NUMERICAL RADIUS NORM FOR BOUNDED MODULE MAPS AND SCHUR MULTIPLIERS

TAKASHI ITOH^{∗} AND MASARU NAGISA^{∗∗}

Abstract. Let T be a bounded linear map from an operator system V to the bounded operators 𝔹(H) on a Hilbert space H. Let w(·) be the numerical radius norm on V and 𝔹(H), and ∥T ∥_{w} = sup{w(T (x))|w(x) ≤ 1}. We define the norm ∥T ∥_{wcb} by sup_{n} ∥T ⊗ id_{n}∥_{w} for a completely bounded map T. It is shown that ∥T ∥_{wcb} =

inf\{∥S∥ | (\begin{matrix}S & T \\ T^{∗} & S \end{matrix}) : 𝕄_{2}(V) ⟶ 𝕄_{2}(𝔹(H)) completely positive \}.

Moreover we give the standard form of a completely bounded module map from the compact operators 𝕂(H) to 𝔹(H) with respect to the numerical radius norm. As an application, we generalize Ando-Okubo’s theorem for Schur multipliers on 𝔹(H).

1. Introduction

Let x be a bounded linear operator on a Hilbert space H. The numerical radius norm of x is given by

w(x) = sup{|(xξ|ξ)| | ξ ∈ H, ∥ξ∥ = 1}.

It is clear that

w(x) ≤ ∥x∥ ≤ 2w(x).

Let 𝔹(H) be the bounded operators on H and V an operator system on a Hilbert space K, that is, *-closed subspace in 𝔹(K) with the identity. If T is a bounded linear map from (V, ∥ · ∥) to (𝔹(H), ∥ · ∥), then T is also bounded with respect to the numerical radius norm from (V, w(·)) to (𝔹(H), w(·)). We define the norm ∥T ∥_{w} for T by

∥T ∥_{w} = sup{w(T (x)) | w(x) ≤ 1, x ∈ V }.

We denote by T ⊗ id_{n} the linear map

𝕄_{n}(𝔹(H)) ⊃ 𝕄_{n}(V) ∋ (x_{ij}) ⟼ (T (x_{ij})) ∈ 𝕄_{n}(𝔹(H)) ≅ 𝔹(H^{n}), where H^{n} means the direct sum \overbrace{H ⊕ ⋯ ⊕ H}^{n}. We introduce the norm ∥T ∥_{wcb} for a completely bounded map T from V to 𝔹(H) by

∥T ∥_{wcb} = sup_{n} ∥T ⊗ id_{n}∥_{w}.

In 1985 [8], Paulsen and Suen introduced the norm |||T ||| for a completely bounded map T from a C^{∗}-algebra A to 𝔹(H) as follows:

|||T ||| = inf\{∥S∥ | (\begin{matrix}S & T \\ T^{∗} & S \end{matrix}) : 𝕄_{2}(A) ⟶ 𝕄_{2}(𝔹(H))

completely positive }.

They proved that if |||T ||| < ∞, then there exist a Hilbert space K, a ∗-representation π : A ⟶ 𝔹(K), an isometry V : H ⟶ K and an operator S ∈ π(A)′ such that |||T ||| = ∥S∥ and T (x) = V^{∗}Sπ(x)V for x ∈ A.

In the present paper, we first show that ∥T ∥_{wcb} = |||T |||. Moreover we show that if T is a completely bounded A-bimodule map from an A-bimodule operator system V to 𝔹(H), then

∥T ∥_{wcb} = inf\{∥S∥ | (\begin{matrix}S & T \\ T^{∗} & S \end{matrix}) : N ⟶ 𝕄_{2}(𝔹(K))

A-bimodule completely positive }.

Next let A be a norm closed unital algebra in 𝔹(H) and set A^{(∗)} = {x^{∗} ∈ 𝔹(H) | x ∈ A}. We show that if T is a completely bounded left A^{(∗)}-, right A-module map from the compact operators 𝕂(H) to 𝔹(H), then there exist {v_{i} | i ∈ I} ⊂ A′, t = (t_{ij}) ∈ 𝔹(ℓ^{2}(I)) such that

∥t∥ = ∥T ∥_{wcb}, ∑_{i∈I} v_{i}^{∗}v_{i} ≤ 1 and T (x) = ∑_{i,j∈I} v_{i}^{∗}t_{ij}xv_{j} for x ∈ 𝕂(H).

This can be seen as a generalization of the Smith’s theorem for completely bounded module maps [Theorem 3.1 in [10], Corollary 10]. As a consequence, we extend the Ando-Okubo’s theorem in [2] for Schur multipliers on 𝔹(H).

2. Numerical radius norm for a bounded linear operator

Let f be a bounded linear functional on 𝔹(H). We denote the operator norm w^{∗}(f) of f by

w^{∗}(f) = sup{|f(x)| | x ∈ 𝔹(H), w(x) ≤ 1}.

It is easily shown that

∥f∥ ≤ w^{∗}(f) ≤ 2∥f∥.

For a σ-weakly continuous linear functional f ∈ 𝔹(H)_{∗} on 𝔹(H), there exists a trace class operator a_{f} such that

f(x) = Tr(a_{f}x) for all x ∈ 𝔹(H).

We identify 𝔹(H)_{∗} with the set of all trace-class operators. We denote by U(H) (resp. P_{1}(H)) the unitary operators on H (resp. the projections of rank 1 on H). Let C be the norm closed convex hull of the set

{e^{iθ}u^{∗}pu | θ ∈ ℝ, u ∈ U(H), p ∈ P_{1}(H)}

in (𝔹(H)_{∗}, ∥ · ∥).

Lemma 1. For any f ∈ 𝔹(H)_{∗}, the followings are equivalent: (1) w^{∗}(f) ≤ 1.

(2) a_{f} ∈ C.

Proof. (1)⇒(2) We assume that a_{f} ∉ C. Then we can choose an element x in 𝔹(H) such that

Re⟨a_{f}, x⟩ > sup{Re⟨c, x⟩ | c ∈ C}.

This means that

|f(x)| ≥ Re(Tr(a_{f}x))
> sup{|Tr(u^{∗}pux)| | u ∈ U(H), p ∈ P_{1}(H)}
= sup{|(xη|η)| | η ∈ H, ∥η∥ = 1}
= w(x).

So we have w^{∗}(f) > 1.

(2)⇒(1) We assume that w(x) ≤ 1. For any positive ε, we can choose projections {p_{i} | 1 ≤ i ≤ n} ⊂ P_{1}(H) and complex numbers {λ_{i} | 1 ≤ i ≤ n} with ∑_{i=1}^{n} |λ_{i}| = 1 such that

|f(x)| = |Tr(a_{f}x)| ≤ | ∑_{i=1}^{n} λ_{i}Tr(p_{i}x)| + ϵ.

Then we have

|f(x)| ≤ ∑_{i=1}^{n} |λ_{i}||Tr(p_{i}x)| + ε ≤ w(x) + ε ≤ 1 + ε.

Therefore we have w^{∗}(f) ≤ 1. □

Let X be a normed space and T a bounded linear operator from X to 𝔹(H). Since the numerical radius norm w(·) on 𝔹(H) is equivalent to the usual operator norm on 𝔹(H), we can regard T as a bounded linear operator from X to (𝔹(H), w(·)) and denote its norm by ∥T ∥_{w}. Also we may consider its transposed operator ^{t}T from (𝔹(H), w(·))^{∗} to X^{∗} and denote its norm by ∥^{t}T ∥_{w^{∗}}. Namely,

∥T ∥_{w} = sup{w(T (x)) | x ∈ X, ∥x∥ ≤ 1},
∥^{t}T ∥_{w^{∗}} = sup{∥^{t}T (f)∥ | f ∈ 𝔹(H)^{∗}, w^{∗}(f) ≤ 1} = ∥T ∥_{w}.

We note that

∥T ∥_{w} = ∥^{t}T |_{𝔹(H)_{∗}}∥_{w^{∗}}.

For a trace class operator a ∈ 𝔹(H), we define f_{a} ∈ 𝔹(H)_{∗} as follows:

f_{a}(x) = Tr(ax) (x ∈ 𝔹(H)).

Proposition 2. Let X be a normed space and T a bounded linear map from X to 𝔹(H). Then we have that ∥T ∥_{w} ≤ 1 if and only if ∥^{t}T (f_{p})∥ ≤ 1 for any p ∈ P_{1}(H).

Proof. We assume that ∥T ∥_{w} ≤ 1. Then we have

∥^{t}T (f_{p})∥ ≤ ∥^{t}T |_{𝔹(H)_{∗}}∥_{w^{∗}}w^{∗}(f_{p})
= ∥T ∥_{w} sup{|Tr(px)| | w(x) ≤ 1} ≤ 1.

The reverse implication is as follows: for a = ∑_{i=1}^{n} λ_{i}p_{i} (∑_{i=1}^{n} |λ_{i}| = 1, p_{i} ∈ P_{1}(H)),

∥^{t}T (f_{a})∥ = ∥ ∑_{i=1}^{n} λ_{i}^{t}T (f_{a})∥ = ∑_{i=1}^{n} |λ_{i}|∥^{t}T (f_{a})∥ ≤ 1.

By Lemma 1, this implies ∥^{t}T |_{𝔹(H)_{∗}}∥_{w^{∗}} ≤ 1. □

Let V be an operator system in 𝔹(H), i.e., V is a self-adjoint subspace of 𝔹(H) and contains the unit. For a unital C*-algebra A, we call V A-bimodule if there exists a unital *-representation π of A on H such that

a · x · b = π(a)xπ(b) ∈ V,

where a, b ∈ A and x ∈ V. Let T be a linear map from V to 𝔹(K). For an A-bimodule operator system V, we call T A-bimodule map if there exists a unital *-representation ϕ of A on K such that

T (a · x · b) = a · T (x) · b = ϕ(a)T (x)ϕ(b),

where a, b ∈ A and x ∈ V. We also call that an action of A on K is locally cyclic if, for any positive integer n and ξ_{1}, ξ_{2}, …, ξ_{n} ∈ K, there exists a vector η ∈ K such that

ξ_{i} ∈ the norm closure of {ϕ(a)η | a ∈ A} (i = 1, ⋯, n).

The following statement is essentially due to R. R. Smith.

Lemma 3. If the action of A on H is locally cyclic, then for any ε > 0 and ξ_{1}, ξ_{2}, …, ξ_{n} ∈ K with ∑_{i=1}^{n} ∥ξ_{i}∥^{2} = 1, there exist η ∈ K and a_{1}, a_{2}, …, a_{n} ∈ A such that

∥η∥ = 1, max_{1≤i≤n} ∥ϕ(a_{i})η − ξ_{i}∥ < ε, ∑_{i=1}^{n} a_{i}^{∗}a_{i} ≤ 1.

Proof. We may assume that ε is a sufficiently small positive number. We can choose a vector ξ ∈ K and b_{1}, …, b_{n} ∈ A such that

∥ϕ(b_{i})ξ − ξ_{i}∥ < ε (1 ≤ i ≤ n).

We put

a_{i} = b_{i}(∑_{i=1}^{n} b_{i}^{∗}b_{i} + δ)^{−1/2}, η = (∑_{i=1}^{n} ϕ(b_{i})^{∗}ϕ(b_{i}) + δ)^{1/2}ξ,

where δ is a sufficiently small positive number. Then we have

∥ϕ(a_{i})η − ξ_{i}∥ < ε, ∑_{i=1}^{n} a_{i}^{∗}a_{i} ≤ 1

and

∥η∥^{2} = ((∑_{i=1}^{n} ϕ(b_{i})^{∗}ϕ(b_{i})+δ)ξ|ξ) < ∑_{i=1}^{n} ∥ξ_{i}∥^{2}+2nε+nε^{2} = 1+2nε+nε^{2}.

Replacing η with (1 + 2nε + nε^{2})^{−1/2}η and ε with a suitable positive number, we can get the desired result. □

In the rest of this paper, we consider the operator norm ∥T ∥_{w} of T viewed as a bounded linear map from (V, w(·)) to (𝔹(K), w(·)), i.e.,

∥T ∥_{w} = sup{w(T (x)) | w(x) ≤ 1, x ∈ V }.

We remark that, for x = (x_{ij}) ∈ 𝕄_{n}(V) ⊂ 𝔹(H^{n}) and a = (a_{kl}) ∈ M_{nm}(A), we can see

a^{∗} · x · a = (∑_{k,l} a_{ki}^{∗} · x_{kl} · a_{lj}) ∈ 𝕄_{m}(V) ⊂ 𝔹(H^{m})

and we get

w(a^{∗} · x · a) ≤ ∥a∥^{2}w(x).

Proposition 4. Let A be a unital C*-algebra, V an A-bimodule operator system and T an A-bimodule bounded linear map from V to 𝔹(H). If the action of A on K is locally cyclic, then we have

∥T ∥_{w} = ∥T ⊗ id_{n}∥_{w}

for any n ∈ ℕ.

Proof. We assume that ∥T ∥_{w} = 1. Then we show that ∥T ⊗ id_{n}∥_{w} ≤ 1.

Let ω_{ξ,ξ} be a vector state on 𝔹(H) where ξ = (ξ_{1} ; ξ_{2} ; ⋮ ; ξ_{n}) and ∑_{i=1}^{n} ∥ξ_{i}∥^{2} = 1.

By proposition 2, it suffices to show that

w^{∗}(^{t}(T ⊗ id_{n})(ω_{ξ,ξ})) ≤ 1.

By the local cyclicity of the action, for any positive ε, we can choose η ∈ H and a_{1}, a_{2}, …, a_{n} ∈ A such that

∥η∥ = 1, max_{1≤i≤n} ∥ϕ(a_{i})η − ξ_{i}∥ < ε, ∑_{i=1}^{n} a_{i}^{∗}a_{i} ≤ 1.

Then we have, for X = (x_{ij}) ∈ 𝕄_{n}(V) with w(X) ≤ 1,

|⟨X,^{t}(T ⊗ id_{n})(ω_{ξ,ξ})⟩| = |⟨(T ⊗ id_{n})(X), ω_{ξ,ξ}⟩|

= | ∑_{i,j=1}^{n}(T (x_{ij})ξ_{j}|ξ_{i})|

≤ | ∑_{i,j=1}^{n}(T (x_{ij})ϕ(a_{j})η|ϕ(a_{i})η)|

+ |(T ⊗ id_{n}(X) (ξ_{1} − ϕ(a_{1})η ; ⋮ ; ξ_{n} − ϕ(a_{n})η) | (ϕ(a_{1})η ; ⋮ ; ϕ(a_{n})η))|

+ |(T ⊗ id_{n}(X) (ξ_{1} ; ⋮ ; ξ_{n}) | (ξ_{1} − ϕ(a_{1})η ; ⋮ ; ξ_{n} − ϕ(a_{n})η))|

≤ | ∑_{i,j=1}^{n}(T (x_{ij})ϕ(a_{j})η|ϕ(a_{i})η)| + 16n\sqrt{n}ε

= | ∑_{i,j=1}^{n}(ϕ(a_{i}^{∗})T (x_{ij})ϕ(a_{j})η|η)| + 16n\sqrt{n}ε

= |⟨T ((a_{1}^{∗} a_{2}^{∗} ⋯ a_{n}^{∗}) · X · (a_{1} ; a_{2} ; ⋮ ; a_{n})), ω_{η,η}⟩| + 16n\sqrt{n}ε

≤ 1 + 16n\sqrt{n}ε,

where we use the following inequalities:

∥(a_{ij})_{i,j=1}^{n}∥ ≤ n max_{i,j} ∥a_{i,j}∥ (a_{ij} ∈ 𝔹(K)), ∥ (ϕ(a_{1})η ; ⋮ ; ϕ(a_{n})η) ∥ ≤ 1

and

∥T (x_{ij})∥ ≤ 2w(T (x_{ij})) ≤ 2w(x_{ij}) ≤ 4∥x_{ij}∥ ≤ 4∥X∥ ≤ 8. Hence we have ∥^{t}(T ⊗ id_{n})(ω_{ξ,ξ})∥ ≤ 1. □

Remark 5. By the similar argument as in the above proof, we can also get the following identity:

sup{w(T (x)) | ∥x∥ ≤ 1, x ∈ V }
= sup{w(T ⊗ id_{n}(X)) | ∥X∥ ≤ 1, X ∈ 𝕄_{n}(V).}

Every completely bounded map from an operator space to 𝔹(H) is also completely bounded with respect to numerical radius norm. We use the following notation:

∥T ∥_{wcb} = sup_{n∈ℕ}∥T ⊗ id_{n}∥_{w}.

3. Main Theorem

Let A be a unital C*-algebra and V an A-bimodule operator system in 𝔹(H). Then the operator system

N = \{(\begin{matrix}x & y \\ z & w \end{matrix}) | x, w ∈ 𝔹(H), y, z ∈ V \}

is also A-bimodule by the action

a · (\begin{matrix}x & y \\ z & w \end{matrix}) · b = (\begin{matrix}π(a)xπ(b) & π(a)yπ(b) \\ π(a)zπ(b) & π(a)wπ(b) \end{matrix}) for a, b ∈ A.

In the rest of this paper, for the simplicity of notations, we write axb instead of π(a)xπ(b).

We remark that the characterization of operators with numerical radius one ([1],[4]). For x ∈ 𝔹(H), we have w(x) ≤ 1 if and only if there exists z ∈ 𝔹(H) such that

(\begin{matrix}1 + z & x \\ x^{∗} & 1 − z \end{matrix}) ≥ 0.

Theorem 6. Let A be a unital C*-algebra, V an A-bimodule operator system in 𝔹(H) and T an A-bimodule completely bounded map from V to 𝔹(K). Then we have

∥T ∥_{wcb}
= min\{∥S∥ | (\begin{matrix}S & T \\ T^{∗} & S \end{matrix}) : N ⟶ 𝕄_{2}(𝔹(K))

A-bimodule completely positive }

= min\{∥S∥ | (\begin{matrix}S & T \\ T^{∗} & S \end{matrix}) : N ⟶ 𝕄_{2}(𝔹(K)) completely positive \},

where

(\begin{matrix}S & T \\ T^{∗} & S \end{matrix}) (\begin{matrix}x & y \\ z & w \end{matrix}) = (\begin{matrix}S(x) & T (y) \\ T (z^{∗})^{∗} & S(w) \end{matrix}) , for (\begin{matrix}x & y \\ z & w \end{matrix}) ∈ N.

Proof. We assume that ∥T ∥_{wcb} = 1. We define the A-bimodule operator system M and A-bimodule linear map φ_{0} from M to 𝔹(K) as follows:

M = \{(\begin{matrix}a + x & y \\ z & a − x \end{matrix}) | a ∈ A, x ∈ 𝔹(H), y, z ∈ V \},

φ_{0}((\begin{matrix}a + x & y \\ z & a − x \end{matrix})) = a + \frac{1}{2}(T (y) + T (z^{∗})^{∗})

It is clear that φ_{0} is unital. We show that φ_{0} is completely positive. For n ∈ ℕ, let

(\begin{matrix}b + x & y \\ y^{∗} & b − x \end{matrix}) ≥ 0,

where b ∈ 𝕄_{n}(A), x ∈ 𝕄_{n}(𝔹(H)) and y ∈ 𝕄_{n}(V). Replacing b by b + ε1 (ε > 0), we may assume that b is positive invertible. Then we have

(\begin{matrix}1_{n} + b^{−1/2}xb^{−1/2} & b^{−1/2}yb^{−1/2} \\ b^{−1/2}y^{∗}b^{−1/2} & 1_{n} − b^{−1/2}xb^{−1/2} \end{matrix}) ≥ 0

⇒ w(b^{−1/2}yb^{−1/2}) ≤ 1 ⇒ w(b^{−1/2}(T ⊗ id_{n}(y))b^{−1/2}) ≤ 1
⇒ (\begin{matrix}1_{n} + z & b^{−1/2}(T ⊗ id_{n}(y))b^{−1/2} \\ b^{−1/2}(T ⊗ id_{n}(y))^{∗}b^{−1/2} & 1_{n} − z \end{matrix}) ≥ 0

for some z ∈ 𝕄_{n}(𝔹(H))

⇒ (\begin{matrix}b + b^{1/2}zb^{1/2} & T ⊗ id_{n}(y) \\ (T ⊗ id_{n}(y))^{∗} & b − b^{1/2}zb^{1/2} \end{matrix}) ≥ 0
⇒ b + \frac{1}{2}(T ⊗ id_{n}(y) + (T ⊗ id_{n}(y))^{∗}) ≥ 0,

that is,

φ_{0} ⊗ id_{n}((\begin{matrix}b + x & y \\ y^{∗} & b − x \end{matrix})) = b + \frac{1}{2}(T ⊗ id_{n}(y) + (T ⊗ id_{n}(y))^{∗}) ≥ 0.

Therefore φ_{0} is a completely positive A-bimodule map.

Using Wittstock’s Hahn-Banach theorem ([12]), we get the unital completely positive A-bimodule linear map φ from N to 𝔹(K) which is an extension of φ_{0}. By definition, we have

φ((\begin{matrix}x & 0 \\ 0 & 0 \end{matrix})) = φ((\begin{matrix}0 & 0 \\ 0 & x \end{matrix})) for x ∈ 𝔹(H).

We define a unital completely positive A-bimodule linear map S from 𝔹(H) to 𝔹(K) as follows:

S(x) = 2φ((\begin{matrix}x & 0 \\ 0 & 0 \end{matrix})) = 2φ((\begin{matrix}0 & 0 \\ 0 & x \end{matrix})).

Now we show that

(\begin{matrix}S & T \\ T^{∗} & S \end{matrix}) : N ⟶ 𝕄_{2}(𝔹(K))

is also completely positive.

Let (\begin{matrix}x & y \\ y^{∗} & z \end{matrix}) ≥ 0, where x, z ∈ 𝕄_{n}(𝔹(H)) and y ∈ 𝕄_{n}(V). Then we

have

(\begin{matrix}x & 0 & 0 & y \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ y^{∗} & 0 & 0 & z \end{matrix}) ≥ 0

and

0 ≤φ ⊗ id_{2n}((\begin{matrix}x & 0 & 0 & y \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ y^{∗} & 0 & 0 & z \end{matrix}))

= (\begin{matrix}φ ⊗ id_{n}((\begin{matrix}x & 0 \\ 0 & 0 \end{matrix})) & φ ⊗ id_{n}((\begin{matrix}0 & y \\ 0 & 0 \end{matrix})) \\ φ ⊗ id_{n}((\begin{matrix}0 & 0 \\ y^{∗} & 0 \end{matrix})) & φ ⊗ id_{n}((\begin{matrix}0 & 0 \\ 0 & z \end{matrix})) \end{matrix})

=\frac{1}{2} (\begin{matrix}S ⊗ id_{n}(x) & T ⊗ id_{n}(y) \\ T ⊗ id_{n}(y)^{∗} & S ⊗ id_{n}(x) \end{matrix})
=\frac{1}{2}((\begin{matrix}S & T \\ T^{∗} & S \end{matrix}) ⊗ id_{n})((\begin{matrix}x & y \\ y^{∗} & z \end{matrix})).

Hence we have (\begin{matrix}S & T \\ T^{∗} & S \end{matrix}) is completely positive and ∥S∥ = 1.

Therefore we obtain the following inequality:

∥T ∥_{wcb}

≥ inf\{∥S∥ | (\begin{matrix}S & T \\ T^{∗} & S \end{matrix}) : N ⟶ 𝕄_{2}(𝔹(K))

A-bimodule completely positive }

≥ inf\{∥S∥ | (\begin{matrix}S & T \\ T^{∗} & S \end{matrix}) : N ⟶ 𝕄_{2}(𝔹(K)) completely positive \}

To show the rest inequality, we assume that ∥S∥ = 1 and

(\begin{matrix}S & T \\ T^{∗} & S \end{matrix}) : N ⟶ 𝕄_{2}(𝔹(K))

is completely positive. For y ∈ 𝕄_{n}(V) with w(y) ≤ 1, we have

(\begin{matrix}1_{n} + z & y \\ y^{∗} & 1_{n} − z \end{matrix}) ≥ 0 for some z ∈ 𝔹(H)

⇒ (\begin{matrix}1_{n} + S ⊗ id_{n}(x) & T ⊗ id_{n}(y) \\ T ⊗ id_{n}(y)^{∗} & 1_{n} − S ⊗ id_{n}(z) \end{matrix}) ≥ 0
⇒ w(T ⊗ id_{n}(y)) ≤ 1.

Then we have ∥T ∥_{wcb} ≤ 1. This implies the desired result.

□

Remark 7. In [8], V. I. Paulsen and C. Y. Suen introduced the norm |||T ||| for a completely bounded map T from a C*-algebra A to 𝔹(K) as follows:

|||T ||| = inf\{∥S∥ | (\begin{matrix}S & T \\ T^{∗} & S \end{matrix}) : 𝕄_{2}(A) ⟶ 𝕄_{2}(𝔹(K))

completely positive }.

By the injectivity of 𝔹(K), we can get

|||T ||| = ∥T ∥_{wcb}.

For a Hilbert space H, we denote the direct sum ⊕_{I}H(≅ H ⊗ ℓ^{2}(I)) of copies of H by H^{I}, and for x ∈ 𝔹(H) we denote the direct sum of copies of x by x^{I} (x⊗id_{ℓ^{2}(I)}). An operator t ∈ 𝔹(H, H^{I}) may be viewed as a column matrix of operators t_{i} ∈ 𝔹(H) where ∑_{i∈I} t_{i}^{∗}t_{i} ∈ 𝔹(H). In the same way, an operator s ∈ 𝔹(H^{I}, H) may be viewed as a row matrix of operators s_{i} ∈ 𝔹(H) where ∑_{i∈I} s_{i}s_{i}^{∗} ∈ 𝔹(H).

For t = (t_{i})_{i∈I} ∈ 𝔹(H, H^{I}) and α = (α_{i})_{i∈I} ∈ ℓ^{2}(I) , we define α · t by

∑_{i∈I} α_{i}t_{i} ∈ 𝔹(H),

which converges in the norm topology. A set of operators t = {t_{i}}, t_{i} ∈ 𝔹(H) with ∑_{i} t_{i}^{∗}t_{i} ∈ 𝔹(H), is said to be strongly independent over a norm closed subspace X of 𝔹(H) if α = 0 whenever α ∈ ℓ^{2} and α · t ∈ X. In the case X = {0}, we will say that the set {t_{i}} is strongly independent.

Lemma 8. [10] Let A be a norm closed unital algebra on H and 𝕂(H) compact operators on H. If s ∈ 𝔹(H^{I}, H) and t ∈ 𝔹(H, H^{I}) satisfy the condition that t = {t_{i}} is strongly independent over the commutant A′ of A and

∑_{i∈I} s_{i}xt_{i}a = ∑_{i∈I} s_{i}xat_{i} for any x ∈ 𝕂(H), a ∈ A,

then we have s_{i} = 0 for all i ∈ I.

We set A^{(∗)} = {x^{∗} ∈ 𝔹(H) | x ∈ A}.

Theorem 9. Let A be a norm closed unital algebra on H and T a completely bounded left A^{(∗)}-, right A-module map from 𝕂(H) to 𝔹(H). Then there exist t = (t_{ij}) ∈ 𝔹(ℓ^{2}(I)) and {v_{i} | i ∈ I} ⊂ A′ such that

∥t∥ = ∥T ∥_{wcb}, ∑_{i∈I} v_{i}^{∗}v_{i} ≤ 1

T (x) = ∑_{i,j∈I} v_{i}^{∗}t_{ij}xv_{j} (x ∈ 𝕂(H)).

Proof. We may regard T as a normal completely bounded A^{(∗)}-A-module map from 𝔹(H) to 𝔹(H). By Theorem 2.10 in [8] and Therorem 6, there exist a *-representation π of 𝕂(H) on a Hilbert space K, an isometry w : H ⟶ K and an operator s ∈ π(𝕂(H))′ such that

∥T ∥_{wcb} = ∥s∥, T (·) = w^{∗}sπ(·)w.

Since all irreducible representations of 𝕂(H) are unitarily equivalent to the identity representation, we may assume that

K = H ⊗ ℓ^{2}(I) = H^{I}, π(x) = x^{I}, s = (s_{ij}1_{𝔹(H)})_{i,j∈I} (s_{ij} ∈ ℂ),

w = (w_{i})_{i∈I} ∈ 𝔹(H, H^{I}),

T (x) = w^{∗}sx^{I}w = ∑_{i,j∈I} w_{i}^{∗}s_{i,j}xw_{j} for x ∈ 𝕂(H).

We decompose ℓ^{2}(I) into the following three subspaces:

H_{1} = {α ∈ ℓ^{2}(I) | α · w = 0}
H_{2} = {α ∈ H_{1}^{⊥} | α · w ∈ A′}
H_{3} = (H_{1} ⊕ H_{2})^{⊥}.

Then we have ℓ^{2}(I) = H_{1} ⊕ H_{2} ⊕ H_{3} and choose an orthonormal base {α^{i} | i ∈ I_{j}} of H_{j} (j = 1, 2, 3). We may consider I = I_{1} ∪ I_{2} ∪ I_{3} (the disjoint union). We define the unitary operator u = (u_{i,j})_{i,j∈I} in 𝔹(H^{I}) ≅ 𝕄_{I,I}(𝔹(H)) as follows:

u_{i,j} = α^{i}(j)1_{𝔹(H)}.

We put

t = (t_{ij}) = usu^{∗}, v_{i} = α^{i} · w ∈ 𝔹(H).

Then we have ∥t∥ = ∥s∥, v = (v_{i}) ∈ 𝔹(H, H^{I}) and

T (x) = w^{∗}sx^{I}w = w^{∗}sx^{I}u^{∗}uw = w^{∗}u^{∗}(usu^{∗})x^{I}uw

= v^{∗}tx^{I}v = ∑_{j∈I}(∑_{i∈I} v_{i}^{∗}t_{ij})xv_{j}.

By definition we can see that

v_{i} = 0 for i ∈ I_{1},
v_{i} ∈ A′ for i ∈ I_{2},
{v_{i} | i ∈ I_{2}} is strongly independent

and {v_{i} | i ∈ I_{3}} is strongly independent over A′ (see [10]). Then we have

T (x) = ∑_{j∈I}(∑_{i∈I} v_{i}^{∗}t_{ij})xv_{j} = ∑_{j∈I_{2}∪I_{3}}( ∑_{i∈I_{2}∪I_{3}} v_{i}^{∗}t_{ij})xv_{j}.

Since T has the right A-module property and {v_{i} | i ∈ I_{2}} ⊂ A′, it holds

∑_{j∈I_{3}}( ∑_{i∈I_{2}∪I_{3}} v_{i}^{∗}t_{ij})xv_{j}a = ∑_{j∈I_{3}}( ∑_{i∈I_{2}∪I_{3}} v_{i}^{∗}t_{ij})xav_{j}

for any x ∈ 𝕂(H), a ∈ A. By Lemma 8, we have ∑_{i∈I_{2}∪I_{3}} v_{i}^{∗}t_{ij} = 0 for j ∈ I_{3}, that is,

∑_{i∈I_{3}} v_{i}^{∗}t_{ij} = − ∑_{i∈I_{2}} v_{i}^{∗}t_{ij} ∈ A′ (j ∈ I_{3}).

The strongly independece over A′ of {v_{i} | i ∈ I_{3}} implies t_{ij} = 0 for i, j ∈ I_{3} and the strongly independece of {v_{i} | i ∈ I_{2}} implies t_{ij} = 0 for i ∈ I_{2}, j ∈ I_{3}. Thus we have that t_{ij} = 0 for i ∈ I_{2} ∪ I_{3}, j ∈ I_{3} and

T x = ∑_{j∈I_{2}}( ∑_{i∈I_{2}∪I_{3}} v_{i}^{∗}t_{ij})xv_{j} = ∑_{i∈I_{2}∪I_{3}} v_{i}^{∗}x(∑_{j∈I_{2}} t_{ij}v_{j}).

By the left A^{(∗)}-modularity of T, we can get the relation

∑_{i∈I_{3}} v_{i}^{∗}a^{∗}x(∑_{j∈I_{2}} t_{ij}v_{j}) = a^{∗} ∑_{i∈I_{3}} v_{i}^{∗}x(∑_{j∈I_{2}} t_{ij}v_{j}),

that is,

∑_{i∈I_{3}}(∑_{j∈I_{2}} v_{j}^{∗}\overline{t_{ij}})xav_{i} = ∑_{i∈I_{3}}(∑_{j∈I_{2}} v_{j}^{∗}\overline{t_{ij}})xv_{i}a,

for any x ∈ 𝕂(H), a ∈ A. By the same argument as above we can get t_{ij} = 0 for i ∈ I_{3}, j ∈ I_{2} and

T x = ∑_{j∈I_{2}}(∑_{i∈I_{2}} v_{i}^{∗}t_{ij})xv_{j}.

If we replace v = (v_{i}) by v_{i} = 0 for i ∈ I_{1} ∪ I_{3}, we can get the desired result. □

The following is due to Smith in [10]. We show it from the view point of Theorem 9.

Corollary 10. Let A and B be norm closed unital algebras on H and T a completely bounded left A- right B-module map from 𝕂(H) to 𝔹(H). Then there exist {a_{i} | i ∈ I} ⊂ A′ and {b_{i} | i ∈ I} ⊂ B′ such that

T (x) = ∑_{i∈I} a_{i}xb_{i}, ∥ ∑_{i∈I} a_{i}a_{i}^{∗}∥∥ ∑_{i∈I} b_{i}^{∗}b_{i}∥ = ∥T ∥_{cb}^{2}

for x ∈ 𝕂(H).

Proof. We define the left action of A ⊕ B^{(∗)} and the right action of A^{(∗)} ⊕ B on 𝕄_{2}(𝕂(H)) and the left A ⊕ B^{(∗)}-, right A^{(∗)} ⊕ B-module completely bounded map T̃ from 𝕄_{2}(𝕂(H)) to 𝕄_{2}(𝔹(H)) as follows:

(a_{1} ⊕ b_{1}^{∗}) (\begin{matrix}x & y \\ z & w \end{matrix}) (a_{2}^{∗} ⊕ b_{2}) = (\begin{matrix}a_{1}xa_{2}^{∗} & a_{1}yb_{2} \\ b_{1}^{∗}za_{2}^{∗} & b_{1}^{∗}wb_{2} \end{matrix})

T̃((\begin{matrix}x & y \\ z & w \end{matrix})) = (\begin{matrix}0 & T (y) \\ 0 & 0 \end{matrix})

where x, y, z, w ∈ 𝕂(H) and a_{1}, a_{2} ∈ A, b_{1}, b_{2} ∈ B. Then we show that ∥T̃∥_{wcb} = ∥T ∥_{cb}.

Let x ∈ 𝔹(H) and its polar decompositin x = v|x|. Then we have

w((\begin{matrix}0 & x \\ 0 & 0 \end{matrix})) = w((\begin{matrix}v & 0 \\ 0 & 1 \end{matrix}) (\begin{matrix}0 & |x| \\ 0 & 0 \end{matrix}) (\begin{matrix}v^{∗} & 0 \\ 0 & 1 \end{matrix}))

=w((\begin{matrix}0 & |x| \\ 0 & 0 \end{matrix})) = sup\{w((\begin{matrix}0 & λ \\ 0 & 0 \end{matrix})) | λ ∈ σ(|x|)\} = \frac{1}{2}∥x∥.

By the relation

1 ≥ w((\begin{matrix}x & y \\ z & w \end{matrix})) ≥ \frac{1}{2}∥ (\begin{matrix}x & y \\ z & w \end{matrix}) ∥ ≥ \frac{1}{2}∥y∥ = w((\begin{matrix}0 & y \\ 0 & 0 \end{matrix})),

we can get the following:

∥T̃ ⊗ id_{n}∥_{w}
= sup\{w(T̃ ⊗ id_{n}((\begin{matrix}X & Y \\ Z & W \end{matrix} ))) | X, Y, Z, W ∈ 𝕄_{2}(𝕂(H)),

w((\begin{matrix}X & Y \\ Z & W \end{matrix} )) ≤ 1\}

= sup\{w((\begin{matrix}0 & T ⊗ id_{n}(Y) \\ 0 & 0 \end{matrix})) | Y ∈ 𝕄_{2}(𝕂(H)), w((\begin{matrix}0 & Y \\ 0 & 0 \end{matrix} )) ≤ 1\}
= sup\{\frac{1}{2}∥T ⊗ id_{n}(Y)∥ | Y ∈ 𝕄_{2}(𝕂(H)), ∥Y ∥ ≤ 2\} = ∥T ⊗ id_{n}∥.

Applying the preceding theorem for T̃, there exist t = (t_{ij}), {a_{i} | i ∈ I} ⊂ A′ and {b_{i} | i ∈ I} ⊂ B′ such that

∥t∥ = ∥T̃∥_{wcb}, ∑_{i∈I}(a_{i}^{∗} ⊕ b_{i})^{∗}(a_{i}^{∗} ⊕ b_{i}) ≤ 1

T̃((\begin{matrix}x & y \\ z & w \end{matrix})) = ∑_{i,j∈I}(a_{i}^{∗} ⊕ b_{i})^{∗}t_{ij} (\begin{matrix}x & y \\ z & w \end{matrix}) (a_{j}^{∗} ⊕ b_{j}).

This means

∥t∥ = ∥T ∥_{cb}, T (y) = ∑_{i,j∈I} a_{i}t_{ij}yb_{j}.

Set ã_{j} = ∑_{i∈I} a_{i}t_{ij} ∈ A′, then we have

T (y) = ∑_{i∈I} ã_{i}yb_{i},

∥T ∥_{cb}^{2} ≤ ∥ ∑_{i∈I} ã_{i}ã_{i}^{∗}∥∥ ∑_{i∈I} b_{i}^{∗}b_{i}∥ ≤ ∥T ∥_{cb}^{2}.

□

4. Application

For a Hilbert space H, we choose a completely orthonormal system {e_{i} | i ∈ I}. We denote by ℓ^{∞} the maximal abelian subalgebra of 𝔹(H) generated by {e_{i} ⊗ e_{i} | i ∈ I}, where (e_{i} ⊗ e_{j})(ξ) = (ξ|e_{j})e_{i} for ξ ∈ H. Let T be a bounded ℓ^{∞}-bimodule map from 𝕂(H) to 𝔹(H). By the module property of T, we have the I × I-matrix a = (a_{ij}) over ℂ such that

T (e_{i} ⊗ e_{j}) = a_{ij}(e_{i} ⊗ e_{j}).

Since the set {a_{ij}}_{i,j∈I} is bounded, we can define the bounded linear operator a_{T} from ℓ^{1} to ℓ^{∞} given by

a_{T} ((λ_{j})_{j∈I}) = (∑_{j∈I} a_{ij}λ_{j})_{i∈I} for (λ_{j})_{j∈I} ∈ ℓ^{1}.

We will extend the Ando-Okubo’s theorem in [2]. The following theorem is a variant of the characterization of Schur multipliers in [9].

Theorem 11. Let T be an ℓ^{∞}-bimodule bounded linear map from 𝕂(H) to 𝔹(H). Then the following conditions are equivalent:

(1) ∥T ∥_{w} ≤ 1.

(2) ∥T ∥_{wcb} ≤ 1.

(3) There exists a completely positive contraction S from 𝕂(H) to

𝔹(H) such that (\begin{matrix}S & T \\ T^{∗} & S \end{matrix}) : 𝕄_{2}(𝕂(H)) ⟶ 𝕄_{2}(𝔹(H)) is ℓ^{∞}-

bimodule completely positive.

(4) There exsit a bounded linear operator v from ℓ^{1} to ℓ^{2} and b ∈ 𝔹(ℓ^{2}) such that a_{T} = v^{∗}bv and ∥v∥^{2}∥b∥ ≤ 1, i.e.,

\begin{matrix}ℓ^{1} & ⟶^{a_{T}} & ℓ^{∞} \\ ^{v}↓ & & ↑^{v^{∗}} \\ ℓ^{2} & ⟶_{b} & ℓ^{2} \end{matrix} and inf ∥v∥∥b∥∥v^{∗}∥ ≤ 1.

(5) There exist {ξ_{i} | i ∈ I} ⊂ H and b ∈ 𝔹(H) such that

∥ξ_{i}∥ ≤ 1, ∥b∥ ≤ 1, a_{ij} = (bξ_{j}|ξ_{i}).

Proof. (1) ⇒ (2) Since the action of ℓ^{∞} on H is locally cyclic, we have ∥T ∥_{w} = ∥T ∥_{wcb} by Proposition 4 .

(2) ⇔ (3) It follows by Theorem 6.

(2) ⇒ (4) By Theorem 9, there exist b = (t_{ij}) ∈ 𝔹(ℓ^{2}(I)) and {v_{i}|i ∈ I} ⊂ ℓ^{∞} such that ∥b∥ = ∥T ∥_{wcb}, ∑_{i∈I} v_{i}^{∗}v_{i} ≤ 1 and T (x) = ∑_{i,j∈I} v_{i}^{∗}t_{ij}xv_{j}. Take the I × I-matrix v = (v_{ij}) over ℂ given by

v_{i}(e_{j}) = v_{ij}e_{j},

where {e_{i}}_{i∈I} is the canonical basis for ℓ^{2}(I) ≅ H. We also regard {e_{i}}_{i∈I} as the canonical basis for ℓ^{1}(I). Then we can get the following identification:

v = (v_{ij})_{i,j∈I} ∈ 𝔹(ℓ^{1}, ℓ^{2}) and v^{∗} = (\overline{v_{ji}})_{i,j∈I} ∈ 𝔹(ℓ^{2}, ℓ^{∞}).

Since ∑_{j} \overline{v_{ij}}v_{ij} ≤ 1 for all i ∈ I, v becomes a contraction. Then we have

a_{kl}(e_{k} ⊗ e_{l}) = T (e_{k} ⊗ e_{l})
= ∑_{i,j∈I} v_{i}^{∗}t_{ij}(e_{k} ⊗ e_{l})v_{j} = (∑_{i,j∈I} \overline{v_{ik}}t_{ij}v_{jl})e_{k} ⊗ e_{l},

that is, a_{T} = v^{∗}bv.

(4) ⇒ (5) We may assume that ∥v∥ = 1. We regard {e_{i}}_{i∈I} as the canonical basis for ℓ^{1} and also as its dual basis for ℓ^{∞}. We put ξ_{i} = ve_{i}. Then we get the desired condition.

(5) ⇒ (1) Let x ∈ 𝔹(H) have a matrix representation (x_{ij}) and w(x) ≤ 1. Then we have, for η = ∑_{i} α_{i}e_{i} ∈ H with ∥η∥ = 1,

|(T (x)η|η)| = |((bξ_{j}|ξ_{i})x_{ij})_{ij}η|η)|

= | ∑_{i,j}(bξ_{j}|ξ_{i})x_{ij}α_{j}\overline{α_{i}}|

= |((x_{ij})_{ij}b^{I}ζ|ζ)|,

where ζ = (α_{i}ξ_{i})_{i∈I} ∈ H^{I}. Since ∥ζ∥ ≤ 1 and (x_{ij}) and b^{I} doubly commute (cf.[3]), we have

|(T (x)η|η)| ≤ w((x_{ij})b^{I}) ≤ w(x)∥b^{I}∥ ≤ 1.

Hence we obtain ∥T ∥_{w} ≤ 1. □

References

[1] T. Ando, On the structure of operators with numerical radius one, Acta Sci. Math. (Szeged), 34(1973), 11–15.

[2] T. Ando and K. Okubo, Induced norms of the Schur multiplier operator, Linear Algebra Appl. 147(1991), 181–199.

[3] K. E. Gustafson and D. K. M. Rao, Numerical Range, Universitext, Springer-Verlag 1996.

[4] U. Haagerup, Injectivity and decomposition of completely bounded maps, Lecture Notes in Math., Springer-Verlag, 1132(1985), 170–222.

[5] U. Haagerup, Decomposition of completely bounded maps on operator algebras, unpublished manuscript.

[6] T. Itoh and M. Nagisa, Schur products and module maps on B(H), Publ. RIMS Kyoto Univ. 36, (2000), 253–268.

[7] V. I. Paulsen, Completely bounded maps and dilations, Pitman Res. Math. Ser. 146, Longman Sci. & Tech. 1986.

[8] V. I. Paulsen and C. Y. Suen, Commutant representations of completely bounded maps, J. Operator Theory 13(1985), 87–101.

[9] G. Pisier, Similarity problems and completely bounded maps, 2nd, expanded edit., Lecture Notes in Math. 1618, Springer-Verlag 2001.

[10] R. R. Smith, Completely bounded module maps and the Haagerup tensor product, J. Funct. Anal. 102, (1991), 156–175.

[11] M. Takesaki, Theory of operator algebras I, Encyclopaedia of Mathematical Sciences, 124. Operator Algebras and Non-commutative Geometry, 5. Springer-Verlag, 2002.

[12] G. Wittstock, Ein operatorwertiger Hahn-Banach Satz, J. Funct. Anal. 40, (1981), 127–150.

∗Department of Mathematics, Gunma University, Gunma 371-8510, Japan

E-mail address: itoh@edu.gunma-u.ac.jp

∗∗Department of Mathematics and Informatics, Chiba University, Chiba 263-8522, Japan

E-mail address: nagisa@math.s.chiba-u.ac.jp