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A note on covariant dynamical semigroups
No. 2
Vol. 32 (lYY3)
A NOTE ON COVARIANT
DYNAMICAL
SEMIGROUPS
A. S. HOLEVO Steklov Mathematical Institute. Vavilova 42, 117966 Moscow, Russia (Received August 24, 1992)
It is shown that in the standard representation of the generator of a norm continuous dynamical semigroup, which is covariant with respect to a unitary representation of an amenable group. the completely positive part can always be chosen covariant and the Hamiltonian commuting with the representation. The structure of the generator of a translation covariant dynamical semigroup is described.
1. Let A be a von Neumann algebra in a separate Hilbert space H, {Qt; t > 0) a normcontinuous dynamical semigroup in A. Let G be a locally-compact group and g + U, a continuous unitary representation of G in ‘FI such that lJ,dU, c A for all g E G. The semigroup pit is covutiunt with respect to the representation U, if &[u;Xu,]
= U;@t[XIUij
(1)
for all X E A, g E G and t 2 0. The generator C of Qt is a weak*-continuous dissipative map satisfying the corresponding covariance condition ,c[v,*XU,] As is well known, the generator represented in the form
completely
= U,*C[X]U,.
of a norm-continuous
(4 dynamical
,c[X] = @[Xl - @[I] 0 x - i[X, H] )
semigroup
can be
(3)
where @ is a normal completely positive map from d into itself, H = H” E A, [A, B] is a commutator, and A o B is the Jordan product of the operators A, B [l, 21. In this note we show that if l is covariant in the sense of [2], in the decomposition (3) @ can always be chosen covariant, and H commuting with iYq. This is not obvious and in fact it is not so in the non-norm‘continuous case, as seen from the example C[X] = -;[4,[4,Xll
= qXq-q20X,
where ;Ft = L2(R), G = R, q$(z) = cc$(cc), and U,+(z) = ti(z + g); $ E ‘Ft. In applications dynamical semigroups describe irreversible dynamics of an open system, and the covariance condition (1) reflects the corresponding symmetry of the surrounding and of the total Hamiltonian. Therefore the covariance condition is a physically sensible restriction in determining possible forms of the quantum master equation. This problem
212
A.S.HOLEVO
has been exhaustively studied for the finite-dimensional representations of subgroups G of the orthogonal group O(3) corresponding to the spatially isotropic surrounding (see [3] and [4]). Another important case, G = RV, arises in connection with the study of translation invariant open quantum systems in the weak coupling limit [5]. By applying our general theorem to the case of a locally compact Abelian group, we obtain the form of the bounded covariant generator which generalizes the one given in that paper. The non-norm continuous case will be considered in detail in a subsequent paper. The covariance condition turns out to be quite helpful in the construction of new broad classes of non-norm continuous dynamical semigroups. 2. We introduce the von Neumann algebra Au = {Us; g E G}’ THEOREM. Let the group G be amenable, and let C be the generator of a norm-continuous dynamical semigroup, satisfying the covariance condition (2). Then it can be represented in the form (3), where @ is a normal completely positive map, satisfying qv;x&J
= U,*@[X]U,
(4)
forallgEG,XEdandH=H*Edndu. Proof Modifying the GNS construction from [6], consider the algebraic tensor product k = ‘FI@ A, generated by elements cp@ X; ‘p E 7-&X E A, and define the inner product in K; by
(‘p 8 Xl@ @ Y) = ((PlDC]X,
04,) >
(5)
where D9[X, Y] = !F[x*Y] - x*!&[Y] - S[X]“Y + X”!F[I]Y.
(6)
Put 7r[Y](cpG%x)=cp@Yx-x~@Yy, V,(cp @ X) = u,cp @ u4Xu;
1
23[X]p = p@X.
By factorizing and completing i with respect to the inner product (5), we obtain the Hilbert space K. One checks that 7r extends to a normal *-representation of A in K, V, to a continuous unitary representation of G in K, B[X] to a bounded weak*-continuous linear map from A to L(‘Ft, K), satisfying D[YX] = 7r[Y]z?[X] + L?[Y]X, B[x]*B[Y]
= DL[X, Y] .
(7) (8)
Moreover, the covariance condition (2) implies 7r[U,fXU,] = v,*7r[x]v,, a[u;xu,]
= v;a[x]u,.
(9) (10)
A NOTE ON COVARIANT DYNAMICAL SEMIGROUPS
As shown in [2], a map satisfying the cocycle equation (7) can be represented form f?[X] = r[X]A - AX,
213
in the
where A E M = cl{B[X]Y; X, Y E d} c L(R, K) and cl means weak*-closure. Introducing the operator-valued function A(g) = V,*AU, - A,
gEG>
we see from (9) (10) that A(g) intertwines the representations
r[X] and X, i.e.
4X1&) = 4g)X.
(11)
Moreover, from the definition, A(g) satisfies the cocycle equation A(&) = Vh*A(g)U/z+ A(h),
g,hEG.
(12)
Thus, A(g) is a bounded weak*-continuous cocycle, intertwining the representations r[X] and X. Since G is amenable we can use the averaging pro_cedure of [7]. By averaging (11) and (12) over g E G, we see that there is an operator A E M such that 7r[X]lT = xx and A(h) = -V;AIJh
+ 3;
then putting A0 = A + 2, we get V,Ao = A,,U,
(13)
and a[X] = r[X]A,, - AoX. The map @[Xl = A,*r[X]A”
(14)
is a bounded weak*-continuous completely positive map from A to itself, which is covariant by (13). Since DC[X, Y] = D@[X, Y] by (8), we get as in [I, 21 C[X] = @[Xl - @[I] 0 x - i[X, H], where @[I] = AiAo E A and H = H* E A. From covariance of L and @ it follows that the derivation X + i[X, H] is also covariant, whence [U,“HU, -H, X] = 0 for all X E A. This means that the function H(g) = U,*Hi& - H
assumes values in the centre Z of A. The function H(g) satisfies the cocycle equation H(gh) = U;H(g)Uh
+ H(h).
Again averaging over g E G we deduce that there is an operator g E 2 such that H(h) = -U,$lJ,,
+ H.
214
A.S.HOLEVO
Then HO = H + p satisfies The theorem is proved. Note obtained and (13) of @ and
l$HoU,
= HO, i.e. HO E A
n dr: and [H, X] = [HO, X].
that for the covariant completely positive map in the decomposition (3) we the Stinespring type representation (14) satisfying the covariance conditions (9) (cf. [8]). In the case A = C(N) we can get further information on the structure L. Without loss of generality we can assume 7r[X] = x @ 10
K = 31:@%, (see, e.g. [6]). From the condition
where D, is a continuous {e,} in 7-&jwe can write
(9) it follows that
representation
of G in ‘Ha. By introducing
an orthonormal
basis
where (A 8 e)$ = A$ C%e for + E ‘Ft, and the series C, ATA, weakly”-converges. intertwining relation (13) then reads
The
AO = c
which means that the set {Ak} is a tensor D,. We then have the representation = ~(A;xA,
c[x] where 3.
A,i are the components Assuming
A = C(E),
4
operator
[9] with respect
- A;A,
o x) - qx, H] ,
of the tensor consider
@ e.j,
to the representation
operator.
the case of a separable
locally-compact
Abelian
group G. Let G be the dual group, (X,g) the value of the character X E G on the element g E G. For simplicity we assume that the representation g -+ U, extends to an imprimitivity system in ‘FI. In this case there is a continuous unitary representation X --f VA of G in ‘H, satisfying
the Weyl relation
%Vx = (~,s)VxK7,
gEG,
LEG.
(15)
THEOREM. Let C be the gene&or of a norm continuous dynamical semigroup in C(R), covariant with respect to the representation U,. Then
C[X] =
J ~{Lk(x)*v,*xv&.(x)6 k - &(X)*Lk(X>
0 X}p(dX)
- i[x, HI,
(16)
where H = H* E Au, p is a positive u-finite meusure on c?, und Lk(X) are weak*-measurable
A NOTE ON COVARIANT
DYNAMICAL
21.5
SEMIGROUPS
functions with values in Au, such that the integral
SC b4~)*~k(~)PL(~~) k
G
weakly*-converges. Proof By the Stone-Naimark posed into the direct integral
theorem,
the representation
D9 in tit) can be decom-
where p is a a-finite measure on g. Let d(X) = dim’Ftx, and let {ek.(X)} be a measurable field of orthogonal Then A E L(Fl, FL@ Ho) can be represented in the form
where J C Ak(X)*Ak(X)p(dX) e k=l
C =
weakly*-converges.
d(X) s x{Ak.(X)*XA,.(A) ^ k=l G
and we always can assume From the fact that
Moreover,
- Ak(X)*A,(X)
d(X) = cc by proper
bases in tie.
extension
0 X}p(HX) - i[X, H] ,
of functions
I&(X).
(& 8 D,q)A = AU,. it follows
Putting
L,+(X) = V;Ak(X),
we obtain
from (15)
[G%(X), U,] = 0, that is Lk.(X) E Au. Combining
these results we obtain
The condition that & extends when it extends to a generalized spectral type of the representation
the relation
(16).
to an imprimitivity system can be relaxed to the case imprimitivity system, which holds if and only if the l&. is absolutely continuous with respect to the Haar
measure in c [lo], but the general form of the covariant generator becomes then somewhat more complicated. In fact, (16) is covariant if VA is any function satisfying (15) and not necessarily a unitary representation of c. The following example, related to [ll], illustrates this remark.
216
A. S. HOLEVO
Let N = z njn)(nl be the number operator and W = 2 Jn+ l)(nj the shift operator n=O n=O in ‘H = 12. Then C$4 U, = eidN , where C$E [0,27r), is a unitary representation of the unit circle G, satisfying the generalized Weyl relation U$Wm = ei4”W”U,,
m=O,l,...
Let x(n) be a nonnegative bounded sequence; then the relation C[X] = q%@jW*XW&@j
- X(N) o X
defines a generator of a dynamical semigroup in ‘l-t,which is covariant with respect to the representation C$+ U,. The restriction of this semigroup onto the Abelian subalgebra of diagonal operators generated by N is the Markov semigroup of the pure birth process with intensities x(n). REFERENCES Lindblad G.: Commun. Math. Phys. 48 (1976), 119-130; Gorini V., Kossakowski A. and Sudarshan E.C.G.: J. Math. fhys. 17 (1976), 821-825. PI Christensen E. and Evans D.E.: .T. London Math. Sot. 20 (1979), 358-368. [31 Gorini V., Verri M. and Sudarshan E.C.G.: Lect. Notes Phys. 135 (1980) 95-103. 141 Artem’ev A.Yu.: Tear. Mat. Fiz. 79 (1989), 323-333. Botvich D.D., Malychev V.A. and Manita A.D.: Helv. Phys. Actu 64 (1991) 1072-1092. PI 161 Holevo A.S.: Lect. Notes Math. 1303 (1988) 128-147. K.R. and Schmidt K.: Lect. Notes Math. 272 (1972). t71 Parthasarathy Scutaru H.: Rep. Math. Phys. 16 (1979) 79-87. PI [91 Barut A.O. and Raczka R.: Theory of Group Representations and Applications, Vol. 1, Warszawa, Polish Scientific Publishers (1977). UOI Holevo A.S.: Lect. Notes Math. 1055 (1984) 153-172. Fagnola F.: Sankhya 53 Series A (1991). [Ill
[ll
Vol. 32 (lYY3)
A NOTE ON COVARIANT
DYNAMICAL
SEMIGROUPS
A. S. HOLEVO Steklov Mathematical Institute. Vavilova 42, 117966 Moscow, Russia (Received August 24, 1992)
It is shown that in the standard representation of the generator of a norm continuous dynamical semigroup, which is covariant with respect to a unitary representation of an amenable group. the completely positive part can always be chosen covariant and the Hamiltonian commuting with the representation. The structure of the generator of a translation covariant dynamical semigroup is described.
1. Let A be a von Neumann algebra in a separate Hilbert space H, {Qt; t > 0) a normcontinuous dynamical semigroup in A. Let G be a locally-compact group and g + U, a continuous unitary representation of G in ‘FI such that lJ,dU, c A for all g E G. The semigroup pit is covutiunt with respect to the representation U, if &[u;Xu,]
= U;@t[XIUij
(1)
for all X E A, g E G and t 2 0. The generator C of Qt is a weak*-continuous dissipative map satisfying the corresponding covariance condition ,c[v,*XU,] As is well known, the generator represented in the form
completely
= U,*C[X]U,.
of a norm-continuous
(4 dynamical
,c[X] = @[Xl - @[I] 0 x - i[X, H] )
semigroup
can be
(3)
where @ is a normal completely positive map from d into itself, H = H” E A, [A, B] is a commutator, and A o B is the Jordan product of the operators A, B [l, 21. In this note we show that if l is covariant in the sense of [2], in the decomposition (3) @ can always be chosen covariant, and H commuting with iYq. This is not obvious and in fact it is not so in the non-norm‘continuous case, as seen from the example C[X] = -;[4,[4,Xll
= qXq-q20X,
where ;Ft = L2(R), G = R, q$(z) = cc$(cc), and U,+(z) = ti(z + g); $ E ‘Ft. In applications dynamical semigroups describe irreversible dynamics of an open system, and the covariance condition (1) reflects the corresponding symmetry of the surrounding and of the total Hamiltonian. Therefore the covariance condition is a physically sensible restriction in determining possible forms of the quantum master equation. This problem
212
A.S.HOLEVO
has been exhaustively studied for the finite-dimensional representations of subgroups G of the orthogonal group O(3) corresponding to the spatially isotropic surrounding (see [3] and [4]). Another important case, G = RV, arises in connection with the study of translation invariant open quantum systems in the weak coupling limit [5]. By applying our general theorem to the case of a locally compact Abelian group, we obtain the form of the bounded covariant generator which generalizes the one given in that paper. The non-norm continuous case will be considered in detail in a subsequent paper. The covariance condition turns out to be quite helpful in the construction of new broad classes of non-norm continuous dynamical semigroups. 2. We introduce the von Neumann algebra Au = {Us; g E G}’ THEOREM. Let the group G be amenable, and let C be the generator of a norm-continuous dynamical semigroup, satisfying the covariance condition (2). Then it can be represented in the form (3), where @ is a normal completely positive map, satisfying qv;x&J
= U,*@[X]U,
(4)
forallgEG,XEdandH=H*Edndu. Proof Modifying the GNS construction from [6], consider the algebraic tensor product k = ‘FI@ A, generated by elements cp@ X; ‘p E 7-&X E A, and define the inner product in K; by
(‘p 8 Xl@ @ Y) = ((PlDC]X,
04,) >
(5)
where D9[X, Y] = !F[x*Y] - x*!&[Y] - S[X]“Y + X”!F[I]Y.
(6)
Put 7r[Y](cpG%x)=cp@Yx-x~@Yy, V,(cp @ X) = u,cp @ u4Xu;
1
23[X]p = p@X.
By factorizing and completing i with respect to the inner product (5), we obtain the Hilbert space K. One checks that 7r extends to a normal *-representation of A in K, V, to a continuous unitary representation of G in K, B[X] to a bounded weak*-continuous linear map from A to L(‘Ft, K), satisfying D[YX] = 7r[Y]z?[X] + L?[Y]X, B[x]*B[Y]
= DL[X, Y] .
(7) (8)
Moreover, the covariance condition (2) implies 7r[U,fXU,] = v,*7r[x]v,, a[u;xu,]
= v;a[x]u,.
(9) (10)
A NOTE ON COVARIANT DYNAMICAL SEMIGROUPS
As shown in [2], a map satisfying the cocycle equation (7) can be represented form f?[X] = r[X]A - AX,
213
in the
where A E M = cl{B[X]Y; X, Y E d} c L(R, K) and cl means weak*-closure. Introducing the operator-valued function A(g) = V,*AU, - A,
gEG>
we see from (9) (10) that A(g) intertwines the representations
r[X] and X, i.e.
4X1&) = 4g)X.
(11)
Moreover, from the definition, A(g) satisfies the cocycle equation A(&) = Vh*A(g)U/z+ A(h),
g,hEG.
(12)
Thus, A(g) is a bounded weak*-continuous cocycle, intertwining the representations r[X] and X. Since G is amenable we can use the averaging pro_cedure of [7]. By averaging (11) and (12) over g E G, we see that there is an operator A E M such that 7r[X]lT = xx and A(h) = -V;AIJh
+ 3;
then putting A0 = A + 2, we get V,Ao = A,,U,
(13)
and a[X] = r[X]A,, - AoX. The map @[Xl = A,*r[X]A”
(14)
is a bounded weak*-continuous completely positive map from A to itself, which is covariant by (13). Since DC[X, Y] = D@[X, Y] by (8), we get as in [I, 21 C[X] = @[Xl - @[I] 0 x - i[X, H], where @[I] = AiAo E A and H = H* E A. From covariance of L and @ it follows that the derivation X + i[X, H] is also covariant, whence [U,“HU, -H, X] = 0 for all X E A. This means that the function H(g) = U,*Hi& - H
assumes values in the centre Z of A. The function H(g) satisfies the cocycle equation H(gh) = U;H(g)Uh
+ H(h).
Again averaging over g E G we deduce that there is an operator g E 2 such that H(h) = -U,$lJ,,
+ H.
214
A.S.HOLEVO
Then HO = H + p satisfies The theorem is proved. Note obtained and (13) of @ and
l$HoU,
= HO, i.e. HO E A
n dr: and [H, X] = [HO, X].
that for the covariant completely positive map in the decomposition (3) we the Stinespring type representation (14) satisfying the covariance conditions (9) (cf. [8]). In the case A = C(N) we can get further information on the structure L. Without loss of generality we can assume 7r[X] = x @ 10
K = 31:@%, (see, e.g. [6]). From the condition
where D, is a continuous {e,} in 7-&jwe can write
(9) it follows that
representation
of G in ‘Ha. By introducing
an orthonormal
basis
where (A 8 e)$ = A$ C%e for + E ‘Ft, and the series C, ATA, weakly”-converges. intertwining relation (13) then reads
The
AO = c
which means that the set {Ak} is a tensor D,. We then have the representation = ~(A;xA,
c[x] where 3.
A,i are the components Assuming
A = C(E),
4
operator
[9] with respect
- A;A,
o x) - qx, H] ,
of the tensor consider
@ e.j,
to the representation
operator.
the case of a separable
locally-compact
Abelian
group G. Let G be the dual group, (X,g) the value of the character X E G on the element g E G. For simplicity we assume that the representation g -+ U, extends to an imprimitivity system in ‘FI. In this case there is a continuous unitary representation X --f VA of G in ‘H, satisfying
the Weyl relation
%Vx = (~,s)VxK7,
gEG,
LEG.
(15)
THEOREM. Let C be the gene&or of a norm continuous dynamical semigroup in C(R), covariant with respect to the representation U,. Then
C[X] =
J ~{Lk(x)*v,*xv&.(x)6 k - &(X)*Lk(X>
0 X}p(dX)
- i[x, HI,
(16)
where H = H* E Au, p is a positive u-finite meusure on c?, und Lk(X) are weak*-measurable
A NOTE ON COVARIANT
DYNAMICAL
21.5
SEMIGROUPS
functions with values in Au, such that the integral
SC b4~)*~k(~)PL(~~) k
G
weakly*-converges. Proof By the Stone-Naimark posed into the direct integral
theorem,
the representation
D9 in tit) can be decom-
where p is a a-finite measure on g. Let d(X) = dim’Ftx, and let {ek.(X)} be a measurable field of orthogonal Then A E L(Fl, FL@ Ho) can be represented in the form
where J C Ak(X)*Ak(X)p(dX) e k=l
C =
weakly*-converges.
d(X) s x{Ak.(X)*XA,.(A) ^ k=l G
and we always can assume From the fact that
Moreover,
- Ak(X)*A,(X)
d(X) = cc by proper
bases in tie.
extension
0 X}p(HX) - i[X, H] ,
of functions
I&(X).
(& 8 D,q)A = AU,. it follows
Putting
L,+(X) = V;Ak(X),
we obtain
from (15)
[G%(X), U,] = 0, that is Lk.(X) E Au. Combining
these results we obtain
The condition that & extends when it extends to a generalized spectral type of the representation
the relation
(16).
to an imprimitivity system can be relaxed to the case imprimitivity system, which holds if and only if the l&. is absolutely continuous with respect to the Haar
measure in c [lo], but the general form of the covariant generator becomes then somewhat more complicated. In fact, (16) is covariant if VA is any function satisfying (15) and not necessarily a unitary representation of c. The following example, related to [ll], illustrates this remark.
216
A. S. HOLEVO
Let N = z njn)(nl be the number operator and W = 2 Jn+ l)(nj the shift operator n=O n=O in ‘H = 12. Then C$4 U, = eidN , where C$E [0,27r), is a unitary representation of the unit circle G, satisfying the generalized Weyl relation U$Wm = ei4”W”U,,
m=O,l,...
Let x(n) be a nonnegative bounded sequence; then the relation C[X] = q%@jW*XW&@j
- X(N) o X
defines a generator of a dynamical semigroup in ‘l-t,which is covariant with respect to the representation C$+ U,. The restriction of this semigroup onto the Abelian subalgebra of diagonal operators generated by N is the Markov semigroup of the pure birth process with intensities x(n). REFERENCES Lindblad G.: Commun. Math. Phys. 48 (1976), 119-130; Gorini V., Kossakowski A. and Sudarshan E.C.G.: J. Math. fhys. 17 (1976), 821-825. PI Christensen E. and Evans D.E.: .T. London Math. Sot. 20 (1979), 358-368. [31 Gorini V., Verri M. and Sudarshan E.C.G.: Lect. Notes Phys. 135 (1980) 95-103. 141 Artem’ev A.Yu.: Tear. Mat. Fiz. 79 (1989), 323-333. Botvich D.D., Malychev V.A. and Manita A.D.: Helv. Phys. Actu 64 (1991) 1072-1092. PI 161 Holevo A.S.: Lect. Notes Math. 1303 (1988) 128-147. K.R. and Schmidt K.: Lect. Notes Math. 272 (1972). t71 Parthasarathy Scutaru H.: Rep. Math. Phys. 16 (1979) 79-87. PI [91 Barut A.O. and Raczka R.: Theory of Group Representations and Applications, Vol. 1, Warszawa, Polish Scientific Publishers (1977). UOI Holevo A.S.: Lect. Notes Math. 1055 (1984) 153-172. Fagnola F.: Sankhya 53 Series A (1991). [Ill
[ll