Isomorphisms of noncommutative log-algebras

Бюллетень науки и практики / Bulletin of Science and Practice

UDC 517.98                                         

Let M be the von Neumann algebra, ц the faithful normal finite trace on M, S(M, ц) — *-algebra of measurable operators associated with M.

Consider the set L i _og(M, ц)={T eS(M, ц): ц(log(1+|T|))<те} and the function ||T|| i _og=

μ(log(1+|T|)) on L log (M, μ). In the work ([1] Lemma 4.1 and 4.3.) the following properties of the function ||'|| iog have been proved.

Lemma 1. Let S, Te S(M, ц). Then

• a)    ||T|| iog >0, provided T/0;

• b)    ||αT|| log ≤||T|| log for all scalars α with |α|≤1;

• c)    If TeL_log(M, ц), then

limHHU = 0 a ^ 0

• d)    ||S+T|| log ≤||S|| log +||T|| log ;

• e)    ||S-T|| log <||S|| log +||T|| log .

It follows from properties a), b), c), d) that the function ||-||_log is an F-norm on the space L log (M, μ), and property e) imply that the space L log (M, μ) is a topological algebra with respect to topology generated by the metric p(S,T)=||S-T||_log ([1], corollary 4.6). Let's call algebra L_log(M, ц) log - algebra.

In the present paper, we determine the necessary and sufficient condition the isomorphism of the log-algebras constructed by various the faithful normal finite trace on various von Neumann algebras.

Ease of Use

Let M be a von Neumann algebra with faithful normal finite traces ц and v. It follows from 1 D the inequality log|fz)|< -|fz)p that Lp(Q,v)cLlog(Q,v) for pe(0,x). And it follows from the inequalities k1logac< logbc < k2logac that the finiteness of the value ^ log ( 1 + \f(z)\)dv does not depend on the choice of the base of the logarithm. Here k1 is a sufficiently small number and k2 is a sufficiently large number.

Let ц and v be faithful normal finite traces on the von Neumann algebra M, denote by □ = ~ the Radon Nikodim derivative of trace v with respect to ц, such a central positive operator from L1(M,ц) for which the equality v(x )= ц( hx) holds for all x eM [2]. From here we get ц( h )=v(1), i. e. h e L1(M,ц). Moreover, there exists a measurable operator □

¹ = — [3]. dM

Prepare Your Paper Before Styling

Proposition 2. L_log(M, ц)с L_log(M,v) if and only if h eM.

Proof. Let heM and feLlog(M, ц), i.e. ^ log( 1 + |f (z)\)d^ < w. Since h is central, the algebra of measurable operators generated by the operators h and f will be commutative. Therefore, in this case S(M, ц) can be identified with the function space on Q. Then

[ log(1 + \f(z)\)dv= f (□ log(1 + \f(z)\))d^< a                      a

< II □ IIw f log(1 + |f(z)|)dp < w

a

Hence f eL log (M, ц), i.e. L log (M, ц)с L log (M,v).

Conversely, let 0< h eL 1 (M, ц)\M. Then it is possible to construct an infinite sequence of sets

M_n={ z eQ: n < h ( z )< n +1}. Now consider the subset of natural numbers N₀={neN: ц(M n )}. Let us redesignate the elements of the set N 0 as follows N 0 ={n 1 ,n 2 ,…}, n k k+1 .

Consider the function

1 g(z} = k^M^)''zQM^ g(z) = o,z E Q\Uk МПк.

Let's put f ( z )= e^g -1, then

∞∞ fniog( 1+|f(z)|)dg=fffMMf;)=%f<

However

log(1 + \f(z)\)dg = v(log(1 + \f(z)\')') = X □ (z))log(1+f(z))

a

∞∞∞

^MJ   nfc1

^ g ~klk2^(Mnk)° k=1        K    k=1k=1

From (1) and (2) it follows that f eL_log(M, ц), and f ?L i _og(M, ц), i.e. L i _og(M, ц), is not a subset of L_log(M, v), for h eL₁(M,ц)\M. So from L_log(M, ц)с L_log(M,v) if and only if, when h eM.

Let h be the Radon-Nikodim derivative of the faithful normal finite trace of v with respect to the faithful normal finite trace of ц The von Neumann algebra M is hence finite. Therefore, by virtue of Theorem 1 [4], h and h ^-1 are elements of the algebra of measurable elements. Now from the equality v( x )= ц( hx ) we get v( h ^-1 x )=ц( h ^-1 hx )=ц(х), i.e. h ^-1 is the derivative of Radon-Nikodim of trace ц with respect to v. Therefore, from Proposition 2 we obtain.

Corollary 3. L_log(M, ц)=L_log(M, v) if and only if h , h ^-1eM.

Let M and N be noncommutative von Neumann algebras with faithful normal finite traces μ and v, respectively. Let a:M >N be an isomorphism from M to N. Then the functional ц ° a-1 will be an faithful normal finite trace on N.

Definition 4. Traces ц and v are said to be equivalent if there exists a *-isomorphism a:M>N such that one of the following equivalent conditions is satisfied:

• ( i )    L log (N,v)=L log (N, ц ° a^-1);

dv   du ° a4

• - T ,          ^e N

• ( ii )    d u ° ^a       d ^v        .

The equivalence of conditions ( i ) and ( ii ) follows from Corollary 3.

Theorem 5. The algebras L_log(M, ц) and L_log(N, v) are *-isomorphic if and only if ц and v are equivalent.

Proof. Let ц and v be equivalent, i. e., there exists a ^isomorphism a: M >N, for which condition (i) is satisfied. The *-isomorphism a: M >N, extends to the *-isomorphism a’ onto the algebra of measurable functions L0(Q). In this case, using the continuity of a' with respect to the topology of convergence in measure, we obtain that a'(Llog(M,ц))=Llog(N,ц°a-1).                                         (3)

By virtue of condition ( i ), we have

L log (N,ц°a^-1)=L log (N,v).                                            (4)

From (3) and (4) we obtain that Llog(M, ц) and Llog(N,v) are *-isomorphic.

Conversely, let a' be a *-isomorphism from L_log(M, ц) to L_log(N,v). Then a' translates bounded elements from L log (M, ц) into bounded elements from L log (N,v), i. e. the restriction of a' to M is a *-isomorphism from M to N. Moreover, the *-isomorphism from M to N satisfies the condition that traces ц and v are equivalent.