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A criterion is given for a compact connected subgroup of Gl(n,C) to be isomorphic to a direct product of unitary groups. It implies that a compact connected subgroup of rank n in Gl(n, C) is isomorphic to a direct product of unitary groups. The paper gives a generalization of some of the results in [3]. Let G be a compact connected subgroup of Gl (n, C). We denote by L(G) the Lie algebra of G and set H(G) = iL(G). The rank of G is the dimension of a maximal torus in G (see [1, p. 93]). THEOREM. Let G be a compact connected subgroup of rank k in Gl(n, C). Suppose there exist r > k orthogonal idempotents a,, ..., ar in H(G). Then r = k and G is isomorph-ic (as a Lie group) to a direct product of unitary groups: G _ U(nl) x ... x U(nm) with n + m+ nm =k. PROOF. By [2, p. 176, Theorem 1] G is similar to a subgroup of U(n). Hence we may assume that G is a subgroup of U(n). Thus the operators in H(G) C End(Cn) are hermitian. Since a1, ..., ar commute we see that T = {exp(it1 a1 + * * * + itrar)Iti, * . ., tr E R} is a torus in G of dimension r. Clearly r = k and T is a maximal torus. If a E H(G) then exp(ita) E G (t E R) and is contained in some conjugate of T (see [1, p. 89]), i.e. exp(ita) E u-1Tu = u*Tu for some u E G. It follows that a = t1u* a,u + .+ ttu*a u. Since a2 = t2u*au + ... + t2u*aru and u*asu E H(G) fo = r r thata s for s = 1, ..., r we see that a E H(G). Let b E H(G), too. Since ab + ba = (a + b)2 a2 b2 we see that ab + ba E H(G). Also, ab ba E iH(G) since ia, ib E L(G). Thus ab E H(G) + iH(G). Let A(G) = H(G) + iH(G). It follows that A(G) is an algebra. Clearly, it is a finite dimensional C*-algebra. By the Wedderburn decomposition there exist central idempotents el, ..., em E A(G) =A such that A = Ael @ .. * *Aem and Aes is isomorphic to End(Xs) for some finite dimensional vector space Xs over C (s= l,...,m). The ideal Aes is closed, hence selfadjoint and a C*-subalgebra of A. Clearly, eS is the identity on Aes and hence es* = eS. Consider the group V of unitary elements in Aes. The isomorphism AesEnd(Xs) defines a (continuous) representation of V on Xs. Using once more [2, p. 176, Theorem 1] we equip Xs with an inner product such that the isomorphism maps V into the unitary Received by the editors May 26, 1975. AMS (MOS) subject classifications (1970). Primary 22E15, 22E60; Secondary 46L05, 46L20. ? American Mathematical Society 1976

Lie groups isomorphic to direct products of unitary groups