z*z+
2i w+-w_ -% ( and taking into account (5.6) we get
))
* ;( 20+-w_) (
=I >
$( w+ - w-) = J_.
(5.8)
Combining (5.7) and (5.8) we get w* = zfiJI_z’z.
(5.9)
Since JI-z*zA
= i(w+ - w_)rA
= (w+ - w_)A
is dense in A, z is the z-transform of an element T affiliated with A. T is symmetric. Indeed, combining (5.5) and (5.9) we get (5.2). Clearly, c is the Cayley transform of T. Q.E.D. Now we are able to prove THEOREM
z-transfom
5.2. Let T be a symmetric element afiliated with a C’-algebra of T and E +, E- be elements of M(A) introduced by E’
= z*z * (zdm
- d-z*)
- zz*.
A, z be the (5.10)
OPERATOR
THEORY
IN THE C*-ALGEBRA
FRAMEWORK
367
Then 1. Ei and E- are orthogonal projections: E** = E* = (E*)2. Moreover, E’ # I # E-. 2. T is selfadjoint if and only if E’ = 0. 3. The set of ail selfadjoint extensions of T is in one to one correspondence with the set of all elements v E M(A) such that v*v = Ef and vu* = E-. 4. The set of all symmetric extensions of T is in one to one correspondence with the set of all partial isometrics v E M(A) such that v*v < E’ and vu* 5 E-. Proof Ad 1. Using the relations derived in the proof of Proposition 5.1 one can easily verify that E+ = I - c;cT, E- = I-c& where CT is the Cayley transform of T and the statements follow (CT is a partial isometry). Ad 2. If T is selfadjoint then z* = z and (cf. (5.10)) E* = 0. Conversely, if E’ = 0 Therefore, z* commutes with z*z and then z*z = zz* and zdm = d-z*. zdm = z*vm. Remembering that D(T) = dr-z*zA is dense in A we get z = z* and the selfadjointness of T follows (cf. (0.2)). It follows immediately from the above considerations that a symmetric element is selfadjoint if and only if its Cayley transform is unitary. Ad 4. Let v E M(A) be a partial isometry such that v*v < E+ and vu* I E-. Then v*v < I- C+T and vu* < I- cTc& Therefore, (UC;)*(+)
= cTv*2)c; 5 cT(I - c;cT)c;
(V*cr)*(V*c~) =
CGVV*C*
5
C>(I
-
CTC&)CT
= 0, =
0
and vc; = V*cT = 0.
(5.11)
Using these relations one can easily prove that c = CT = v is a partial isometry. Indeed, c*c = I - (E+ - v*v), CC*= I-(E-
-VU*)
are orthogonal projections. Let b E c;A. Then b = c&a (where a E A), vb = vc;a = 0 (cf. (5.11)) and cb = CTb. Moreover, b = c$?rc$a = (c; + ZI*)CTC$U= c*cTc$a and b E c*A. Therefore, (c - I)c*A > (CT - I)c$A is dense in A, c is the Cayley transform of a symmetric SQA (see Proposition 5.1) and using (5.4) we get T c S. Conversely, assume that 5’ is a symmetric extension of T. By virtue of (5.4) for any b E CGA there exists b’ E c:A such that i(cT - I)b = i(cs - I)b’, (CT + I)b = (cs + I)b’.
368
S. L.WORONOWICZandK.NAPI6RKOWSKI
Solving these equations we get b’ = b and csb’ = crb. It shows that c c;A
(5.12)
c&& = c&.
(5.13)
c;A
and We know that C$SC$ = c& hence czcsa = a for any a E c;A. By virtue of (5.12) the same relation holds for a E +A. In particular c~csc>cr = C&CT.Therefore, c+r I c&s - c$cs - c$-CT is an orthogonal projection. Let 21= cs(c$,$
- c>cr).
Then V*V= c$s - c$cr. It shows that w is a partial isometry and V*V< I - c1;c~ = E+. Using (5.13) one can verify that WV*= csc; - crc$ 5 E-. To end the proof of Statement 4 we notice that by virtue of (5.13) v =
c,c;c, -
c&c,
= cs -
CT.
Ad 3. According to the remark at the end of the proof of Statement 2 the extension and only if cs is unitary. Clearly, this is the case if and only if V*V and vu* = I - c,c;. = E-. Q.E.D.
s is selfadjoint if = I - c$cT = E+
COROLLARY 5.3. Let A be a C*-algebra such that M(A) 0 and I. Then every symmetric TVA is selfadjoint,
Remembering
contains no projections
except
that E* = I - w *wg we get the following enhancement
COROLLARY 5.4.Let A be a C-algebra such that each isometty unitary. Then each symmetric TVA is selfadjoint.
belonging to M(A)
Let TVA. We say that T is positive if x*Tx > 0 for any x E D(T). z-transform of T. One can easily check that T is positive if and only if v-z
is
Let z be the
2 0.
In particular any positive element is symmetric. PROPOSITION 5.5.Every positive element T ashated joint extension. Proof: Let z be the z-transform
with a C*-algebra admits a selfad-
of T. For any X E C such that IX] = 1, %J+ 2 0 we set WA= z + A&-=-Z.
Then wl;w~ = I + (2%x)v’mz 2 I. It shows that Sp(w;wx) c [l, cc#. Therefore, (cf. [7, Problem 691) Sp(wxwi) c (0) u [l, m[. Let f be a continuous function on [0, m[ such that f(0) = 1 and f(t) = 0 for t 2 1. We set E(X) = f(wxw;,). Then E(X) is an orthogonal projection belonging to M(A); E(A) depends continuously The existence of a partial isometry 2) such that ‘U*V= E+ on X and E(fi) = E*.
369
OPERATORTHEORYINTHEC*-ALGEBRAFRAMEWORK
and vu* = EC*-algebras.
follows now from the following lemma well known in the K-theory
of
LEMMA 5.6.Let e, f be orthogonalprojections belonging to M(A) such that lie - fll < 1. Then there exists u E M(A) such that u*u = e and uu* = f. Proof Apply Proposition
0.2 to the element c = ef.
Q.E.D.
Remark: At the moment we are not able to show that any positive element admits a positive selfadjoint extension. Let T be a positive element affiliated with C*-algebra A. We endow D(T) with the norm
lll4ll = dIW(T + Wll,
2
E WY.
Let E)(T) be the completion of D(T) with respect to this norm, DF = E)(T) n D(T*) and TF = T*[D~. TF is called the Friedrichs extension of T. PROBLEM: Is TF affiliated with A? It seems that the answer is negative in general. However, if A is balanced (i.e. any left multiplier is a multiplier: L&I(A) = M(A)) then TFVA for any positive TVA. In this case TF is a positive selfadjoint extension of T. 6. Tensor product of affiliated elements
In this section we shall prove the following THEOREM 6.1.Let Ai be a C*-algebra and TivAi (i = 1,2). Then there exists unique Ti 8 TzrtA1 @ A2 such that D(Tl) glatg D(T2) is a core of Tl @ T2 and (TI @
T2)(w
@ a2)
=
Tlal
8 T2a2
(6.1)
for any al E D(Tl) and ~42E D(T2). Moreover, (TI @IT2)* = T; @ T;.
(6.2)
We shall use the following: PROPOSITION 6.2.Let C be a unital C*-algebra; J c C be a closed ideal; bl, b2, . . . , bN be Hermitian, mutually commuting elements of C, A c RN be the joint spectrum of (bI, b2, . . . , bN) and f, g E C(A). Assume that (f(A)
= 0 *
(s(X) = 0)
(6.3)
for any A E A. Then g(bl, b2, . . . , bN)J is contained in the closure of f (bl, b2, . . . , bN) J. Proof: Due to (6.3) the closed ideal of C(_4) generated by f contains g (cf. [5, Lemma 2.9.41). It means that for any E > 0 there exists h, E C(n) such that
19(X)- f(xMx)l for all X E A. Let z E J.
I 6
Then gE = h,(bl, b2,. . . , bN)z E J, using the above inequality we get Q.E.D. f (bl, b2,. . . , bN)yell 5 e and the statement follows.
lldh, b, . . . >blv)s -
370
S. L. WORONOWICZ
and K. NAPIbRKOWSKI
We shall use Proposition 6.2 in the following situation: C = M(Al @A*); J = Al @AZ; bl = z;z, @zI and b, = I @ z;z~, where zi and z2 are z-transforms of Ti and T2 resp. Then A c [0, 112 and the functions f(xi, X2) = (1 - X1)(1 - X2) + X1X2 and g(Xi, X2) = ,,/(l - X,)(1 - XZ) satisfy the relation (6.3). Indeed, the only points, where f vanishes, are (1,0) and (0,l). Notice that in this case g(bl, b&J = (dECJm) (Al CCA?) 3 D(Tl) @n/qD(Tz) is dense in Al 8 AZ. Therefore, we have LEMMA
Replacing LEMMA
6.3. The range of (I - z;z,) @ (I - ~$22) + z;zI @ ~33 is dense in Al @ AZ. Tl and T2 by T; and T; resp. we get 6.4. The range of (I - ZIZ;) @ (I - z2z;) + zlz; c%z2z; is dense in Al ~3AZ.
Proof if Theorem 6.1: We shall use Theorem
2.3 of [17]. Let
CL=JI-Z1ZT@d_, b = c = z1 @ x2, d=dw@,/‘-,
Q=
(;I
;:).
Clearly a, b, c, d E iU(Al @JAZ). By direct computation one many check that ab = cd. Next a(Al cz A2) contains a(Al @‘alg A2) = D(T;) glalg D(T,*) and d(Al @3AZ) contains d(Al @lalSA2) = D(T,) Q,D(Tz). Both these sets are dense in Al @ AZ. Finally 0 a2 + bb*
QQ* =
and using Lemmas 6.3 and 6.4 we see that the range of Q is dense in (Al EIA~)CB(AI@A2). By virtue of Theorem 2.3 of [17] there exists an element TI $3T2qA1 @ A2 such that d(Al @ AZ) is a core of TL @ T2 and (T, @ Tz)dx = bx
(6.4)
that Al galg A2 is dense in Al ~3 A2 we see that for all x E Al ECAZ. Remembering if al E D(TI) and d(Ai @alg A2) = D(Tl) @al9 D(T2) is a core of TI ~3 T2. Moreover, a2 E D(T2) then al = ,/-x1, a2 = d-x:2 and inserting in (6.4) x = xi 8 x2 we get (6.1). Let x, y E A, @ AZ. By virtue of (0.1) x E D((Tl @ T2)*) and y = (TI @ T~)*x if and only if ((TI @ T2)c)*x = x*y
(6.5)
for all c E D(T, 8 T2). We may assume that c is of the form c = dc’, where c’ E Al 8 A2 (d(A, C$AZ) is a core of TLc$T2). Then (TL ~3T2)c = bc’ and relation (6.5) is equivalent to b*x = d*y.
(6.6)
the other hand, replacing Tl and T2 and T; and T,* resp. (and consequently a by d and c by b*) and using Statement 2 of Theorem 2.3 of [17] we see that (6.6) is equivalent Q.E.D. to x E D(T; @ T;) and y = (T;” ~3T;)x. This way formula (6.2) is proved. On
OPERATOR THEORY IN THE C*-ALGEBRA
FRAMEWORK
371
REFERENCES (11 [21
[31 141 I51 [61 171 [81 [91 [lOI 1111 [121 [131 [141 [151
[161 [171
0. Bratteli: Derivations, Dissipations and Group Actions on (Y-algebras, Springer Lecture Notes in Mathematics 1229 (1986). N. Tatsuuma: Duality theorem for locally compact groups and some related topics, Algebres d’operateurs et leurs applications en physique mathematique (Proc. Colloq. Marseille, 1977), pp. 387-408, Colloques Internat. CNRS. 274, CNRS, Paris 1979. N.I. Achiezer and I.M. Glazman: Theory of Linear Operators in Hilbert Spaces, Nauka, Moskva 1966. S. Baaj and P. Jungl: C. R. Acad. Sci. Parts, S&e I, 296 (1983) 875-878, see also S. Baaj: Multiplicateur non born&, These 3eme Cycle, Universite Paris VI, I1 Decembre 1980. J. Dixmier: Les C*-algebres et leur representations, Gauthier-Villars, Paris 1967. L. Girding: Proc. Nat. Acad. Sci. USA 33 (1947) 331-332. P.R. Halmos: A Hilbert Space Problem Book, Springer-Verlag, Berlin-Heidelberg-New York, 1974. T. Kato: Perturbation Theory for Linear Operators, Springer-Verlag, Berlin-Heidelberg-New York 1966. K. Maurin: Methods of Hilbert Spaces, Warszawa 1967. E. Nelson: Ann. of Math. 70 (1959), 572-615. E. Nelson and W.F. Stinespring: Amer. J. Math. 81 (1959) 547-560. G.K. Pedersen: C*-Algebras and their Automorphism Groups, Academic Press, London-New York-San Francisco 1979. W. Szymadski: Proc. Amer. Math. Sot. 107, No. 2 (1989) 353-359. J.M. Vallin: Proc. Lond. Math. Sot. (3) 50 (1985), No. 1. S.L. Woronowicz: “Pseudospaces, pseudogroups and Pontryagin duality”, Proceedings of the International Conference on Mathematical Physics, Lausanne 1979, Lecture Notes in Physics, Vol. 116, Springer, BerlinHeidelberg-New York. S.L. Woronowicz: Duality of Locally Compact Groups, CPT CNRS Marseille, June 1983, preprint CPT83ip.1519. S.L. Woronowicz: Commun. Math. Phys. 136 (1991) 399-432.