1. introduction in their seminal works ([27], see also [26]), Murray and von Neumann defined three types of von Neumann algebras (namely, type I, type II and type III) according to the properties of their projections

1. INTRODUCTION

In their seminal works ([27], see also [26]), Murray and von Neumann defined three types of von Neumann algebras (namely, type I, type II and type III) according to the properties of their projections. They showed that any von Neumann algebra is a sum of a type I, a type II, and a type III von Neumann subalgebras. This classification was shown to be very important and becomes the basic theory for the study of von Neumann algebras (see, e.g., [201]). Since a -algebra needs not have any projection, a similar classification for -algebras seems impossible. There is, however, an interesting classification scheme for -algebras proposed by Cuntz and Pedersen in [14], which captures some features of the classification of Murray and von Neumann.

The classification theme of -algebras took a drastic turn after an exciting work of Elliott on the classification of $A F$-algebras through the ordered -theory, in the sense that two $A F$ algebras are isomorphic if and only if they have the same ordered -theory ([16]). Elliott

then proposed an invariant consisting of the tracial state space and some -theory datum of the underlying -algebra (called the Elliott invariant) which could be a suitable candidate for a complete invariant for simple separable nuclear -algebras. Although it is known recently that it is not the case (see [37]), this Elliott invariant still works for a very large class of such -algebras (namely, those satisfying certain regularity conditions as described in [18]). Many people are still making progress in this direction in trying to find the biggest class of -algebras that can be classified through the Elliott invariant (see, e.g., [17, [उ5]). Notice that this classification is very different from the classification in the sense of Murray and von Neumann.

In this article, we reconsider the classification of -algebras through the idea of Murray and von Neumann. Instead of considering projections in a -algebra , we consider open projections and we twist the definition of the finiteness of projections slightly to obtain our classification scheme.

The notion of open projections was introduced by Akemann (in [1]). A projection in the universal enveloping von Neumann algebra (i.e. the biduals) of a -algebra (see, e.g., [36. §II.2]) is an open projection of if there is an increasing net of positive elements in with in the -topology. In the case when is commutative, open projections of are exactly characteristic functions of open subsets of the spectrum of . In general, there is a bijective correspondence between open projections of and hereditary -subalgebras of (where the hereditary -subalgebra corresponds to an open projection is ; see, e.g., 30 ). Characterisations and further developments of open projections can be found in, e.g., [2, . Since every element in a algebra is in the closed linear span of its open projections, it is reasonable to believe that the study of open projections will provide fruitful information about the underlying -algebra. Moreover, because of the correspondence between open projections (respectively, central open projections) and hereditary -subalgebras (respectively, closed ideals), the notion of strong Morita equivalence as defined by Rieffel (see [33] and also $[11,34]$ is found to be very useful in this scheme.

One might wonder why we do not consider the classification of the universal enveloping von Neumann algebras of -algebras to obtain a classification of -algebras. A reason is that for a -algebra , its bidual always contains many minimum projections (see, e.g., [1. II.17]), and hence a reasonable theory of type classification cannot be obtained without serious modifications. Furthermore, are usually very far away from , and information of might not always be respected very well in ; for example, and have isomorphic biduals, but the structure of their open projections can be used to distinguish them (see, e.g., Example [2.1 and also Proposition ).

As in the case of von Neumann algebras, in order to give a classification of -algebras, one needs, first of all, to consider a good equivalence relation among open projections. After some thoughts and considerations, we end up with the "spatial equivalence" as defined in Section 2, which is weaker than the one defined by Peligrad and Zsidó in [\$1] and stronger than the ordinary Murray-von Neumann equivalence. One reason for making this choice is that it is precisely the "hereditarily stable version of Murray-von Neumann equivalence"

that one might want (see Proposition a ), and it also coincides with the "spatial isomorphism" of the hereditary -subalgebras (see Proposition a) ).

Using the spatial equivalence relation, we introduce in Section 3 , the notion of -finite -algebras. It is shown that the sum of all -finite hereditary -subalgebra is a (not necessarily closed) ideal of the given -algebra. In the case when the -algebra is or , this ideal is the ideal of all finite rank operators on . Moreover, through finiteness, we define type , type , type as well as -semi-finite -algebras, and we study some properties of them. In particular, we will show that these properties are stable under taking hereditary -sub algebras, multiplier algebras, unitalization (if the algebra is not unital ) as well as strong Morita equivalence. We will also show that the notion of type coincides precisely with the discreteness as defined in [\$1].

In Section 4, we will compare these notions with some results in the literature and give some examples. In particular, we show that any type I -algebra (see, e.g., [30]) is of type ; any type II -algebra (as defined by Cuntz and Pedersen) is of type ; any semifinite -algebras (in the sense of Cuntz and Pedersen) is -semi-finite; any purely infinite -algebra (in the sense of Kirchberg and Rørdam) with real rank zero and any separable of type III (as introduced by Cuntz and Pedersen). Using our arguments for these results, we also show that any purely infinite -algebra is of type III. Moreover, a von Neumann algebra is a type , a type , a type or a -semi-finite -algebra if and only if is, respectively, a type I, a type II, a type III, or a semi-finite von Neumann algebra.

In Section 5, we show that any -algebra contains a largest type closed ideal , a largest type closed ideal , a largest type closed ideal as well as a largest semi-finite closed ideal . It is further shown that is an essential ideal of , and is an essential ideal of . On the other hand, is always a -semi-finite -algebra, while is always of type if one sets . We also compare , and with and , respectively.

In the Appendix, we give a very general classification scheme and observe that most of the results in the main body are actually true in a more general context. In particular, we show that many results in the main body remain valid if one replaces type , type , type and -semi-finiteness with discreteness, type II, type III and semi-finiteness, respectively.

Notation 1.1. Throughout this paper, is a non-zero -algebra, is the multiplier algebra of is the center of , and is the bidual of . Furthermore, Proj is the set of all projections in , while is the set of all open projections of A. All ideals in this paper are two-sided ideals (not assumed to be closed unless specified).

If and is a subspace of , we set , and denote by the norm closure of . For any , we set her to be the hereditary -subalgebra of (note that if is a partial isometry, then her ). When is understood, we will use the notation instead. Moreover, is the right support projection of a norm one element ,

i.e. is the -limit of and is the smallest open projection in with .

Acknowledgement: The authors would like to thank Prof. Larry Brown, Prof. Edward Effros and Prof. George Elliott for giving us some comments.

2. SPATIAL EQUIVALENCE OF OPEN PROJECTIONS

In this section, we will consider a suitable equivalence relation on the set of open projections of a -algebra. Let us start with the following example, which shows that the structure of open projections is rich enough to distinguish and , while they have isomorphic biduals (see Proposition 2,3)(b) below for a more general result).

Example 2.1. The sets of open projections of and can be regarded as the collections and , of open subsets of and of open subsets of the one point compactification of , respectively. As ordered sets, and are not isomorphic. In fact, suppose on the contrary that there is an order isomorphism . Then is a proper open subset of . Let and with . As is a minimal element, it is a singleton set. Thus, , which gives the contradiction that .