A Note on Idempotents in Finite Aw*-factors

We prove that the value of the quasi-trace on an idempotent element in a AW*-factor of type II1 is the same as the dimension of its left (or right) support. It is a long standing open problem (due to Kaplansky) to prove that an AW*factor of type II1 is in fact a von Neumann algebra. A remarkable answer, in the affirmative, was found by Haagerup ([Ha]), who proved that if an AW*-factor A is generated by an exact C*-algebra, then A is indeed a von Neumann algebra. The main object, that was investigated in connection with Kaplansky’s problem, is the quasi-trace, whose construction we briefly recall below. One starts with an AW*-factor of type II1, say A. Denote by P(A) the collection of projections in A, that is P(A) = {p ∈ A : p = p = p}. A key fact is then the existence of a (unique) dimension function D : P(A) → [0, 1] with the following properties: • D(p) = D(q) ⇐⇒ p ∼ q; • if p ⊥ q, then D(p+ q) = D(p) +D(q); • D(1) = 1. The symbol “∼” denotes the Murray-von Neumann equivalence relation (p ∼ q ⇔ ∃x ∈ A with p = xx and q = xx), while “⊥” denotes the orthogonality relation (p ⊥ q ⇔ pq = 0; this implies that p+ q is again a projection). Once the dimension function is defined, it is extended to self-adjoint elements with finite spectrum. More explicitly, if a ∈ A is self-adjoint with finite spectrum, then there are (real) numbers α1, . . . , αn and pairwise orthogonal projections p1, . . . , pn, such that a = ∑n k=1 αkpk. We then define d(a) = ∑n k=1 αkD(pk). For an arbitrary self-adjoint element a ∈ A, one can approximate uniformly a with a sequence (an)n≥1 ∈ {a} ′′ of elements with finite spectrum. (Here {a} stands for the AW*-subalgebra generated by a and 1.) It turns out that the limit q(a) = limn→∞ d(an) is independent of the particular choice of (an)n≥1. Finally, for an arbitrary element x ∈ A, one defines Q(x) = q(Rex) + iq(Imx), where Rex = 1 2 (x+ x ) and Imx = 1 2i (x− x ). The map Q : A → C, defined this way, is the unique one with the properties: (i) Q is linear, when restricted to abelian C*-subalgebras of A; (ii) Q(xx) = Q(xx) ≥ 0, for all x ∈ A; (iii) Q(x) = Q(Rex) + iQ(Imx), for all x ∈ A; 1991 Mathematics Subject Classification. Primary 46L10; Secondary 46L30.