We prove that BUp has only two standard H-structures. The two most familiar H-multiplications on BU arise from the direct sum of complex vector bundles and from the tensor product of virtual line bundles; BU with these multiplications will be denoted by BU and BUe respectively [2]. Both multiplications are standard, define a standard H-space (X, e, ,u) to be an H-space whose rational Pontrjagin ring H*(X, Q) induced by ,u is both associative and graded commutative. We consider H-spaces localized at a fixed prime p. THEOREM 1.1. Let m be a standard H-multiplication on BU. Then BUpm is H-equivalent toU or U . We recall that two H-spaces (X, e, ,u) and (Y, f, v) are H-equivalent if there exists a homotopy equivalence k: X -* Y such that v(k X k) k,u. The two classes of multiplications on BUp in Theorem 1.1 can easily be distinguished for the Frobenius map (: H2(BUM, Z/pZ) -H2p(BUm, Z/pZ) defined by ((x) = xP is nontrivial when m = ED but is the zero homomorphism if m = ?. Theorem 1.1 completes results about standard multiplications on BUp proved in [4]. The strongest result in this context was given in ?4 of [4]. X is assumed to have the homotopy type of a connected CW-complex with finite skeleta. THEOREM 1.2. Let X be a standard H-space andp an odd prime. Assume that (a) H*(X, Z/pZ) is a polynomial algebra, (b) Dim QH2'(X, Z/pZ) S 1 for all i and the equality holds at least for 1 < i p 1, (c) t: H2(X, Z/pZ) -H2p(X, Z/pZ) is nontrivial. Then Xp R BUp' as an H-space. We can now add a companion result for B Up. THEOREM 1.3. Let X be a standard H-space andp an odd prime. Assume that (a) H*(X, Z/pZ) is a polynomial algebra, (b)' Dim QH2i(X, Z/pZ) S 1 for all i and the equality holds at least for 1 < i < P, Received by the editors January 23, 1984. 1980 Mathematics Subject Classification. Primary 55R35; Secondary 55N15. 'Author supported by Kuwait University Research grant SM102. ?1985 American Mathematical Society 0002-9939/85 $1.00 + $.25 per page