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In this paper, we prove that if S is an n-dimensional subspace of L(H), then S is ([i ] + 1)-reflexive, where [n ] denotes the greatest integer not n larger than '. By the result, we show that if 1?( ) = E Ai( )Bi is an 2=1 elementary operator on a C*-algebra A, then 'D is completely positive if and only if 'D is ([n1 ] + 1)-positive. In this paper, let H denote a complex Hilbert space. Let H(') denote the direct sum of n copies of H. For T E L(H), we write T(') for the operator on H(') which is the direct sum of n copies of T; the notation is extended to a subset of L(H) by S(n) = {T(n) E L(H(n)): T E S}. If S is a subspace of L(H), S is called n-reflexive if S(n) = ref (S(n)) =_ {T(n) E L(H(n)): T(n)X E [S(n)X], for all x E H (n)}, where [] denotes norm closed linear span. By the definiton, we have that if S is m-reflexive, then S is n-reflexive for n > m. A separating vector for a subspace S of L(H) is a vector x E H such that T 4 Tx, T E S, is an invective map. For x, y E H, let x 0 y denote the rank-one operator u | 4 (u, x)y. Let A denote a C*-algebra. Then A is called primitive, if A has a faithful irreducible representation on some Hilbert space. An elementary operator AP on A n is a linear mapping of the form AP: T F4 AiTBi, where {Ai},nL1 and {Bi},nU1 are i=l subsets of A. In this paper, we assume that all elementary operators are nonzero. A linear map 4J on A is positive (resp. hermitian-preserving) if 4)(T) is positive (resp. hermitian) for all positive (resp. hermitian) T in A. We define 4'n = 4{I 0 In: Mn(A) -4 Mn(A) by 4) 0 In((Tij)nxn) = (4)(Ti ))nxn. 4) is said to be n-positive if 4J 0 In is positive. If 4J is n-positive for all n, then 4J is said to be completely positive. In [4], Magajna states the following problem: For each positive integer r determine the smallest k = k(r) such that all rdimensional subspaces of L(H) are k-reflexive. In [4], Magajna proves k < r. In this paper, we prove that if S is an n-dimensional subspace of L(H), then S is ([n] + 1)-reflexive. Also by this result, we study complete positivity of elementary operators on a C*-algebra A. We prove that if n 4 ) = Ai( )Bi is an elementary operator on a C*-algebra A, then 4) is i=l1 Received by the editors July 8, 1996 and, in revised form, May 14, 1997. 1991 Mathematics Subject Classification. Primary 47B47, 47B49; Secondary 46L05.

Let T1,...,T, C B(H) be bounded operators on a Hilbert space NH such that TiT1 + . + TnTnT < I-H. Given a symmetry j on NH, i.e., j2 = j*j = I-, we define the j-symmetric commutant of {Ti,...,Tn to be the operator space {A C B(N): TiA = jATi, i = 1, ..., n}. In this paper we obtain lifting theorems for symmetric commutants. The result extends the Sz.-Nagy-Foias commutant lifting theorem (n = 1, j = I-H), the anticommutant lifting theorem of Sebestyen (n 1, j = -Ia), and the noncommutative commutant lifting theorem (j = Ia). Sarason's interpolation theorem for H' is extended to symmetric commutants on Fock spaces. 1. LIFTING THEOREM FOR SYMMETRIC COMMUTANTS Let F+ be the unital free semigroup on n generators s,... ., sr, and let e be its neutral element. For any o... sik E F+, we define its length 11 := k, and lel = 0. On the other hand, if Ti E B(H), i = 1,.. ., n, we denote T, :Til ... and Te := IH. Let us recall from [Pol], [Po2], and [Po4] a few results concerning the noncommutative dilation theory for n-tuples of operators. A sequence of operators T := [Ti,... , T], Ti E B(H), i = 1, ... , n, is called contractive (or row contraction) if TIT1* + *.. + TnTn < I-. We say that a sequence of isometries V := [VI, ... , 1V] on a Hilbert space IC D H is a minimal isometric dilation of T if the following properties are satisfied: (i VI VI* + + Vn Vn* < IK; (ii Vi* I = Ti* I i = 1, . . n; (iii) IC = VaEF+ VaH. Consider the full Fock space on n generators F2(Hn) := ClE Q Hm Secondary 30E05. The author was partially supported by NSF Grant DMS-9531954. ?2000 American Mathematical Society 1705 This content downloaded from 207.46.13.103 on Thu, 20 Oct 2016 04:32:19 UTC All use subject to http://about.jstor.org/terms

Let H H be a complex Hilbert space and let B ( H ) B(H) be the algebra of all bounded linear operators on H H . For c = ( c 1 , … , c k ) c=(c_{1},\dots ,c_{k}) , where c 1 ≥ ⋯ ≥ c k > 0 c_{1}\ge \cdots \ge c_{k}>0 and p ≥ 1 p\ge 1 , define the ( c , p ) (c,p) -norm of A ∈ B ( H ) A\in B(H) by \[ ‖ A ‖ c , p = ( ∑ i = 1 k c i s i ( A ) p ) 1 p , \|A\|_{c,p}=\left (\sum _{i=1}^{k} c_{i} s_{i}(A)^{p}\right )^{\frac {1}{p}} , \] where s i ( A ) s_{i}(A) denotes the i i th s s -numbers of A A . In this paper we study some basic properties of this norm and give a characterization of the extreme points of its closed unit ball. Using these results, we obtain a description of the corresponding isometric isomorphisms on B ( H ) B(H) .

Two of the most useful inequality formulas for bounded linear operators on a Hilbert space are the Ldwner-Heinz and Reid's inequalities. The first inequality was generalized by Furuta (so called the Furuta inequality in the literature). We shall generalize the second one and obtain its related results. It is shown that these two generalized fundamental inequalities are all equivalent to one another. 1. NOTATIONS AND INTRODUCTION Throughout the paper we use capital letters to denote bounded linear operators acting on a Hilbert space H. T is positive (written T > 0 ) in case (Tx, x) > 0 for all x EH. If S and T are Hermitian, we write T > S in case T-S > O. T = U I T I is the polar decomposition of T with U the partial isometry, whereas I T I is the positive square root of the operator T*T, and U*U is the initial projection. Let I denote the identity operator. Under the polar decomposition of T we recall a well-known relation: I T* Iq= U I T Iq U* for q > 0 [1, p. 752]. The aim of this article is to define and characterize a generalized Reid inequality. Consequently, related inequalities and improved inequalities are given. Recall that if S > 0 and SK is Hermitian, then for x EH, I (SKx, x) I? 11 K 11 (Sx, x) is Reid's inequality [8], and the inequality was sharpened by Halmos [4, pp. 51, 244], where II K II is replaced by the spectral radius of K. Let a, ,3 E [0, 1] with a + ,3 > 1. For x,y EH let us call I (SK I SK Ic+,-1 x,y) I'll K IIc+/31 Sax SI y I an extended Reid's inequality. We shall call, for an obvious reason, the inequality (2) in Theorem 1 below a generalized Reid inequality, whereas (1) is the well-known altered Furuta inequality [2]. Our main result is characterizations of this inequality.

Let 0 < p, q, r E R be real numbers with p + 2r < (1 + 2r)q and 1 < q. Furuta (1987) proved that if bounded linear operators A, B E B(H) P+2r 1 on a Hilbert space H satisfy 0 < B < A, then B q < (BrAPBr) q. This inequality is called the Furuta inequality and has many applications. In this paper, we prove that the Furuta inequality holds in a unital hermitian Banach *-algebra with continuous involution. Let A, B be bounded linear operators on a Hilbert space H. The celebrated Lbwner-Heinz inequality states the following; Theorem A (L6wner-Heinz inequality [4], [5]). Let A, B E B(H) satisfy 0 < B < A. If 0 < p < 1, then BP < AP. For an extension of Theorem A, Furuta obtained the following interesting inequality in [1] and one page elementary proof in [2]. Theorem B (Furuta inequality [1], [2]). Let 0 < p, q, r G R and A, B E B(H) satisfy 0 < B < A. If p + 2r < (1 + 2r)q and 1 < q as shown in the figure, then B q < (BrAPBr) q.