A REMARK ON THE SLICE MAP PROBLEM

It is shown that there exist a (r-weakly closed operator algebra/i, generated by finite rank operators and a a-weakly closed operator algebra/ generated by compact operators such that the Fubini product F/ contains properly /.

In [6] Kraus initiated the slice map problem for a-weakly closed operator spaces.By an operator space we mean a norm closed linear subspace of L(H), the operators of a Hilbert space H.As stated in the introduction of [9], the slice map problem is of interest because a number of questions concerning tensor products of a-weakly closed operator spaces are special cases of the slice map problem [4][5][6][7][8][9].
A a-weakly closed operator space A is said to have Property 5' if AFB AB for any q-weakly closed subspace B [6].Kraus  [9] first gave (r-weakly closed operator spaces not having Property S. Effros et al. [3] also characterized (r-weakly closed operator spaces having Property Sa.One of useful theorems [7, Theorem 2.1] for the slice map problem says that a (r-weakly closed unital operator algebra generated by finite rank operators has Property S (cf.[10]).In this paper, we show that the condition "unital" is essential in the theorem.

MAIN RESULT.
For operator spaces A and B, let A(B denote the norm closed linear span of {a(R)b a 6_ A and b 6_ B}.If A and B are (r-weakly closed, let AB denote the (r-weakly closed linear span of {a(R)b: a 6_ A and b 6_ B}.
Let X and Y be yon Neumann algebras.For g 6_ X., the predual of X, the right slice map R 9 associated with g is a unique bounded linear map from X6Y to Y such that Rg(z (R) y) =< z,g > y.For h 6_ Y., the left slice map Lh from XY to X is a unique bounded linear map such that Lh(z (R) y) =< y, h > z.Let A and B be (r-weakly closed linear subspaces of X and M. CHO AND T. HURUYA )", respectively.We define the Fubini product A(FB of A and B by A(gFB {z _ X(Y Rg(z) E B, Ln(x) _ A for every 9 X.,h c= Y.}.The space AFB does not depend on XY [6, Remark 1.2].
Let A be a C*-algebra.If we assume that A acts universally on a Hilbert space H, the second dual A** of A can be identified with the a-weak closure B of A in L(H).In this case, the weak* topology on A** coincides with the (r-weak topology on B.
The following example shows that the condition "containing the identity" is necessary in Theorem 2.1 of [7].
EXAMPLE.There exist a (r-weakly closed operator algebra generated by finite rank operators on a Hilbert space H and a (r-weakly closed operator algebra/} generated by compact operators on H such that F/} contains properly i,b.
PROOF.Let co denote the C*-algebra of all complex sequences that converge to zero.
Oavie [1] constructed a closed linear subspace A0 of co satisfying the following properties: (1)  A0 does not have the approximation property in the sense of Grothendieck; (2) A0 contains a dense linear subspace A1 with the norm topology such that each element has finite support, where each element of co is identified with a function whose domain is the set of all positive integers.
Since c* is *-isomorphic to g, the yon Neumann algebra of all bounded sequences, we assume that c* acts on the Hilbert space g2 in the usual way.Let A denote the (r-weak closure of A0 in c*.For a closed linear subspace Do of co, let D denote the (r-weak closure of Do in c*.We note that (c0+c0)** c*c* and AFD C_ c*c*.Put F(Ao, Do, coeo) {z _ c0c0 ng(z) c= Do, Lh(z) _ Ao for every g _ c$,h E c$}.
By the same argument in the proof of [9, Theorem 5.8] (with a C*-algebra A replaced by an operator space A), we can choose a closed linear subspace Bo of co such that F(Ao, Bo, co'co) contains properly Ao(Bo.Let B be the (r-weak closure of Bo in c$*.Since A f'l co A0 and B fqco B0, we have F(Ao, Bo, Co'co) :3 (A (R)F B)f'l(coco).The opposite inclusion is trivial.
It follows that F(Ao, So, coco)= (AFB)fq(coco).Since AB is identified with the weak* closure of AoBo in (c0c0)**, we have (AB)gl (co(co) AoBo.Hence AFB contains properly A)B.{(0 a) } {(0 b).b_B) SinceA, Let H /2g. Put 0 0 "aeA and / 0' 0 consists of finite rank operators on g, it is easy to see that , is a (r-weakly closed operator algebra generated by finite rank operators on H. Since co consists of compact operators on g2, / is a (r-weakly closed operator algebra generated by compact operators on H. Then and F[" 0 0 (R) 0 0 (R)a" a e AFB 0 0 (R) 0 0 (R) a'a C= AB Hence F/ contains properly /.This completes the proof.
Let K be the C*-algebra of all compact operators on a separable infinite dimensional Hilbert space.An operator space A is said to have the operator approximation property if

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.
However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation