M.Takesaki, Theory of operator algebras. I. Springer-Verlag, New York-Heidelberg, 1979.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....N and 0. It is called singular iff 0 for some normal implies = 0. Any state 2 E(O) has a unique decomposition = n s ; 9) where n and s are positive linear functionals and n is normal while s is singular (in particular, n and s are disjoint, see [KR, Ta] for details) Since normal states are 12 mapped to normal states by V , the uniqueness of this decomposition implies that if is V invariant, then both n and s are V invariant. We say that a NESS 2 ( is normal if its singular part s is zero and purely singular ....

Takesaki, M.: Theory of Operator Algebras I. Springer-Verlag, New York (1979).

....cance of the Radon Nikodym theorem for CP maps at large. The real power of this theorem lies in the fact that it contains the traditional forms of the Radon Nikodym theorem as special cases. In order to see this, we will need the following result (see Corollary IV.3.5 and Proposition IV.3. 9 in [35]) a positive map T from a C algebra A to another C algebra B is automatically completely positive whenever at least one of A and B is Abelian. With this in mind, let us observe that any positive linear functional on a C algebra A is a positive map from A to C , and therefore is CP. When we ....

M. Takesaki, Theory of Operator Algebras I (Springer-Verlag, Berlin, 1979).

....The first two sections of this Appendix are a resume of results from general topology (flavored Polish) pertinent to Borel spaces. The classical references are the books of Kuratowski, Sierpinski and Hausdor#. Modern expositions may be found in the books of Bourbaki [B] Arveson [A] and Takesaki [T]; it is to these that we shall refer the reader. The results are arranged for convenient reference; it is not proposed that they be proved in this order (they are not, in the cited references) Probably the most e#cient preparation for the study of Borel spaces is to read Chapter IX, of [B] ....

Takesaki, M., Theory of operator algebras.I., Springer--Verlag, New York, 1979.

.... 2 N and 0. It is called singular iff 0 for some normal implies = 0. Any state 2 E(O) has a unique decomposition = n s ; 9) where n and s are positive linear functionals and n is normal while s is singular (in particular, n and s are disjoint, see [KR, Ta] for details) Since normal states are mapped to normal states by V , the uniqueness of this decomposition implies that if is V invariant, then both n and s are V invariant. We say that a NESS 2 ( is normal if its singular part s is zero and purely singular if its ....

Takesaki, M.: Theory of Operator Algebras I. Springer-Verlag, New York (1979).

....is injective homomorphism we obtain (5.14) Thus F(I) is a homomorphism. The injectivity follows from I I = 1c. 4. Using the previous step of the proof we see that F(P) is a norm one projection from F( C ) onto its von Neumann subalgebra F(I) F( C) Thus F(P) is a conditional expectation [24]. 19 Corollary 5.5 If F is a functor of second quantisation then for any real Hilbert space 7 and any infinite dimensional real Hilbert space E the algebras F(7 E)o(ic) and F(7 ) are isomorphic , in particular F( E) c) C1. Proof. We can choose E = i2(Z) Let S be the right shift on ....

Takesaki, M.: "Theory of Operator Algebras I", Springer Verlag, New York, 1979.

....we obtain r( xv) r( x)r( v) 5.1a) Thus r(I) is a homomorphism. The injectivity follows from I I = 1: 4. Using the previous step of the proof we see that r(P) is a norm one projection from r( c ) onto its yon Neumann subalgebra r(I) r( c) Thus r(P) is a conditional expectation [22]. Corollary 5.6 If r is a functor of second quantisation then for any infinite dimen sional real Hilbert space 1C we have r( c) r) Cl. Proof. We can choose C = 2( Let S be the right shift. Then S n converges weakly to 0 as n cx . By the previous proposition we have that r( c) z is ....

Takesaki, M.: "Theory of Operator Algebras I", Springer Verlag, New York, 1979.

.... described by a nite quantum probability space of A 0 valued random variables i.e. a von Neumann algebra 1 Partially supported by KBN grant 2P03A05415 1 A, endowed with a tracial normal state together with a subalgebra A 0 and the state preserving conditional expectation P 0 from A to A 0 [22]. The triple (A; A 0 ) is provided with a ltration of subalgebras A I of A, for all closed intervals I of the time axis R. A group (S t ) t2R of automorphisms of (A; acts as a shift on the local algebras S t (A I ) A I t and lets A 0 pointwise invariant. For disjoint intervals I; J the ....

Takesaki, M.: \Theory of Operator Algebras I", Springer Verlag, New York, 1979.

....essential way the results of L uck [Lu] In this case, they also prove the degenerate asymptotic L 2 Morse inequalities. 1. Preliminaries In this section we establish the main notation of the paper, and review some basic facts about Hilbertian A modules. See [F, CFM] for details. We refer to [Di, T] for the necessary de nitions on von Neumann algebras. Let A be a nite von Neumann algebra with a xed nite, normal and faithful trace : A C . We will always assume that this trace is normalized i.e. 1) 1. The involution in A will be denoted . By 2 (A) we denote the completion of ....

M.Takesaki, Theory of operator algebras I, Springer-Verlag, New York, 1979.

....C subalgebra of M. In this case Phi fi j B and Phi fi j M have the same norm. Proof. Assuming in (i) that e GammafiK B 1 Omega is contained in the ball of radius C 0, we shall show that the same is true for e GammafiK M 1 i.e. ii) holds. Let X 2 B. By Kaplanski density theorem [38], there exists a net of operators X i 2 B 1 strongly convergent to X . Since ke GammafiK X i Omega k C, we may assume, possibly restricting to a subnet, that e GammafiK X i Omega weakly converges to j 2 H, kjk C. Now take 2 D(e GammafiK ) We have ( j) lim i ( e GammafiK ....

....C; 1.15) namely the boundedness property with respect to L holds for C. Now, by the following Lemma 1.23, C is irreducible on H, thus L is semibounded by Cor. 1.3. As JLJ = GammaL, L is indeed a bounded operator. We now follow an argument in [10] By the Kadison Sakai derivation theorem (cf. [38]) there exists a selfadjoint element h 2 M, indeed a minimal positive one, such that e ith Xe Gammaith = e itL Xe GammaitL ; X 2 M ; and indeed Delta it h Delta Gammait = h by the canonicity of the minimal positive choice for h. Therefore Delta it h Omega = h Omega Gamma t ....

M. Takesaki, "Theory of Operator Algebras", vol. II, in preparation.

....but at the same time takes nite values on many in nite dimensional spaces. We shall very brie y describe the de nition and the necessary properties of the von Neumann dimension and trace. For more details we refer the reader to [At] C] and textbooks on von Neumann algebras (e.g. D] N] [T]) We shall denote the dimension by dim . It is de ned on the set of all (projective) Hilbert modules and takes values in [0; 1] The simplest Hilbert module is given by a left regular representation of : it is the Hilbert space L 2 consisting 9 of all complex valued L 2 functions on . The ....

M. Takesaki, Theory of operator algebras, I, Springer-Verlag, 1979.

....(5.13) Thus Gamma(I ) is a homomorphism. The injectivity follows from I I = 1K . 4. Using the previous step of the proof we see that Gamma(P ) is a norm one projection from Gamma(K 0 ) onto its von Neumann subalgebra Gamma(I ) Gamma(K) Thus Gamma(P ) is a conditional expectation [22]. Corollary 5.6 If Gamma is a functor of second quantisation then for any infinite dimensional real Hilbert space K we have Gamma(K) O(K) C 1. Proof. We can choose K = 2 (Z) Let S be the right shift. Then S n converges weakly to 0 as n 1. By the previous proposition we have that ....

Takesaki, M.: "Theory of Operator Algebras I", Springer Verlag, New York, 1979.

....a C dynamical system and the mixing property (w.r.t. the unperturbed dynamics) for the states considered here seems to be not satisfied for all observables in the von Neumann algebra (A(W 2 R (Z) 2 ; oe) 00 , see below. We recall the definition of mutually normality according to [31], Cap. III. Let i , i = 1; 2 be two states of a C algebra A with GNS representations i , i = 1; 2 respectively. Then 2 is normal w.r.t. 1 if there exists a normal homomorphism ae of 1 (A) 00 onto 2 (A) 00 such that ae ffi 1 = 2 : In the abelian case, this definition ....

Takesaki M. Theory of operator algebras I, Springer 1979.

....faithfull semifinite normal trace that takes value 1 when evaluated on projections of dimension 1. Let e be any projection in M Omega B(H) equiped with the tensorial product traces 0 ) and let in this case M t be the isomorphism class of (M Omega B(H) e . It is well known ( 9 ] 15] [24] , 23] that this isomorphism class does not depend on the choices made in the selection of the projection e . In particular we obtain in this way an equivalent definition for the fundamental group of M : F(M) ft 0j(9) 2 Aut(M Omega B(H) 0 ( x) t 0 (x) x 2 M Omega B(H) g: ....

) M. Takesaki , Theory of operators algebras I ,II, Springer , New York (1979).

....trace in B Omega B(K) then the map 1 : A(X) Omega A(X) 1 (x Omega ) x Omega admits a (global) continuous cross section when Omega A(X) is endowed with the norm topology. Proof. In this case, since B is finite, UB is complete in the strong ( strong ) operator topology [20]. Moreover, Popa and Takesaki proved in [16] that it admits a geodesic structure in the sense of Michael [13] It has been already remarked that the set function x Omega 7 fxu Omega u Ju J : u 2 UB g is lower semicontinuous in the norm topology. Therefore theorem 5.4 of [13] applies, ....

M. Takesaki, Theory of operator algebras I, Springer Verlag, New York, 1979.

....the nature of the index map in K theory. Prerequisites In addition to the general knowledge possessed by anyone with a graduate education in mathematics, we assume familiarity with the fundamentals of the theory of C algebras, as can be found in the rst chapter of such texts as [Da] or [Ta]. We also make frequent use of some elementary facts concerning the tensor product of C algebras. The reader who is not familiar with the C algebraic tensor product needs to accept on faith that a C algebraic norm can be de ned on the algebraic tensor product of two given C ....

M. Takesaki, Theory of Operator Algebras I, Springer, 1979.

....in particular the term von Neumann algebra is used to denote what at the time (1936) was called a ring of operators . We do not use the notations of [13] we give, however, the location in [13] of the results recalled. A systematic description of the dimension theory can be found e.g. in [17]. A set M of bounded operators on the Hilbert space H is called a von Neumann algebra if it is an algebra, contains the unit operator I , it is closed, i.e. it contains the adjoint of each of its elements, and it is closed in the strong (equivalently: weak) operator topology. Let N be any set ....

M. Takesaki, Theory of Operator Algebras, I. (Springer Verlag, New York, 1979)

....index sets for the proper formulation of our results, the reader will not be misled by thinking of as N. We now review several tensor products which will be needed subsequently. The minimal (also called injective or spatial) tensor product of C algebras A and B is denoted by A Omega min B [30], while M Omega N denotes the von Neumann algebra tensor product of von Neumann algebras M and N [30] The Haagerup tensor product A Omega h B [13, 18] is the completion of the algebraic tensor product A Omega B in the norm kuk h = inf 8 : fl fl fl fl fl fl n X j=1 a j a j fl fl ....

....N. We now review several tensor products which will be needed subsequently. The minimal (also called injective or spatial) tensor product of C algebras A and B is denoted by A Omega min B [30] while M Omega N denotes the von Neumann algebra tensor product of von Neumann algebras M and N [30]. The Haagerup tensor product A Omega h B [13, 18] is the completion of the algebraic tensor product A Omega B in the norm kuk h = inf 8 : fl fl fl fl fl fl n X j=1 a j a j fl fl fl fl fl fl 1=2 fl fl fl fl fl fl n X j=1 b j b j fl fl fl fl fl fl 1=2 9 = 2:13) ....

[Article contains additional citation context not shown here]

M. Takesaki, Theory of operator algebras I, Springer-Verlag, Berlin, 1979. 30

No context found.

M.Takesaki, Theory of operator algebras. I. Springer-Verlag, New York-Heidelberg, 1979.

No context found.

Takesaki, M.: Theory of Operator Algebras I. Springer-Verlag, New York (1979).

No context found.

M. Takesaki, Theory of Operator Algebras I (Springer-Verlag, Berlin, 2001).

No context found.

M. Takesaki, Theory of operator algebras I, Springer-Verlag, 1979.

No context found.

M. Takesaki, Theory of operator algebras, I, Springer--Verlag, New York--Heidelberg, 1979. 34

No context found.

M. Takesaki. Theory of operator algebras, I, Springer Verlag, Berlin, Heidelberg, New York, 1979.

No context found.

M. Takesaki, Theory of Operator Algebras, I. (Springer Verlag, New York, 1979)

No context found.

M. Takesaki, Theory of Operator Algebras I, Springer Verlag, New York, 1979.

First 50 documents

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC