Duality in quotient Banach spaces

C. Ambrozie* and V. Müller**

Let X be a Banach space. A linear manifold M ⊂ X is called paraclosed if there exists a Banach space X_{0} and a bounded operator a : X_{0} → X such that M = aX_{0}. Clearly we can assume that the operator a is injective. Indeed, let X′_{0} = X_{0}/ kera and let a′ : X′_{0} → X be the operator induced by a. Then a′ is injective and a′X′_{0} = aX_{0} = M. Equivalently, M ⊂ X is paraclosed if there is a Banach space topology on M which is stronger then the topology inherited from X (such a topology is necessarily unique by the closed graph theorem).

For basic properties of paraclosed subspaces (called sometimes also operator ranges) see e.g. [2], [3], [4], [6]. Recall that intersection or sum of two paraclosed subspaces is again paraclosed. Although paraclosed subspaces exhibit many properties of closed subspaces, the lattice of all paraclosed subspaces of X is much richer than the lattice of all closed subspaces. For example, each bounded operator S : X → X has a nontrivial invariant paraclosed subspace.

Let X be a Banach space and M ⊂ X a paraclosed subspace. Then the quotient X/M is no longer a topological vector space. However, the operators acting in X/M have still nice spectral properties. Since the results for operators in quotient Banach spaces have interesting consequences even in the spectral theory of bounded operators in Banach spaces, the quotients of Banach spaces (or of other more general types of spaces) attracted attention of several authors, see e.g. [8], [9], [10], [11], [13]. The spectrum of a morphism on a quotient Banach space X/aX_{0} is always nonempty provided that X ≠ aX_{0} [11]. The joint spectrum and the corresponding analytic functional calculus for n-tuples of commuting morphisms were also defined [11], [12].

An important tool in the theory of operators in Banach spaces is the duality. The present paper is an attempt to introduce the duality for operators in quotient Banach spaces. The attempt is successful in an important partial case — for quotient Banach spaces X/M where M is a dense paraclosed subspace of X. Note that in the other extremal case — if M is a closed subspace of X — the quotient X/M is again a Banach space so that the duality theory is well-known.

We show below that each operator acting in X/M induces canonically two operators acting in X/\overline{M} and \overline{M}/M, respectively. Thus these two extremal cases give information also about the general case.

The duality theory for operators in quotient Hilbert spaces was recently presented by Terescenco [5], [7].

All Banach spaces considered here are either real or complex. By an operator between two Banach spaces we mean always a bounded linear operator.

* This paper was written during a visiting scholarship offered to the first author by the Mathematical Institute of the Czech Academy of Sciences, Prague. ** The research was supporteted by grant No. A1019801 of AV ČR.

1991 Mathematics Subject Classification: 47S99, 47A05.

The basic definitions and results concerning quotient Banach spaces can be found in [10], [1]. By a quotient Banach space we mean a linear vector space X/aX_{0} where X and X_{0} are Banach spaces and a : X_{0} → X is an injective operator.

Let X/aX_{0} and Y/bY_{0} be quotient Banach spaces. A linear mapping ϕ : X/aX_{0} → Y/bY_{0} is called a morphism (in the category of quotient Banach spaces) if its lifted graph

G_{0}(ϕ) = {(x, y) ∈ X ⊕ Y : ϕ(x + aX_{0}) = y + bY_{0}}

is a paraclosed subspace of X ⊕ Y. Recall that the sum, as well as the composition of morphisms is also a morphism whenever these operations make sense, see for instance [1], Lemma I.9.3. A morphism that is injective and onto is called an isomorphism. By [10], the inverse of an isomorphism is also an isomorphism.

By a densely defined quotient Banach space we mean the quotient Banach space X/aX_{0} such that aX_{0} is dense in X.

Let X/aX_{0} and Y/bY_{0} be quotient Banach spaces and let T : X → Y be a bounded operator satisfying T (aX_{0}) ⊂ bY_{0}. Then T defines a morphism T̂ : X/aX_{0} → Y/bY_{0} by T̂(x + aX_{0}) = T x + bY_{0}. Morphisms T̂ that arise in this way are called strict.

Set T_{0} = b^{−1}T a : X_{0} → Y_{0}. An easy application of the closed graph theorem implies that T_{0} is a bounded operator. The following diagram commutes

\begin{matrix}X & ⟶^{T} & Y \\ ↑^{a} & & ↑^{b} \\ X_{0} & ⟶^{T_{0}} & Y_{0} \end{matrix}

Clearly T̂ = 0 if and only if T X ⊂ bY_{0}.

Let X/aX_{0} be a densely defined quotient Banach space. Then a^{∗} : X^{∗} → X_{0}^{∗} is also an injective operator. Define the dual of X/aX_{0} by

(X/aX_{0})^{#} = X_{0}^{∗}/a^{∗}X^{∗}.  (1)

Since a^{∗}X^{∗} is an operator range, it is paraclosed. Hence (X/aX_{0})^{#} is also a quotient Banach space.

The duality defined by (1) has properties analogous to the duality of Banach spaces. Let X/aX_{0} and Y/bY_{0} be quotient Banach spaces. Define their direct sum by

X/aX_{0} ⊕ Y/bY_{0} = (X ⊕ Y)/(a ⊕ b)(X_{0} ⊕ Y_{0}).

By induction it is possible to define direct sums of a finite number of quotient Banach spaces.

Clearly the direct sum of a finite number of densely defined quotient Banach spaces is again a densely defined quotient Banach space.

Proposition 1. Let X_{i}/a_{i}X_{i0} (i = 1, …, n) be densely defined quotient Banach space. Then

(⨁_{i}(X_{i}/a_{i}X_{i0}))^{#} ≡ ⨁_{i}( X_{i}/a_{i}X_{i0})^{#} ,

where ≡ denotes the isomorphism of quotient Banach spaces.

Proof. We prove the statement for two densely defined quotient Banach spaces X/aX_{0} and Y/bY_{0}. For n-tuples of quotient Banach spaces the proof remains without any change. We have

((X/aX_{0})⊕(Y/bY_{0}))^{#} = ((X ⊕Y)/(a⊕b)(X_{0}⊕Y_{0}))^{#} = (X_{0}⊕Y_{0})^{∗}/(a⊕b)^{∗}(X ×Y)^{∗} ≡

(X_{0}^{∗}/a^{∗}X^{∗}) ⊕ (Y_{0}^{∗}/b^{∗}Y^{∗}) = (X/aX_{0})^{#} ⊕ (Y/bY_{0})^{#} .

Proposition 2. Let X/aX_{0} be a quotient Banach space. Let J : X → X^{∗∗} and J_{0} : X_{0} → X_{0}^{∗∗} be the canonical embeddings. Then

(i) Ja = a^{∗∗}J_{0}, (ii) JaX_{0} ⊂ a^{∗∗}X_{0}^{∗∗} so that Ĵ: X/aX_{0} → X^{∗∗}/a^{∗∗}X_{0}^{∗∗} is a strict morphism, (iii) if X_{0} is reflexive, then Ĵ is injective, (iv) if X is reflexive then Ĵ is onto.

Proof.

(i) Let x_{0} ∈ X_{0} and f ∈ X^{∗}. Then

⟨Jax_{0}, f⟩ = ⟨ax_{0}, f⟩ = ⟨x_{0}, a^{∗}f⟩ = ⟨J_{0}x_{0}, a^{∗}f⟩ = ⟨a^{∗∗}J_{0}x_{0}, f⟩.

Thus Ja = a^{∗∗}J_{0}. (ii) By (i), JaX_{0} ⊂ a^{∗∗}X_{0}^{∗∗}. (iii) Let X_{0} be reflexive so that J_{0} is onto. Let x ∈ X and Jx ∈ a^{∗∗}X_{0}^{∗∗}, i.e., Jx =

a^{∗∗}J_{0}x_{0} for some x_{0} ∈ X_{0}. Then Jx = Jax_{0} and x = ax_{0} ∈ aX_{0} since J is injective. Hence Ĵ: X/aX_{0} → X^{∗∗}/a^{∗∗}X_{0}^{∗∗} is injective.

(iv) Let X be reflexive so that J is onto. Then Ĵ is also onto.

Our goal is to define the duality for morphisms between densely defined quotient Banach spaces. We start with strict morphisms.

Let X/aX_{0} and Y/bY_{0} be densely defined quotient Banach spaces, let T : X → Y be a bounded operator satisfying T (aX_{0}) ⊂ bY_{0}. Set T_{0} = b^{−1}T a : X_{0} → Y_{0}. Then the following diagram commutes

X^{∗} ⟵^{T^{∗}} Y^{∗}
↓_{a^{∗}}  ↓_{b^{∗}}
X_{0}^{∗} ⟵^{T_{0}^{∗}} Y_{0}^{∗}

so that T_{0}^{∗}(b^{∗}Y^{∗}) ⊂ a^{∗}X^{∗}. Thus T_{0}^{∗} defines a strict morphism T̂_{0}^{∗} : Y_{0}^{∗}/b^{∗}Y^{∗} → X_{0}^{∗}/a^{∗}X^{∗}. Define the dual of the strict morphism T̂ : X/aX_{0} → Y/bY_{0} by

(T̂)^{#} = T̂_{0}^{∗} : (Y/bY_{0})^{#} → (X/aX_{0})^{#} .  (2)

The duality between T̂ and (T̂)^{#} has the expected properties. The next theorem shows also that the definition of the adjoint is correct.

Theorem 3. Let X/aX_{0}, Y/bY_{0} and Z/cZ_{0} be densely defined quotient Banach spaces, let T, S : X → Y and U : Y → Z be operators satisfying T (aX_{0}) ⊂ bY_{0}, S(aX_{0}) ⊂ bY_{0} and U(bY_{0}) ⊂ cZ_{0}. Then

(i) if T̂ = 0 then T̂^{#} = 0, (ii) (T̂ + Ŝ)^{#} = T̂^{#} + Ŝ^{#} , (iii) if α ∈ C then (αT̂)^{#} = αT̂^{#} , (iv) (ŜT̂)^{#} = T̂^{#} Ŝ^{#} .

Proof. (i) Let T̂ = 0 so that T X ⊂ bY_{0}. Define R : X → Y_{0} by Rx = b^{−1}T x (x ∈ X). We show by the closed graph theorem that R is a bounded operator. Indeed, let x_{n} ∈ X, x_{n} → 0 and Rx_{n} → y_{0}. Then T x_{n} → 0 and in the same time T x_{n} = bRx_{n} → by_{0}. Since b is injective, we have y_{0} = 0 and R is continuous.

Further T_{0} = Ra and T_{0}^{∗} = a^{∗}R^{∗} so that T_{0}^{∗}Y_{0}^{∗} ⊂ a^{∗}X_{0}^{∗} and T̂^{#} = T̂_{0}^{∗} = 0. Thus the definition of T̂^{#} is correct.

The remaining statements are clear.

Let X/aX_{0} and Y/bY_{0} be densely defined quotient Banach spaces, let T : X → Y be an operator satisfying T (aX_{0}) ⊂ bY_{0}. Define operators e_{1} : X_{0} → X ⊕ Y_{0} and e_{2} : X ⊕ Y_{0} → Y by e_{1}x_{0} = dx_{0} ⊕ T_{0}x_{0} (x_{0} ∈ X_{0}) and e_{2}(x ⊕ y_{0}) = T x − dy_{0} (x ∈ X, y_{0} ∈ Y_{0}), respectively. Clearly e_{2}e_{1} = 0, e_{1} is injective and e_{2} has a dense range. The properties of the strict morphism T̂ are closely connected with the exactness properties of the complex

0 ⟶ X_{0} ⟶^{e_{1}}X ⊕ Y_{0} ⟶^{e_{2}}Y ⟶ 0.

Note that the complex corresponding to the adjoint T̂^{#} is

0 ⟶ Y^{∗}⟶^{e_{2}^{∗}}Y_{0}^{∗} ⊕ X^{∗}⟶^{e_{1}^{∗}}X_{0}^{∗} ⟶ 0.

Proposition 4.

(i) T̂ is onto ⇔ e_{2} is onto ⇔ e_{2} has closed range, (ii) T̂ is injective ⇔ kere_{2} = Ime_{1} ⇒ Ime_{1} is closed, (iii) T̂^{#} is onto ⇔ e_{1}^{∗} is onto ⇔ e_{1} is bounded below ⇔ Ime_{1} is closed, (iv) T̂^{#} is injective ⇔ kere_{1}^{∗} = Ime_{2}^{∗},

(v) if T̂ is injective then T̂^{#} is onto, (vi) if T̂^{#} is injective then T̂ is onto, (vii) T̂^{#} is an isomorphism ⇔ T̂ is an isomorphism.

Proof. Straightforward.

Our aim now is to define the duality for non-strict morphisms.

Recall that a quotient Banach space X/aX_{0} is called standard if X is an ℓ_{1}-space (over some set A). By [10], each morphism from a standard quotient Banach space to any quotient Banach space is strict. Further each quotient Banach space is isomorphic to a standard one. We repeat the proof of this fact ((i)–(iv) of the following theorem) since we need a little more detailed information.

Theorem 5. Let X/aX_{0} be a quotient Banach space. Then there exists a standard quotient Banach space Z/cZ_{0} and an operator Q : Z → X with the following properties:

(i) QZ = X and Q(cZ_{0}) ⊂ aX_{0}. Thus Q_{0} = a^{−1}Qc : Z_{0} → X_{0} is a bounded operator, (ii) Q̂ : Z/cZ_{0} → X/aX_{0} is a strict isomorphism, (iii) kerQ ⊂ cZ_{0}, (iv) Q_{0} is onto so that Q(cZ_{0}) = aX_{0},

(v) Q(\overline{cZ_{0}}) = \overline{aX_{0}}.

Proof. (i) Let S_{X} be the unit sphere in X and let Z = ℓ_{1}(S_{X}). Let Q : Z → X be the canonically defined surjective operator.

Set Z_{0} = {x_{0} ⊕ z ∈ X_{0} ⊕ Z : dx_{0} = Qz}. Define c : Z_{0} → Z by c(x_{0} ⊕ z) = z. Then c is injective. Indeed, suppose that x_{0} ⊕ z ∈ Z_{0} and c(x_{0} ⊕ z) = z = 0. We have ax_{0} = Qz = 0 so that x_{0} = 0. Thus Z/cZ_{0} is a standard quotient Banach space. Further Q(cZ_{0}) ⊂ aX_{0}. Indeed, let x_{0} ⊕ z ∈ Z_{0} so that ax_{0} = Qz. We have Qc(x_{0} ⊕ z) = Qz = ax_{0} ∈ aX_{0}. Thus Q̂ : Z/cZ_{0} → X/aX_{0} is a strict morphism.

(ii) Since Q is onto, Q̂ is also onto. Suppose that z ∈ Z and Qz ∈ aX_{0}. Then there exists x_{0} ∈ X_{0} such that Qz = ax_{0} and x_{0} ⊕ z ∈ Z_{0}. Hence z = c(x_{0} ⊕ z) ∈ cZ_{0} and Q̂ is injective by Proposition 4. Hence Q̂ is a strict isomorphism.

(iii) Let z ∈ kerQ. Then 0 ⊕ z ∈ Z_{0} so that z = c(0 ⊕ z) ∈ cZ_{0}.

(iv) Let x_{0} ∈ X_{0}. Since Q : Z → X is onto, there is z ∈ Z such that ax_{0} = Qz. Consequently x_{0} ⊕ z ∈ Z_{0} and ax_{0} = Qz = Qc(x_{0} ⊕ z) ∈ Q(cZ_{0}). Hence Q_{0} : Z_{0} → X_{0} is onto.

(v) Since Q(cZ_{0}) ⊂ aX_{0} and Q is continuous, we have also Q(\overline{cZ_{0}}) ⊂ \overline{aX_{0}}.

Conversely, let x ∈ \overline{aX_{0}}. Then there exists a sequence x_{n} ∈ X_{0} such that ax_{n} → x. For each n there is z_{n} ∈ Z such that ax_{n} = Qz_{n} and z ∈ Z such that Qz = x. Thus x_{n} ⊕ z_{n} ∈ Z_{0}, so that z_{n} = c(x_{n} ⊕ z_{n}) ∈ cZ_{0} and Qz_{n} = ax_{n} → x = Qz. By the open mapping theorem there are vectors u_{n} ∈ Z with Qu_{n} = Q(z − z_{n}) and u_{n} → 0. Thus v_{n} = z_{n} −z −u_{n} ∈ kerQ ⊂ cZ_{0} and z −(z_{n} −v_{n}) = −u_{n} → 0 so that z_{n} −v_{n} → z ∈ \overline{cZ_{0}}.

We define now adjoints to non-strict morphisms in densely defined quotient Banach spaces.

Let X/aX_{0} and Y/bY_{0} be densely defined quotient Banach spaces, let ϕ : X/aX_{0} → Y/bY_{0} be a morphism. By Theorem 5 there exist a canonically constructed standard quotient Banach space Z/cZ_{0} and a surjective operator Q : Z → X such that Q̂ : Z/cZ_{0} → X/aX_{0} is a strict isomorphism. Thus ϕQ̂ : Z/cZ_{0} → Y/bY_{0} is strict and we define ϕ^{#} : (Y/bY_{0})^{#} → (X/aX_{0})^{#} by

ϕ^{#} = (Q̂^{#} )^{−1} · (ϕQ̂)^{#} .  (3)

Clearly ϕ^{#} defined by (3) is a morphism. For strict morphisms the definition coincides with the original one. Indeed, let ϕ be strict, i.e., ϕ = T̂ for some operator T : X → Y. Then ϕQ̂ = T̂ Q and the following diagram commutes

Z ⟶^{Q} X ⟶^{T} Y
↑^{c}  ↑^{a}  ↑^{b}
Z_{0} ⟶^{Q_{0}} X_{0} ⟶^{T_{0}} Y_{0}

We have (ϕQ̂)^{#} = Q̂_{0}^{∗}T_{0}^{∗} = Q̂_{0}^{∗}T̂_{0}^{∗} = Q̂^{#} ϕ^{#} and ϕ^{#} = (Q̂^{#} )^{−1} · (ϕQ̂)^{#} . Thus the definition of the adjoint coincides with the definition for strict morphisms.

The properties of adjoints of strict morphisms can be easily generalized for nonstrict morphisms.

Theorem 6. Let X/aX_{0} and Y/bY_{0} be densely defined quotient Banach spaces, let ϕ : X/aX_{0} → Y/bY_{0} be a morphism. Then

(i) if ϕ = 0 then ϕ^{#} = 0, (ii) if ψ : X/aX_{0} → Y/bY_{0} is another morphism then (ϕ + ψ)^{#} = ϕ^{#} + ψ^{#} , (iii) if α ∈ C then (αϕ)^{#} = αϕ^{#} , (iv) if Z/cZ_{0} is a densely defined quotient Banach space and ψ : Y/bY_{0} → Z/cZ_{0} a

morphism then (ψϕ)^{#} = ϕ^{#} ψ^{#} ,

(v) if ϕ is injective then ϕ^{#} is onto; if ϕ^{#} is injective then ϕ is onto, (vi) ϕ^{#} is an isomorphism ⇔ ϕ is an isomorphism, (vii) if ϕ is an isomorphism, then (ϕ^{−1})^{#} = (ϕ^{#} )^{−1}.

Theorem 7. Let X/aX_{0} and Y/bY_{0} be quotient Banach spaces, let ϕ : X/aX_{0} → Y/bY_{0} be a morphism. Let x ∈ \overline{aX_{0}}. Then there exists y ∈ \overline{bY_{0}} such that ϕ(x+aX_{0}) = y+bY_{0}. In other words, the restriction of ϕ to \overline{aX_{0}}/aX_{0} is a morphism from \overline{aX_{0}}/aX_{0} to \overline{bY_{0}}/bY_{0}.

Proof. By Theorem 5, there is a standard quotient Banach space Z/cZ_{0} and a strict morphism Q̂ : Z/cZ_{0} → X/aX_{0} (defined by an operator Q : Z → X). Further ϕQ̂ : Z/cZ_{0} → Y/bY_{0} is strict so that ϕQ̂ is defined by an operator S : Z → Y.

Let x ∈ \overline{aX_{0}}. By Theorem 5 (iv) there is z ∈ \overline{cZ_{0}} such that Qz = x. Set y = Sz. We have

ϕ(x + aX_{0}) = ϕ(Qz + aX_{0}) = ϕQ̂(z + cZ_{0}) = Sz + cZ_{0} = y + cZ_{0}.

Further y = Sz ∈ \overline{bY_{0}}.

Corollary 8. Let X/aX_{0} and Y/bY_{0} be quotient Banach spaces, let ϕ : X/aX_{0} → Y/bY_{0} be a morphism. Then ϕ induces an operator T : X/\overline{aX_{0}} → Y/\overline{bY_{0}} defined by T (x + \overline{aX_{0}}) = y + \overline{bY_{0}} where y + bY_{0} = ϕ(x + aX_{0}).

Theorem 9. Let X_{j}/a_{j}X_{j0}, j = 1, …, n and Y_{i}/b_{i}Y_{i0}, i = 1, …, m be densely defined quotient Banach spaces. Let

ϕ_{i,j} : X_{j}/a_{j}X_{j0} → Y_{i}/b_{i}Y_{i0}, i = 1, …, m, j = 1, …, n

be morphisms. Define the linear mapping

ϕ := [ϕ_{i,j}]_{i,j} : ⨁_{j}(X_{j}/a_{j}X_{j0}) → ⨁_{i}(Y_{i}/b_{i}Y_{i0})

by ϕ( x̂_{j}) := ⊕_{i=1}^{m} ϕ_{i,j}x̂_{j} (x_{j} ∈ X_{j}, j = 1, …, n), where x̂ denotes the class of x modulo the corresponding subspace. Then ϕ is a morphism and

ϕ^{#} = (ϕ_{i,j}^{#})_{i,j}^{t} : ⨁_{i}(Y_{i0}^{∗}/b_{i}^{∗}Y_{i}^{∗}) → ⨁_{j}(X_{j0}^{∗}/a_{j}^{∗}X_{j}^{∗}),

where (·)^{t} stands for the transposed matrix.

Proof. Write X = ⊕_{j}X_{j} and Y = ⊕_{i}Y_{i}. For i ∈ {1, …, m}, j ∈ {1, …, n} denote by [ψ^{(i,j)}] the m × n operator matrix whose entries at (i′, j′) are: ϕ_{i,j} if (i′, j′) = (i, j) and zero otherwise.

We have

G_{0}([ψ^{(i,j)}]) = G_{0}(ϕ_{i,j}) + (⨁_{j′≠j} X_{j′}) ⊕ (⨁_{i′=1}^{m}b_{i′}Y_{i′0})

where we identify G_{0}(ϕ_{i,j}), X_{j′} and b_{i′}Y_{i′0} with the corresponding subspaces of X ⊕ Y. Thus G_{0}(ψ^{(i,j)}) is a paraclosed subspace of X ⊕ Y and ψ^{(i,j)} is a morphism.

Since ϕ = ∑_{i,j}[ψ^{(i,j)}] and the class of morphisms is closed under addition, ϕ is also a morphism.

To show the statement about the adjoints, let Q̂_{j} and P̂_{i} denote the strict morphisms that represent X_{j}/a_{j}X_{j0}, (Y_{i}/b_{i}Y_{i0}, respectively) on standard quotient Banach spaces as stated in Theorem 5. Let θ_{i,j} := P_{i}^{−1}ϕ_{i,j}Q_{j} and θ := [θ_{i,j}]_{i,j} = P^{−1}ϕQ, where

Q := ⊕_{j}Q_{j} and P := ⊕_{i}P_{i}. Then the desired conclusion ϕ^{#} = [ϕ_{i,j}^{#}]_{i,j}^{t} is equivalent

to the equality θ^{#} = [θ_{i,j}^{#}]_{i,j}^{t}. Thus we may suppose that all X_{j} and Y_{i} are standard.

Hence we can assume that ϕ_{i,j} = T̂_{i,j} are strict morphisms. Set T_{0} := [T_{i,j,0}]_{i,j}, where T_{i,j,0} : a_{j}X_{j0} → b_{i}Y_{i0} is the restriction of T_{i,j} to a_{j}X_{j0}.

Now the Banach space adjoint of the operator matrix T_{0} is the transposed matrix of the adjoints [T_{i,j,0}^{∗}]_{j,i}. By factoring through the subspaces ⊕_{j}a_{j}X_{j0} and ⊕_{i}b_{i}^{∗}Y_{i}^{∗} respectively, we obtain the equality in the statement.

Example 10. Let B(H) be the set of all operators acting on a separable Hilbert space H. Denote by K the Banach space of all compact operators endowed with the uniform norm and by C_{1} the space of all trace class operators on H endowed with the trace norm. Let a : C_{1} → K be the natural inclusion. Then K/aC_{1} is a densely defined quotient Banach space.

Further C_{1}^{∗} can be identified with B(H) and K^{∗} with C_{1}; in both cases the duality is given by the formula ⟨T, S⟩ = trT S. Thus (K/aC_{1})^{#} ≡ B(H)/a^{∗}C_{1} where a^{∗} : C_{1} → B(H) is also the natural inclusion.

For A ∈ B(H) denote by L_{A} and R_{A} the operators of left (right) multiplication on B(H) defined by L_{A}S = AS, R_{A}S = SA (S ∈ B(H)). Clearly L_{A} leaves invariant both K and C_{1} so that the restriction L_{A}|K defines a strict morphism acting on K/C_{1}.

It is easy to verify that L̂_{A}|K^{#} = R_{A}. Similarly R̂_{A}|K^{#} = L_{A}.

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Institute of Mathematics of the Romanian Academy PO Box 1–764, RO–70700 Bucharest Romania cambroz@stoilow.imar.ro Mathematical Institute Academy of Sciences of the Czech Republic 115 67 Prague 1, Žitná 25 Czech Republic muller@math.cas.cz