Arens algebras, associated with commutative von Neumann algebras

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Introduction.Let (~, ~, ~c) be a measurable space with a finite measure, = ~, ~c) the Banach space of all p-measurable com- plex functions on Q, integrable with the degree, p E [1,+~).R. Arens [1] introduced and studied the set = n He showed, l$poo . in particular, that is a complete locally-convex metrizable algebra with respect to "t" topology generated by the system of norms = ( 0 3 A 9 | f | p d ) 1/p , p > l.Later G.R. Allan [2] observed that t) is a GB*-algebra with the unit ball Bo = {f E L °° : 1} .Further inves- tigation of properties of the Arens algebra was made by S.J. Bhaft [3,4].He described the ideals of the algebra and considered some classes of homomorphisim of this algebra.B.S. Zakirov [5] showed that is an EW*-algebra and gave an example of two measures, ~c and võ n an atomic Boolean algebra, for which the algebras and Lw(v) are not isomorphic.It is clear that the problem of complete classification of the Arens algebras arises.Speaking more preciesly, what conditions should be imposed on measures p and v for the corresponding Arens algebras to be isomorphic?It is natural to solve this problem in the class of equiv- alent measures.Therefore instead of a measurable space with a measure, one should consider a commutative von Neumann algebra M with faithful normal finite traces p and v on M and study the problem of *-isomorphism of EW *algebras LW(M; = n LP(M; ) and L03C9(M, v) 1~pT he present article gives the complete solution of the mentioned problem, a classification of the normalized Boolean algebras from the book by D.A. Vladimirov [6] being considerably used.All neccessary notations and results from the theory of von Neumann algebras are taken from [7] and the theory of integration on von Neumann algebras is from [8]. . 2. Preliminaries.Let M be an arbitrary von Neumann algebra, a faithful normal finite trace on M, P(M) the lattice of all projections of M. Let K(M, fl) be the *-algebra of all -measurable operators affiliated with M [8) .. , In the commutative case, when M = L~(03A9, 03A3, ) and where (03A9, 03A3, ) is a measurable space, the algebra K(M, p) coincides with the algebra of all measurable complex functions on (0, E, ).
For every set A C K(M, p) we shall denote by Ah (respectively, by A+) the set of all self-adjoint (respectively, positive self-adjoint) operators from A. The partial order in Kh(M, generated by the positive cone K+(M, ~c) will be denoted by x y.
Put M(x) = y x, y E M~ for every x E K+(M, ~C).Let p E [l,oo) and = {x E ~}, where |x| = (~*~)1/2.The set LP(M, ~C) is a subspace in K(M, ~C) and the function ~x~p = (|x|p)1/p is a Banach norm on LP(M, ) [9].Moreover, 1. ~x~p = ~x* ~p = ~xu~p for all x E LP(M, ) and a unitary element uEM; 2. If Ixl |y|, x E K(M, ), y E LP(M, ), then x E LP(M, and ~x~p ~ ~y~p; 3. If x E Lp(M, ), y E' Lq(M, ) with 1 p + 1 q = T, , 1 P, q, r ~, then xy E ) and ~x~p~y~q.From these properties of the norm ~p it follows that the set LW (M, ) = n is a *-subalgebra in K(M, ~C), and M C ~c).It was 1~ps hown in [5] that M = L03C9(M, p) if and only if dimM ~.Furthermore, since LW (M, ~) is a solid *-subalgebra in K(M, ~C) (e.g. the inequality x E K(M, ~), y E ~) implies x E LW(M, ~)), L"(M, ~,) is an EW*-algebra, the bounded part of which coincides with M [10].Now we cite from [6] some information which will be used in the seqnel.Let X be an arbitrary complete Boolean algebra, e E X, Xe = [0, e] = ~g E e}.The minimal cardinality of the set which is dense in Xe in the (o)-topology will be denoted T(Xe).An infinite complete Boolean algebra X is called homogeneous, if T(Xe) = for any non-zero e, g e X.The cardinality of T(X) = r(X.)where I -is the unit of the Boolean algebra X is called a weight of a homogeneous Boolean algebra J~. .Let be a strictly positive countably additive measure on X.If (1) = 1, then the pair (X, ~) is called a normalized Boolean algebra.It was shown in [6] that for any cardinal number T there existed a complete homogeneous normalized Boolean algebra X with the weight r(X) = T.The next theorem gives a criterion of isomorphism of two homogeneous normalized Boolean algebras.
This theorem enables us to describe the class of von Neumann algebras for which the existence of *-isomorphism between the Arens algebras L03C9(M, ) and L"(N, v) is equivalent to isomorphism between M and N. Proposition 1.Let M and N be commutative von Neumann algebras, the Boolean algebras P(M) and P(N) of which are homogeneous, and let Jl and v be faithful normal finite traces on M and N, respectively.The following conditions are equivalent: 1) The Arens algebras ) and L03C9(N, v) are *-isomorphic; 2) The von Neumann algebras M and N -are *-isomorphic; 3) T(P(M)) = T(P(N)).
Proof.Sience L03C9(M, ) and v) are EW*-algebras the bounded parts of which coincide with M and N respectively, restriction on M of any *-isomorphism from ~c) on Lf.AJ(N, v) is a *-isomorphism from M on N. On the other hand if the von Neumann algebras M and N are *-isomorphic, then their Boolean algebras of projectors are also isomorphic and therefore, in this case, T(P(M)) = T(P(N)).Now suppose that T(P(M)) = T(P(N)) and assume '(x) = (x)/ (1), v'(y) = v(y)/v(I), x E M, y E N. According to the theorem 1, there exists an isomorphism of Boolean algebras ~ : X 2014~ Y for which = '(x) for all x EX.This isomorphism extends to a *-isomorphism 03A6 : K(M, ~c) -> K(N, v) (See [11]): At the same time = v'(~(x)) for all x E Since = = we have ~c)) = ~c')) = LP(N, v') = LP(N, v) for all p > l.Hence ~(L"(M, ~c)) = v).
Corolary.Let M and N be non-atomic commutative von Neumann algebras on separable Hilbert spaces, and v faithful normal finite traces on M and N, respectively.Then the Arens algebras L03C9(M, ) and LW(N, v) are *-isomorphic.Proof.At first, show that if M acts on a separable Hilbert space H, then the Banach space (Lr(M, ' ((T) is also separable.To start one should note that in this case the strong topology is metrizable on the unit ball Mi of the algebra M ([12] p.24).In addition, the convergence xa ~ 0 in the strong topology in Mi is equivalent to the convergence --~ 0 ([12] p.130).Thus, for any sequence of {xn} C M and x E M the convergence implies sup ~xn~M oo and ~xn -x~2 ~ 0, where ~.~M is a C*-norm in M. Hence, on any ball Mn = {x E M|~x~M ~ n} the strong topology coincides with the topology induced from L2(M, Since H is separable, there exists a countable set Xn C M which is dense in Mn in ' 00 the strong topology ([13], p.568).Hence the countable set X = U Xn is n=l.
There is one thing left to say: the (o)-topology in (P(M), coincides with the topology induced from (L2(M, Therefore, the P(M) is a non-atomic Boolean algebra which is separable in the (o)-topology.Hence it is homogeneous [6].Similarly, P(N) is a non-atomic Boolean algebra and (P(M)) = T(P(N)).According to the proposition 1, the Arens algebras L03C9(M, ) and v)are *-isomorphic.Let (X, ~c) be an arbitrary complete non-atomic normalized Boolean algebra.It was shown in [6] that there is a sequence ~en ~ of non-zero pair- wise disjoint elements for which the Boolean algebras [0, en] are homoge- neous and Tn = T([0, en]) n =1, 2, ... This collection is determined uniquely and the matrix ~ Tl 72 ... ~(02) ...y is called the passport of the Boolean algebra (X, ~c) The following theorem will be used for investigation of isomorphisms of Arens algebra.
Theorem 2 ~6~. .Let (X, ~c) and (Y, v) be complete non-atomic normalized Boolean algebras.The following conditions are equivalent.I. .There exists an isomorphism p J~ 2014~ Y for which = for all x ~ X.
2. The passports of the Boolean algebras (X, J.l) and (Y, coincide. 3. Main results.A von Neuman algebra M is called a-finite if it admits at most countable family of orthogonal projections.On any a-finite von Neumann algebra M, , there exists a normal state, in particular, if M is commutative, then its Boolean algebra of projections P(M) is a normed one.
The next theorem discribes the class of commutative a-finite von Neumann algebras M for which the Arens algebras L03C9(M, ) and L03C9(M, v) are *-isomorphic for any faithful normal finite traces of ~t and v on M. Theorem 3.For a commutative a-finite von Neumann algebra M the following conditions are equivalent: 1. .The Arens algebras L03C9(M, ) and L03C9(M, v) are *-isomorphic for any faithful normal finite traces p and v on M.
2. M = Mo + 03A3ni=1 Mi, where Mo is a finite-dimentional commutative von Neumann algebra, Mi is an infinite-dimensional commutative von Neumann algebra in which the lattice of projections P(Mi) is a homo- geneous Boolean algebra and Ti = (P(Mi) Ti+1, , i = 1, ... , n -1 (the summand Mo are ~z ~ Mi may be absent).
Proof. 1) --~ 2). .Let A be the set of all atoms in P(M) and e = sup A.
Since Mo = l~, the *-isomorphism ~ is generated by some bijection 7r of the set of natural numbers.It means that ~(x) = ~(~2n~) _ ~2'~tn~~ = y E ~c).In particular, oo o0 n =1 which is wrong.Hence, a set A is either finite or empty.Now suppose that in the Boolean algebra P((I-e)M) there is a count- able set {en} of disjoint elements, for which the algebras Xn = P(enM) are homogeneous and Tn = T(Xn) Choose two faithful normal finite traces ~c and v on M such that = n-2, v(en) = e-~~ and = for all x E Mo.Let 03A6 be a *-isomorphisms from LW(M, v) on Then ~((Ie)M) = (Ie)M and, since weights Tn are different, 03A6(enM) = en(M) (See [6]).does not belong to L03C9(M, v).The obtained contradiction implies that the set {en ~ is at most count- able.
Using theorem 3, it is easy to construct an example of a non-atomic commutative von Neumann algebra M with traces p and v, such that the Arens algebras L03C9(M, ) and L03C9(M, v) are isomorphic, while there is no *-isomorhism c~ from M on M, for which v o ~p = ~c.Indeed, assume that M = Ml + M2, where are non-atomic commutative a-finite von Neumann algebras in which the lattice of projections form homogeneous Boolean algebras and r(.P(Mi)) r(jP(M2)).Identify M1 with the subal- gebra eiMi and M2 with (I -ei)Mi , e1 E P(M).Let be an arbitrary faithful normal finite trace on At, = 1.Assume that v ( x ) = p (( e 1 ) -1 ( x e 1 ) + q ( ( I -e 1 ) ) -1 ( x ( 1 -e1)), x E M, p, q > 0, p + q = 1.It is evident that v is a faithful normal finite trace on M. Choose p and q such that (e1) ~ v(e1) = p, -el) ṽ(I -el) = q.According to the theorem 2, there is no *-isomorphism ~ : : M --~ M for which v o cp = /~.At the same time, according to the theorem 3, the Arens algebras and v) are *-isomorphic.Now, let us find out when the Arens algebras coincide for different traces.Let and v be two faithful normal finite traces on a commutative von Neumann algebra M. Denote by h === ~ the Radon-Nikodim derivate of the trace ~ relative v, i.e. h is the element from L+(M, v) for which (x) = v(hx) for all x E M.
It is clear that the element x from K(M, ) belongs to L1(M, ) if and only if hx E L1(M, v).In this case the equality p(x) = v(hx) holds.Proposition 2. .L03C9(M, v) C L03C9(M, ) if only if h E U LP(M, v), 1~p~w here LOO(M, v) is identified with M.
Proof.Let L"(M, v) C C Then = v(hx) for all x E L03C9(M, v), and is a positive linear functional on L03C9(M, v).Since L03C9(M, v) is a complete metrizable locally-convex algebra with respect to the t-topology generated by the system of norms {~x~p = (v(|x|p))1/p} p~1 (see [3]) and involution in L03C9(M, v) is continuous in this topology, is con- tinuous [14].It was shown in [3] that the dual space of (LtJJ(M, v)t) may be identified with U LP(M, v).Hence one can find such y E LP(M, v) for for some p (1,~], then is a t- continuous linear functional on and therefore = oo for any x E and q > 1; we recall that |x|q v) for all .r 6v) and q > 1.Thus, q~1 The following criterion of coincidence of the algebras and v) arises from the proposition 2. Remarks. 1.In the example constructed after theorem 3 p) = L~(M, x/) since Now everything is ready to obtain the criterion of *-isomorphism of the Arens algebras p) and x/).Let M be an arbitrary non-atomic commutative a-finite von Neumann algebra.According to [6] the Boolean algebra P(M) of projections M possesses uniquely determined collection of non-zero pairwise disjoint elements for which the Boolean algebras Xn = {e 6 P(M) : e ~ en} are homogeneous and (Xn+1).
Assume that the collection is infinite otherwise all Arens algebras are *-isomorphic (see theorem 3).oo lf , q = oo .
Therefore, according to the theorem 4 h E U LP(N, v), and h~1 E 1p~Ũ Lp(N, ), where h is the Radon-Nikodym's derivative of the trace 1p~r elative the trace v, being considered in N.So using the equality hen = n03BD-1nen, n = 1, 2, ..., the required inequalities follow from the condition 2).
Remarks 2. Repeating the argument from the proof of the theorem 5, it is easy to obtain the following criterion of *-isomorphism of the Arens algebras ~c) and v) : Let and v be faithful normal finite traces on a infinite dimensional atomic commutative von Neumann algebra N, the set of all atoms in P(N), n = (qn), Vn = v(qn), , n = 1 > 2 ....Then, , the Arens algebras and v) are *-isomorphic only in the case when there are such p, q ~ (1, ~) and permutation 7r of a set of natural numbers, that 00 00 ~' 1~ oo, in th case p, q E (1, ) n =1 _ and sup | n03BD-1n| [ oo if p = oo, sup |03BDn -1n| oo if q = oo .n>1 n>1 3. Any von Neumann algebra M is represented as M = Mi + M2, where M is an atomic von Neumann algebra and M2 is a non-atomic von Neumann algebra.Moreover, if ~ is a *-automorphism of M, then = Mi and ~(M2) -M2.Therefore theorem 5 and Remark 2 give criterion of isomorphism of Arens algebras for arbitrary commutative a-finite von Neumann algebras Choose x E K((I -e)M, v) such that xen = 2nen.Then x E e)M, v), = x and oo ~(1~'(x)I) = 1 ~'~nw2 = ~ñ=1 i.e.

Theorem 4 .
Let , v be faithful normal finite traces on a commutative von Neumann algebra M. Then p