1. Introduction

Let be a probability space. Denote by , the set of all (classes of) complex-valued measurable functions on . Let stand for the measure topology in . The classical Banach principle can be stated as follows:

Classical Banach Principle. Let be a Banach space, and let be a sequence of continuous linear maps. Consider the following properties:

(i) the sequence converges almost everywhere (a.e) for every

(ii) a.e for every ;

(iii) holds, and the maximal operator is continuous at 0;

(iv) the set is closed in X.

Then the implications are always true. If in addition, there exists a dense subset , such that the sequence converges a.e. for every , then all the four conditions (i) – (iv) above are equivalent.

The Banach Principle above was often applied in the case where . However, in the case p = the uniform topology on appears to be too strong for the classical Banach Principle to be effective in . For example, continuous functions are not uniformly dense .

Bellow and jones [3], using the fact that the unit ball is complete in , suggested to consider the measure topology on by replacing by , since is not a linear space, however, some geometrical implications do occurred but was resolved by Bellow and Jones. A non-commutative version of the Banach Principle for was proposed by Chilin and Litvinov, while the non-commutative notions of Banach Principle for measurable operators affiliated to a semi-finite von Neumann algebra (W^*-algebra) were established by Goldstein, M; Litvinov, S and Litvinov, S; Mukhamedov, F. Then it was refined and applied in [5], [6], [13]. In [5] the notion of uniform equicontinuity of a sequence of functions into was introduced. The objective of the paper is to give a comparative survey notes on non-commutative extension of the Banach Principle for that was suggested in [3], [7], which extend the results in [8] to the case of JW-algebras without direct summand of the type .

2. Preliminaries

Let be a semi-finite von Neumann algebra of bounded operators acting on a complex Hilbert space , and let denote the algebra of all bounded linear operators on . A densely-defined closed operator in is said to be affiliated to if . We denote by the complete lattice of projections in . Let be a faithful normal semi-finite trace on Denote by the orthogonal complemented projection for the projection , where is the identity of . An operator affiliated to is called if for every there exist a projection with such that belongs to the domain of the operator . Let the set of all operator affiliated to . Denote the uniform norm in .

If we set

then, the topology defined on by the family of neighborhoods of zero is called the measure topology ([14], [15])

Theorem 1: is a complete metrizable topological *-algebra

Proof: see ([14], [15]) for the details of the proof.

Let be a semifinite of without a direct summand of type and be the complete lattice of projections in , and be a faithful normal semifinite trace on . Let be the von Neumann enveloping algebra of the Jordan algebra . Then can be uniquely extended to a faithful normal semifinite trace on , for which we will use the same symbol . A self adjoint operator is called affiliated to a if all its spectral projections belong to . An operator affiliated to is called if for all there exist with such that belongs to the domain of the operator . Let be the set of all operators affiliated to .

Remark: A sequence is said to converge almost uniformly (a.u) to if for all there exist with such that

Proposition 1: For the following conditions are equivalent

(i) converges a.u. in ;

(ii) For all there exist with such that as

Proof: see [7] for details

The following theorem is a non-commutative notion of Riesz theorem.

Theorem 2: If and then for some subsequence

The proof of this theorem can be seen in [9] and [15].

A sequence is said to converge bilaterally with square almost uniformly (b.s.a.u.) to if given there is with

Then the following proposition holds;

Proposition 2: For the following conditions are equivalent.

(i) converges a.u. in ;

(ii) Given , there is with

(iii) converges b.s.a.u.in

(iv) Given , there is with

Proof: conditions (i) are trivial. For (iv) : From

so we can see that b.s.a.u. fundamentalness of a sequence in a reversible JW-algebra is equivalent to a.u. fundamentalness of the same sequence in its von Neumann enveloping algebra M = M(A). Thus the statement follows from proposition 1 above and hence the proved.

The Riesz theorem above will now take the following form

Theorem 3: If and then for some subsequence

Proof: Directly from propositions 1 and 2 respectively.

3. Uniform Equicontinuity for Sequences of Maps into

Let E be any set. If and are such that , then we denote

The following fact should be noted.

Corollary: Let be a semigroup, be sequence of additive maps. Assume that is such that for every there exist a sequence and a projection with such that

(i) converges a.u. as for every k;

(ii)

Then the sequence converges a.u. in L.

Proof: Fix and let and with be such that the two conditions hold. Now pick and let be such that then by proposition 1, there is a projection with and a positive integer for which the inequality holds whenever. If we define , then and for all . Therefore by proposition 1, the sequence converges a.u. in L.

The following definitions hold:

Definition: A sequence is said to be equicontinuous at if, given , there is a neighborhood U of in (X,t) such that for every and every one can find a projection with satisfying

Now let be a topological space, and let and be such that

Definition: Let and be as above. Let The sequence will be called uniformly equicontinuous at on E if, given, there is a neighborhood U of in such that for every there exists a projection satisfying

4. Bilateral with Square Uniform Equicontinuity for Sequences of Maps into

Definition: Let be a topological space, and be such that for . A sequence is called bilaterally with square equicontinuous at in such that i.e. for all for all one can find a projection with satisfying .

Definition: Let A sequence is called bilaterally with square uniformly equicontinuous at on E, if for all there is a neighborhood of in such that i.e. for all and for all one can find a projection with satisfying

We can now define a bilaterally sequence with square uniformly equicontinuous as follows:

Definition: Let . A sequence is called bilaterally with square uniformly equicontinuous at on E, if for all there is a neighborhood U of in such that for all there is with satisfying

We then have the following result

Proposition 3: Let the sequence and as in definition above. Then,

(i) is equicontinuous at on E into iff it is bilaterally with square equicontinuous at on E into

(ii) is uniformly equicontinuous at on E into iff it is bilaterally with square uniformly equicontinuous at on E into

Proof: Directly follows from proposition 2 and arguments in [7]

Also in [7] it has been established that for any the sets

and

are complete.

Therefore, is a complete metrizable topological algebra, and is a complete metrizable topological Jordan subalgebra of Hence, it is easy to see that the set is complete as well.

5. Main Results

In the algebra, let’s consider the following conditions. Let now for a sequence of function then

(i)almost uniform convergence of for every

(ii) uniform equicontinuity at 0 on ;

(iii) closedness in of the set

With these conditions, one can study relationships among the conditions and. Following the classical scheme, one more condition can be added, namely, a non-commutative version of the existence of the maximal operator as follows: given and there is with .. This condition may be called a pointwise uniform boundedness of on . It can be easily verified that implies , but does not guarantee . However, if is additive for every , then follows from while if is closed in then is equivalent to the closedness of in .

In the JW-algebras, the above conditions can also be extended as follows: Let . For a sequence then

(i) Bilateral with square almost uniform convergence of for every

(ii) Bilateral with square uniform equicontinuity at 0 on

(iii) Closedness in of the set

We can then discuss relationships among the conditions and as discussed in the case of the *-algebras. These are summarized below in the following theorems and whose proofs are obtained directly in [7] and the arguments in this paper.

Theorem 4: Let be a sequence of positive continuous linear maps with . Then the sequence is also

Theorem 5: A sequence of additive maps is as well

Theorem 6: Let be a sequence of positive continuous linear maps such that . If a sequence is being the conditions are equivalent.

6. Conclusions

The results in this paper has shown how Banach Principle for semifinite Von Neumann (W^*-algebras) algebras was extended to the case of without direct summand of type I₂. We can extend these results to the case of bilateral almost uniform convergence on semifinite von Neumann algebras and semifinite without direct summand of type I₂. These results can further be extended to obtain Stochastic Banach Principle, and then apply it to obtain some new Ergodic type theorems for Jordan algebras.