Isometries between C*-algebras

Cho-Ho Chu and Ngai-Ching Wong^{∗}

Abstract Let A and B be C*-algebras and let T be a linear isometry from A into B. We show that there is a largest projection p in B^{∗∗} such that T (·)p : A ⟶ B^{∗∗} is a Jordan triple homomorphism and

T (ab^{∗}c + cb^{∗}a)p = T (a)T (b)^{∗}T (c)p + T (c)T (b)^{∗}T (a)p

for all a, b, c in A. When A is abelian, we have ∥T (a)p∥ = ∥a∥ for all a in A. It follows that a (possibly non-surjective) linear isometry between any C*-algebras reduces locally to a Jordan triple isomorphism, by a projection.

1 Introduction

In his seminal paper [10], Kadison showed that a surjective linear isometry T between unital C*-algebras A and B is of the form T (·) = uη(·) where u is a unitary element in B and η is a Jordan *-isomorphism. This result remains true in the non-unital case although the unitary element u generally comes from B ⊕ ℂ [13]. In both cases, T preserves the Jordan triple product:

T (ab^{∗}c + cb^{∗}a) = T (a)T (b)^{∗}T (c) + T (c)T (b)^{∗}T (a)

for all a, b, c ∈ A. In infinite-dimensional holomorphy, C*-algebras, and the larger class of JB*-triples, arise as tangent spaces to bounded symmetric domains and it has been shown in [11] that the geometry of these domains is completely determined by the Jordan triple structures of these spaces. Indeed, a bijective linear map T between two JB*-triples is an isometry if, and only if, it preserves the Jordan triple product:

T {a, b, c} = {T (a), T (b), T (c)}

To appear in Revista Matematica Iberoamericana.

Mathematics Subject Classification 2000. 46L05, 46B04, 46L70, 32M15.

Keywords : C*-algebra, JB*-triple, isometry, Banach manifold.

∗Supported by Taiwan National Science Council Grants: 86-2115-M110-002, 37128F.

as shown in [11, Proposition 5.5] (see also [3, 4, 6, 16]). By polarization, T preserves the Jordan triple product if, and only if,

T {a, a, a} = {T (a), T (a), T (a)}.

The Jordan triple product in a C*-algebra is given by

\{a, b, c\} = \frac{1}{2}(ab^{∗}c + cb^{∗}a)

and in particular, the above characterization of surjective linear isometries between JB*-triples extends Kadison’s result as well as giving it a geometric perspective. It also highlights the importance of the Jordan triple product in the study of isometries of C*-algebras.

It is natural to ask to what extent the above triple-preserving property of a linear isometry persists if it is not surjective. We address this question in this paper. Let T : A ⟶ B be a linear isometry, possibly non-surjective. We study T locally. Without surjectivity, the C^{∗}-algebra and affine geometric techniques of [10, 4] can not be used directly to obtain conclusive results. Nevertheless, we show there is a largest projection p ∈ B^{∗∗}, called the structure projection of T, such that T (A)p is a Jordan subtriple of B^{∗∗} and the map

T (·)p : A ⟶ T (A)p

is a triple homomorphism with T {a, a, a}p = {T (a), T (a), T (a)}p for all a ∈ A. The structure projection p is closed but the map T (·)p need not be injective. When A is abelian, we study the structure projection p in some detail, motivated by the question of the local behaviour of T, and show that the map T (·)p is isometric which also extends Holsztynski’s result in [8] for non-surjective isometries between continuous function spaces (see also [9]). It follows that, for any A and B, the isometry T is reduced locally to a triple isomorphism by a projection in the sense that, for any a ∈ A, there is a closed projection p_{a} ∈ B^{∗∗} such that the map T (·)p_{a} is a triple isomorphism from the Jordan subtriple Z_{a} of A, generated by a, into B^{∗∗} and

T {x, y, z}p_{a} = {T (x), T (y), T (z)}p_{a}

for all x, y, z ∈ Z_{a}. Although T (A)p could be zero if A is nonabelian, we give conditions for T (A)p to be non-zero in this case.

This work was carried out during the second author’s visit at Goldsmiths College, University of London. He would like to thank colleagues there for their warm hospitality. We wish to thank Professor L.G. Brown for a useful discussion and for drawing our attention to the norm identity in Remark 4.4. We also thank the referee for many helpful suggestions.

2 Isometries of C*-algebras and their ranges

Throughout the paper, an isometry between Banach spaces is not assumed to be surjective. We first recall that a JB*-triple Z is a complex Banach space equipped with a Jordan triple product {·, ·, ·} : Z^{3} ⟶ Z which is symmetric and linear in the outer variables, and conjugate linear in the middle variable such that for a, b, c, x, y ∈ Z, we have

(i) {a, b, {c, x, y}} = {{a, b, c}, x, y} − {c, {b, a, x}, y} + {c, x, {a, b, y}};

(ii) the map z ∈ Z ↦ {a, a, z} ∈ Z is hermitian with nonnegative spectrum;

(iii) ∥{a, a, a}∥ = ∥a∥^{3}.

A closed subspace of a JB*-triple is called a subtriple if it is closed with respect to the triple product. A linear map T : Z ⟶ W between JB*-triples is called a triple homomorphism if it preserves the triple product in which case, the range T (Z) is a subtriple of W and the kernel J of T is a triple ideal of Z, that is, {Z, Z, J} + {Z, J, Z} ⊂ J. We refer to [2, 17, 18, 20] for expositions as well as recent surveys of JB*-triples and symmetric Banach manifolds. In the sequel, we write a^{(3)} = {a, a, a}. We note that a norm-closed subspace Z of a C*-algebra is a JB*-triple if a ∈ Z implies aa^{∗}a ∈ Z, in which case Z is called a JC*-triple and the triple product is given by triple polarization

2{a, b, c} = ab^{∗}c + cb^{∗}a
= \frac{1}{8} ∑_{α^{4}=β^{2}=1} αβ(a + αb + βc)(a + αb + βc)^{∗}(a + αb + βc).

In C*-algebras, the closed triple ideals are the closed algebra two-sided ideals [7, p.350].

We begin with a simple example of a linear isometry T : A ⟶ B between abelian C*-algebras which is not a triple homomorphism.

Example 2.1. Let C(Ω) and C(Ω ∪ {β}) be the C*-algebras of continuous functions on the closed unit disc Ω ⊂ ℂ and Ω ∪ {β} respectively, where β ∈ ℂ\Ω. Define T : C(Ω) ⟶ C(Ω ∪ {β}) by

(T f)(x) = \{ \begin{matrix}f(x) & if x ∈ Ω \\ \frac{1}{2}(f(1) + f(0)) & if x = β. \end{matrix}

Then T is a linear isometry and T (C(Ω)) = {h ∈ C(Ω ∪ {β}) : 2h(β) = h(1) + h(0)} which is not a subtriple of C(Ω ∪ {β}). So T is not a triple isomorphism onto its range. Nevertheless, we have T (f^{(3)}) = T (f)^{(3)} if f(1) = f(0) = 0.

Let T : A ⟶ B be a linear isometry between C*-algebras. Although the range T (A) need not be a subtriple of B, we show in Proposition 2.2 below that T (A), cut down by a projection, is always a subtriple of B^{∗∗}. This result will be used to study T locally later. In Example 2.1, such a projection is given by the characteristic function of Ω in C(Ω ∪ {β}).

We need some notation first. We denote by T^{∗∗} the second dual map of T and for convenience, we often write T a for T (a). The identity of a unital C*-algebra will be denoted by 111. Given a C*-algebra A, we denote its closed unit ball by A_{1}, and by A_{1}^{∗} the closed unit ball of the dual A^{∗}. Let Q(A) = {φ ∈ A_{1}^{∗} : φ ≥ 0} be the quasi-state space which is weak* compact and convex. Every weak* closed face of Q(A) containing zero is of the form F (p) = {φ ∈ Q(A) : φ(111 − p) = 0} for some closed projection p ∈ A^{∗∗}, called the support projection of the face (cf. [5, 15] or [14, 3.11.10]). The polar decomposition of a functional ψ ∈ A^{∗} is denoted by ψ(·) = v^{∗}|ψ|(·) = |ψ|(v^{∗}·) where v^{∗} is a partial isometry in A^{∗∗}.

For each φ in Q(A), we let (π_{φ}, H_{φ}, ω_{φ}) be the Gelfand-Naimark-Segal representation of A induced by φ. As usual, we also denote by π_{φ} the extended representation of A^{∗∗} on the Hilbert space H_{φ} (see, for example, [14, p. 60]). For simplicity, we write xω_{φ} for π_{φ}(x)ω_{φ} in H_{φ} whenever x ∈ A^{∗∗}. Thus we have xω_{φ} = 0 if, and only if, φ(x^{∗}x) = 0. Further, we have φ(x^{∗}x) = 0 for all φ ∈ F (p) if, and only if, xp = 0 (cf. [14, §3.10] and [1, Corollary 3.5]). We note that if φ is a pure state with support projection p, then F (p) = [0, 1]φ.

Proposition 2.2. Let A and B be C*-algebras and let T : A ⟶ B be a linear isometry. Then there is a largest projection p in B^{∗∗} such that

(i) T (·)p : A ⟶ B^{∗∗} is a triple homomorphism;

(ii) T {a, b, c}p = {T a, T b, T c}p for all a, b, c in A.

Further, p is a closed projection and (T a)^{∗}(T b)p = p(T a)^{∗}(T b) for all a, b in A.

Proof. Let

F_{1} = ⋂_{a∈A_{1}}{φ ∈ Q(B) : (T a^{(3)})ω_{φ} = (T a)^{(3)}ω_{φ}}
= ⋂_{a∈A_{1}} {φ ∈ Q(B) : φ ((T a^{(3)} − (T a)^{(3)})^{∗}(T a^{(3)} − (T a)^{(3)})) = 0} .

Then F_{1} is a weak* closed face of Q(B) containing zero. For a in A_{1}, we define a weak* continuous affine map Φ_{a} : Q(B) ⟶ Q(B) by

Φ_{a}(φ)(·) = φ ((T a)^{∗}(T a) · (T a)^{∗}(T a)) .

For n = 1, 2, …, the sets

F_{n+1} = {φ ∈ F_{n} : Φ_{a}(φ) ∈ F_{n}, ∀a ∈ A_{1}} = ⋂_{a∈A_{1}} F_{n} ∩ Φ_{a}^{−1}(F_{n})

form a decreasing sequence of weak* closed faces of Q(B). The intersection F = ⋂_{n=1}^{∞} F_{n} is a weak* closed face of Q(B) containing zero. Let p be the closed projection in B^{∗∗} supporting F:

F = F (p) = {φ ∈ Q(B) : φ(111 − p) = 0}.

For each a in A_{1} and φ in F, we have

Φ_{a}(φ)(·) = φ ((T a)^{∗}(T a) · (T a)^{∗}(T a)) ∈ F,

and consequently,

⟨p(T a)^{∗}(T a)ω_{φ}, (T a)^{∗}(T a)ω_{φ}⟩ = Φ_{a}(φ)(p) = Φ_{a}(φ)(1) = ∥(T a)^{∗}(T a)ω_{φ}∥^{2}.

Hence

p(T a)^{∗}(T a)ω_{φ} = (T a)^{∗}(T a)ω_{φ}, ∀φ ∈ F = F (p)

and therefore

p(T a)^{∗}(T a)p = (T a)^{∗}(T a)p.

It follows that

p(T a)^{∗}(T a) = (T a)^{∗}(T a)p, ∀a ∈ A.

By polarization, we have

p(T a)^{∗}(T b) = (T a)^{∗}(T b)p  (2.1)

for all a, b ∈ A. To verify (i), we note that

(T a^{(3)})ω_{φ} = (T a)^{(3)}ω_{φ}, ∀φ ∈ F.

This gives

(T a^{(3)})p = (T a)^{(3)}p.

By triple polarization and (3.1), we get

T {a, b, c}p = {T a, T b, T c}p = {(T a)p, (T b)p, (T c)p}.

Finally, if q is a projection in B^{∗∗} satisfying conditions (i) and (ii), then

F (q) = {φ ∈ Q(B) : φ(111 − q) = 0} ⊆ F_{n}, n = 1, 2, …

since Φ_{a}(F (q)) ⊆ F (q) for a ∈ A_{1} and it is evident that F (q) ⊆ F_{1}. Therefore F (q) ⊆ F (p) and q ≤ p. The last assertion has been shown in (2.1).

Remark 2.3. (a) Although the above result only requires T to be contractive, all subsequent applications of the result, including the next two remarks, requires T to be isometric.

(b) In the above proof, if T is surjective or T (A) is a subtriple of B, then F_{1} = Q(B) and p = 111.

(c) For an arbitrary projection p ∈ B^{∗∗}, conditions (i) and (ii) above are independent of each other in general and they need not imply (2.1). Consider, for instance, the identity map T : A ⟶ A, for which (ii) is satisfied by any projection, but only the central projections in A^{∗∗} satisfy (i) and (2.1). Nevertheless, if T^{∗∗}(111) is unitary, then (i) implies (2.1) and hence (ii), for any projection p ∈ B^{∗∗}. Indeed, if T^{∗∗}(111) = 111, then T commutes with involution and, by weak*-continuity of the triple product and (i), we have T {111,111, a}p = {111p,111p, T (a)p} which gives T (a)p = pT (a)p = pT (a) for a = a^{∗} and hence for all a ∈ A. For unitary T^{∗∗}(111), the map T^{∗∗}(111)^{∗}T^{∗∗} is unital and the preceding statement gives pT (a)^{∗}T (b) = p(T^{∗∗}(111)^{∗}T (a))^{∗}(T^{∗∗}(111)^{∗}T (b)) = (T^{∗∗}(111)^{∗}T (a))^{∗}(T^{∗∗}(111)^{∗}T (b))p = T (a)^{∗}T (b)p. If B is abelian, then of course (i) and (ii) are equivalent.

Definition 2.4. We denote by p_{T} the projection for the isometry T in Proposition 2.2 and call it the structure projection of T.

We give the following examples of structure projections p_{T}. Let M_{n} be the C*- algebra of n × n matrices.

Example 2.5. Let T : M_{2} ⟶ M_{3} be defined by

T (\begin{matrix}a & b \\ c & d \end{matrix}) = (\begin{matrix}a & b & 0 \\ c & d & 0 \\ 0 & 0 & a \end{matrix}) .

Then T is a unital linear isometry and T (M_{2}) is not a subtriple of M_{3}. The structure projection p_{T} is given by

p_{T} = (\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix}) .

We note that Morita [12] has shown that a linear isometry T : M_{n} ⟶ M_{n} is of the form T (x) = uxv or T (x) = ux^{t}v for some unitary u, v ∈ M_{n} where x^{t} denotes the transpose of x.

Example 2.6. Let A = C[0, 1], B = C([0, 1] ∪ {2}) and define T : A ⟶ B by

(T f)(x) = \{ \begin{matrix}f(x) & for x ∈ [0, 1] \\ ∫_{0}^{1} f(y)dy & for x = 2. \end{matrix}

Then T is a unital linear isometry, T (A) = {h ∈ B : h(2) = ∫_{0}^{1} h(y)dy} has codimension 1 in B and it is not a subtriple of B. We have p_{T} = χ_{[0,1]}, the characteristic function of [0, 1], which is in B.

Example 2.7. Let T : ℂ ⟶ M_{2} be defined by

T (a) = (\begin{matrix}0 & \frac{a}{2} \\ a & 0 \end{matrix}) .

Then T is an isometry and T (ℂ) is not a subtriple of M_{2}. Also T (1) is not unitary and T (ℂ) contains no nontrivial positive element. Its structure projection p_{T} is given by

p_{T} = (\begin{matrix}1 & 0 \\ 0 & 0 \end{matrix})

which does not commute with T (a) for a ≠ 0. Also T (a^{(3)}) ≠ T (a)^{(3)} for all non-zero a ∈ ℂ.

Example 2.8. Let K(H) be the C*-algebra of compact operators on a Hilbert space H with an orthonormal basis {e_{1}, e_{2}, …}, and B(H) the algebra of bounded operators on H. Define a linear isometry T : c_{0} ⟶ K(H) by

T (x) = \frac{x_{1}}{2} e_{1} ⊗ e_{1} + x_{1}e_{3} ⊗ e_{2} + \frac{x_{2}}{2} e_{5} ⊗ e_{3} + x_{2}e_{7} ⊗ e_{4} + ⋯
= \frac{1}{2} ∑_{n=1}^{∞} x_{n}e_{4n−3} ⊗ e_{2n−1} + ∑_{n=1}^{∞} x_{n}e_{4n−1} ⊗ e_{2n}

where x = (x_{n}) ∈ c_{0} and (e_{i} ⊗ e_{k})(·) = ⟨·, e_{k}⟩e_{i}. We have

x^{(3)} = (x_{1}^{(3)}, x_{2}^{(3)}, …),

T (x^{(3)}) = \frac{1}{2} ∑_{n=1}^{∞} x_{n}^{(3)}e_{4n−3} ⊗ e_{2n−1} + ∑_{n=1}^{∞} x_{n}^{(3)}e_{4n−1} ⊗ e_{2n},

and

T (x)^{(3)} = \frac{1}{8} ∑_{n=1}^{∞} x_{n}^{(3)}e_{4n−3} ⊗ e_{2n−1} + ∑_{n=1}^{∞} x_{n}^{(3)}e_{4n−1} ⊗ e_{2n}

by orthogonality. Hence, for any projection q in K(H)^{∗∗} = B(H),

T (x^{(3)})q = T (x)^{(3)}q

if, and only if,

(∑_{n=1}^{∞} x_{n}^{(3)}e_{4n−3} ⊗ e_{2n−1})q = 0.

This happens for all x in c_{0} exactly when qe_{2n−1} = 0 for n = 1, 2, . . .. Therefore the structure projection p_{T} is the orthogonal projection onto span{e_{2}, e_{4}, …} and we have

∥T (x)p_{T} ∥ = ∥x∥ and p_{T} (T x) = 0

for all x in c_{0}.

Remark 2.9. Let T : A ⟶ B be a linear isometry between C*-algebras. Let B be a C*-subalgebra of B̃, with common approximate identity, and regard B^{∗∗} as a subalgebra of B̃^{∗∗}. Then the structure projection p̃_{T} of the isometry T : A ⟶ B̃ is the same as p_{T}. Evidently, we have p_{T} ≤ p̃_{T}. Suppose p_{T} ≠ p̃_{T}. Choose a state ψ ∈ B̃^{∗} such that ψ(p_{T}) < ψ(p̃_{T}). Then the state

φ(·) = \frac{ψ(p̃_{T} · p̃_{T})}{ψ(p̃_{T})}

is in the closed face F (p̃_{T}) of Q(B̃) supported by p̃_{T}. This means, by the proof of Proposition 2.2, that

Φ_{b}^{n}(φ)((T a^{(3)} − (T a)^{(3)})^{∗}((T a^{(3)} − (T a)^{(3)})) = 0 (a, b ∈ A_{1}, n = 0, 1, 2, …)

where Φ_{b}^{0}(φ) = φ and Φ_{b}^{n} is the nth iterate of Φ_{b}. The restriction φ|_{B} is a state of B and clearly the above identity remains true when φ|_{B} replaces φ, that is, φ|_{B} ∈ F (p_{T}) ⊆ Q(B) which gives the contradiction

1 = φ(p_{T}) = \frac{ψ(p̃_{T}p_{T} p̃_{T})}{ψ(p̃_{T})} = \frac{ψ(p_{T})}{ψ(p̃_{T})}.

So p_{T} = p̃_{T}.

We note that, for a linear isometry T : A ⟶ B between C*-algebras, the triple homomorphism T (·)p_{T} = 0 if, and only if, T^{∗∗}(111)p_{T} = 0. This follows from the weak* continuity of the triple product and the identity

T (a)p_{T} = T^{∗∗}(a)p_{T} = T^{∗∗}{111,111, a}p_{T} = {T^{∗∗}(111)p_{T} , T^{∗∗}(111)p_{T} , T (a)p_{T} }.

We study various necessary and sufficient conditions for T (·)p_{T} ≠ 0 in the next two sections. The above identity also shows that T^{∗∗}(111)p_{T} is a partial isometry in B^{∗∗}.

3 Isometries from abelian C*-algebras

In this section, we study the structure projection of a linear isometry on an abelian C*-algebra. This is motivated by the intention to study a linear isometry locally, that is, to study its restriction on a subtriple generated by an element. We show in Theorem 3.10 below that when A is abelian, the structure projection p_{T} of an isometry T from A into any C*-algebra B is large enough to make the triple homomorphism T (·)p_{T} an isometry. Consequently, a linear isometry T on any C*-algebra reduces locally to a triple isomorphism via a projection, as shown in Corollary 3.12. We also give an alternative construction of p_{T} in Proposition 3.14 when the codomain B is a dual C*-algebra. We prove some lemmas first.

Definition 3.1. Let T : A ⟶ B be a linear map between C*-algebras. For each φ in A^{∗} with ∥φ∥ = 1, let

A_{φ} = {a ∈ A : φ(a) = ∥a∥ = 1}.

Similarly, for each ψ in B^{∗} with ∥ψ∥ = 1, let

B_{ψ} = {b ∈ B : ψ(b) = ∥b∥ = 1}.

If A_{φ} ≠ ∅, we define

Q_{φ} = {ψ ∈ B^{∗} : ∥ψ∥ = 1 and T (A_{φ}) ⊆ B_{ψ}}.

Lemma 3.2. Let T : A ⟶ B be a linear isometry between C*-algebras. For φ in A^{∗} with ∥φ∥ = 1 and A_{φ} ≠ ∅, the set Q_{φ} is a non-empty weak* closed face of B_{1}^{∗}.

Proof. We first note that Q_{φ} is an intersection of non-empty weak* closed faces of B_{1}^{∗}:

Q_{φ} = ⋂_{a∈Aφ}{ψ ∈ B_{1}^{∗} : ψ(T a) = 1}.

We show these faces have finite intersection property. To this end, let a_{1}, a_{2}, …, a_{n} be in A_{φ} and let a = ∑_{i=1}^{n} a_{i}. Since φ(a) = n, we have ∥T a∥ = ∥a∥ = n. Therefore, there is a norm one functional ψ in B^{∗} such that ψ(T a) = n. It follows that ∑_{i=1}^{n} ψ(T a_{i}) = n and so ψ(T a_{i}) = 1 for i = 1, 2, …, n. Consequently, we have ψ ∈ ⋂_{i=1}^{n}(T a_{i})^{−1}{1}.

Lemma 3.3. Let T : A ⟶ B be a linear isometry between C*-algebras, and let φ ∈ A^{∗} with ∥φ∥ = 1 and A_{φ} ≠ ∅. Then for any a ∈ A_{φ} and ψ ∈ Q_{φ} ⊆ B_{1}^{∗} with polar decomposition ψ = v^{∗}|ψ|, we have

(i) ∥(T a)ω_{|ψ|}∥ = 1;

(ii) (T a)ω_{|ψ|} = vω_{|ψ|} and (T a)^{∗}vω_{|ψ|} = ω_{|ψ|} in H_{|ψ|}.

Proof. Given a ∈ A_{φ} and ψ ∈ Q_{φ}, we have T a ∈ B_{ψ} and therefore,

1 = ψ(T a) = |ψ|(v^{∗}(T a))
= ⟨v^{∗}(T a)ω_{|ψ|}, ω_{|ψ|}⟩ = ⟨(T a)ω_{|ψ|}, vω_{|ψ∥}⟩ = ⟨ω_{|ψ|}, (T a)^{∗}vω_{|ψ∥}⟩ .

Since ∥vω_{|ψ|}∥ = 1 and ∥(T a)ω_{|ψ|}∥ ≤ ∥T a∥ = 1, we have ∥(T a)ω_{|ψ|}∥ = 1 and (T a)ω_{|ψ|} = vω_{|ψ|}. Similarly, we have (T a)^{∗}vω_{|ψ|} = ω_{|ψ|}.

In the remaining lemmas of this section, we assume that A is an abelian C*-algebra and is identified with the algebra C_{0}(X) of continuous functions on a locally compact Hausdorff space X, vanishing at infinity. Fix a linear isometry T : C_{0}(X) ⟶ B, where B is any C*-algebra. We write

A_{x} = A_{δ_{x}} = {f ∈ C_{0}(X) : f(x) = ∥f∥ = 1};
Q_{x} = Q_{δ_{x}} = {ψ ∈ B^{∗} : ∥ψ∥ = 1 and T (A_{x}) ⊆ B_{ψ}}

where δ_{x} is the point mass at x. Note that A_{x} ≠ ∅ for all x in X.

We let Q = ⋃_{x∈X} Q_{x} and define |Q_{x}| = {|ψ| : ψ ∈ Q_{x}}, |Q| = ⋃_{x∈X} |Q_{x}|.

Lemma 3.4. Given x ≠ x′ in X, we have |Q_{x}| ∩ |Q_{x′}| = ∅.

Proof. We first show that Q_{x} ∩ Q_{x′} = ∅. Suppose, otherwise, that there exists ψ ∈ Q_{x} ∩ Q_{x′}. Then T A_{x} ⊆ B_{ψ} and T A_{x′} ⊆ B_{ψ}. Let f ∈ A_{x} and f′ ∈ A_{x′} with ff′ = 0. Since T is an isometry and ∥f + f′∥ = 1, we have ∥T f + T f′∥ = 1. But ψ(T f) = ψ(T f′) = 1 implies ∥T f + T f′∥ ≥ 1 + 1 = 2 which is a contradiction.

Now suppose there exists ψ ∈ |Q_{x}| ∩ |Q_{x′}| with ψ = |φ| = |φ′| and φ ∈ Q_{x}, φ′ ∈ Q_{x′}. Let φ = v^{∗}|φ| and φ′ = v′^{∗}|φ′| be the polar decompositions. By Lemma 3.3, given f in C_{0}(X), we have

f ∈ A_{x} ⟹ (T f)ω_{ψ} = vω_{ψ};
f ∈ A_{x′} ⟹ (T f)ω_{ψ} = v′ω_{ψ}.

We can choose an f in A_{x} ∩A_{x′} which then gives vω_{ψ} = v′ω_{ψ}. Consequently, for every a in A we have

φ(a) = ψ(v^{∗}a) = ⟨aω_{ψ}, vω_{ψ}⟩_{ψ} = ⟨aω_{ψ}, v′ω_{ψ}⟩_{ψ} = ψ(v′^{∗}a) = φ′(a).

Hence φ = φ′ ∈ Q_{x} ∩ Q_{x′} which is impossible.

Definition 3.5. Define σ : |Q| ⟶ X by

σ(|ψ|) = x for ψ ∈ Q_{x}.

Let P (B) be the set of all pure states of B. The following lemma shows that |Q| ∩ P (B) ≠ ∅.

Lemma 3.6. σ(|Q| ∩ P (B)) = X.

Proof. Consider the isometry T from A = C_{0}(X) onto T (A). The adjoint map T^{∗} sends the set ∂T (A)_{1}^{∗} of extreme points in the closed unit ball of T (A)^{∗} onto the extreme points of the closed unit ball of C_{0}(X)^{∗}. In particular, for each x in X, there is a ψ in ∂T (A)_{1}^{∗} with T^{∗}ψ = δ_{x}. Let ψ̃ be an extreme point in B_{1}^{∗} extending ψ. Let ψ̃ = v^{∗}|ψ̃| be the polar decomposition of ψ̃. Then ψ̃(T f) = T^{∗}ψ(f) = f(x) for all f in C_{0}(X) which implies that ψ̃ ∈ Q_{x} and |ψ̃| ∈ |Q_{x}| ∩ P (B). Hence σ(|ψ̃|) = x.

Let q = ⋁{p_{φ} : φ ∈ |Q| ∩ P (B)} be the atomic projection in B^{∗∗} supporting all pure states in |Q| where p_{φ} is the minimal projection in B^{∗∗} supporting the pure state φ. Note that q depends on T.

Lemma 3.7. For all f in C_{0}(X), we have ∥(T f)q∥ = ∥T f∥.

Proof. Let ∥f∥ = |f(x)| > 0 for some x in X. Then \frac{f}{f(x)} ∈ A_{x} and \frac{T f}{f(x)} ∈ B_{ψ} for some ψ ∈ Q_{x} with |ψ| ∈ |Q| ∩ P (B) by Lemma 3.6. It follows from Lemma 3.3 that ∥(T f)ω_{|ψ|}∥ = ∥f∥ = ∥T f∥. So ∥T f∥ ≥ ∥(T f)q∥ ≥ ∥(T f)p_{|ψ|}∥ ≥ ∥(T f)ω_{|ψ|}∥ = ∥T f∥.

Lemma 3.8. Let φ = |ρ| for some ρ in Q with polar decomposition ρ = v^{∗}φ. Let f ∈ C_{0}(X). If f(σ(φ)) = 0, then (T f)ω_{φ} = (T f)^{∗}vω_{φ} = 0.

Proof. Without loss of generality, we may assume that ∥f∥ = 1. By Urysohn’s Lemma, it suffices to show that if f vanishes in a neighborhood of σ(φ) in X, then (T f)ω_{φ} = (T f)^{∗}vω_{φ} = 0. For this, we choose g in A_{σ(φ)} such that fg = 0. Then

∥g∥ = 1 = g(σ(φ))

and

∥f + g∥ = 1 = (f + g)(σ(φ)).

By Lemma 3.3, we have

(T g)ω_{φ} = vω_{φ} = T (f + g)ω_{φ}

and

(T g)^{∗}vω_{φ} = ω_{φ} = (T (f + g))^{∗}vω_{φ}.

Consequently (T f)ω_{φ} = (T f)^{∗}vω_{φ} = 0.

Lemma 3.9. Let ψ ∈ Q have polar decomposition ψ = v^{∗}φ where φ = |ψ|. Then for all f in C_{0}(X), we have (T f)ω_{φ} = f(σ(φ))vω_{φ} and (T f)^{∗}vω_{φ} = \overline{f(σ(φ))}ω_{φ}.

Proof. Recall that σ(φ) = x if ψ ∈ Q_{x}. Pick h ∈ C_{0}(X) such that h(σ(φ)) = 1 = ∥h∥, that is, h ∈ A_{σ(φ)}. Since

(f − f(σ(φ))h)(σ(φ)) = 0,

Lemma 3.8 gives

T (f − f(σ(φ))h)ω_{φ} = (T (f − f(σ(φ))h))^{∗}vω_{φ} = 0.

Therefore

(T f)ω_{φ} = f(σ(φ))(T h)ω_{φ} = f(σ(φ))vω_{φ}

since (T h)ω_{φ} = vω_{φ} by Lemma 3.3. Similarly, we have, by Lemma 3.3 again,

(T f)^{∗}vω_{φ} = \overline{f(σ(φ))}(T h)^{∗}vω_{φ} = \overline{f(σ(φ))}ω_{φ}.

We are now ready to prove that T (·)p_{T} is an isometry if A is abelian.

Theorem 3.10. Let T : A ⟶ B be a linear isometry between C*-algebras and let A be abelian. Let p_{T} ∈ B^{∗∗} be the structure projection of T. Then we have

∥(T a)p_{T} ∥ = ∥a∥ (a ∈ A).

Proof. Let q ∈ B^{∗∗} be the atomic projection, determined by T, in Lemma 3.7. We show that T (·)q is a triple homomorphism from A = C_{0}(X) onto T (A)q. Let φ ∈ |Q| ∩ P (B) with φ = |ψ| for some ψ ∈ Q. Let ψ = v^{∗}φ be the polar decomposition. By Lemma 3.9, we have

(T f^{(3)})ω_{φ} = f^{(3)}(σ(φ))vω_{φ} = f(σ(φ))\overline{f(σ(φ))}f(σ(φ))vω_{φ} = (T f)^{(3)}ω_{φ}.

Hence, by the definition of q, we have

(T f^{(3)})q = (T f)^{(3)}q

for every f in C_{0}(X), and hence the map T (·)q is a triple homomorphism. On the other hand, using Lemma 3.9 again, we get

(T g)^{∗}(T f)ω_{φ} = \overline{g(σ(φ))}f(σ(φ))ω_{φ}

which gives q(T g)^{∗}(T f)ω_{φ} = (T g)^{∗}(T f)ω_{φ} since qω_{φ} = ω_{φ}. Therefore q(T g)^{∗}(T f)q = (T g)^{∗}(T f)q and q commutes with (T g)^{∗}(T f) for all f, g in C_{0}(X). It follows that q satisfies condition (ii) in Proposition 2.2 and so q ≤ p_{T} by maximality of p_{T}. By Lemma 3.7, T (·)q is an isometry which implies that T (·)p_{T} is such also.

Remark 3.11. When B is abelian, Theorem 3.10 gives a result of Holsztynski [8, 9] as a special case.

Given any element a in a C*-algebra or, more generally, a JB*-triple A, the (closed) subtriple Z_{a} of A generated by a is linearly isometric (and hence triple isomorphic) to an abelian C*-algebra [11, Corollary 1.15]. Applying the above theorem to the restriction of a linear isometry to Z_{a}, we obtain the following local result on linear isometries between C*-algebras.

Corollary 3.12. Let T : A ⟶ B be a linear isometry, where A is a JB*-triple and B is a C*-algebra. Then for every a ∈ A, there is a largest projection p_{a} ∈ B^{∗∗}, which is closed, such that T (·)p_{a} : Z_{a} ⟶ B^{∗∗} is an isometry and a triple homomorphism satisfying

T {x, y, z}p_{a} = {T x, T y, T z}p_{a}

for all x, y, z ∈ Z_{a}.

Remark 3.13. (a) Clearly, p_{T} ≤ p_{a}, but it can happen that p_{T} ≠ p_{a} = 111. In Example 2.1, we have p_{T} ≠ 111 and if a ∈ C(Ω) satisfies a(0) = a(1) = 0, then every b ∈ Z_{a} also satisfies b(0) = b(1) = 0 since {f ∈ C(Ω) : f(0) = f(1) = 0} is a (closed) subtriple of C(Ω) containing a. Therefore T restricts to a triple isomorphism on Z_{a}, in other words, p_{a} = 111.

(b) The condition T {a, a, a} = {T a, T a, T a} alone need not imply that p_{a} = 111. This amounts to saying that the condition T (a^{(3)}) = T (a)^{(3)} need not imply T (a^{(2n+1)}) = (T a)^{(2n+1)} for all n. Consider the unital isometry T in Example 2.6 and the function

f(x) = \frac{25}{4} − \frac{63}{4}x^{2}

in C[0, 1]. A simple calculation gives

(T f)(2) = ∫_{0}^{1} f(x)dx = 1

and

T (f^{(3)})(2) = ∫_{0}^{1} f^{(3)}(x)dx = ∫_{0}^{1} (\frac{25}{4} − \frac{63}{4}x^{2})^{3} dx = 1.

Therefore, we have T (f^{(3)}) = (T f)^{(3)}, but T (f^{(5)}) ≠ (T f)^{(5)} since

T (f^{(5)})(2) = ∫_{0}^{1} f^{(5)}(x)dx = −\frac{20959168}{11264} ≠ 1 = (T f)^{(5)}(2).

In the proof of Theorem 3.10, the two maps T (·)q and T (·)p_{T} are actually equal if B is a dual C*-algebra. We show this in the next proposition as well as giving an exact formula relating q and p_{T}.

A C*-algebra B is called a dual C*-algebra if I^{⊥⊥} = I for all closed one-sided ideals I of B, where for any closed left (resp. right) ideal I (resp. J) of B, we define I^{⊥} = {b ∈ B : Ib = {0}} (resp. J^{⊥} = {b ∈ B : bJ = {0}}). It is known that a C*-algebra B is dual if and only if every maximal abelian subalgebra of B is generated by minimal projections, or equivalently, B is a c_{0}-sum of algebras of compact operators on Hilbert spaces (cf. [19, p.157]). Therefore, a unital dual C*- algebra is finite-dimensional. Given a dual C*-algebra B, the minimal projections in B are also minimal in B^{∗∗}, and every singular state of B^{∗∗} vanishes on B.

Given b in B^{∗∗}, we denote by r(b) the right support projection of b which is the smallest projection in B^{∗∗} satisfying br(b) = b. If T is a linear isometry from a C*-algebra A into B, then for the partial isometry T^{∗∗}(111)p_{T}, we have r(T^{∗∗}(111)p_{T}) = p_{T} T^{∗∗}(111)^{∗}T^{∗∗}(111)p_{T}.

Proposition 3.14. Let p_{T} be the structure projection of T : A ⟶ B in Theorem 3.10 and q the projection in its proof. Let B be a dual C*-algebra. Then we have

(i) T (·)p_{T} = T (·)q;

(ii) q is the right support projection of T^{∗∗}(111)p_{T};

(iii) p_{T} = q + 111 − r(T A) where r(T A) = ⋁{r(T (a)) : a ∈ A}.

Proof. (i) We note that q ≤ p_{T} from the proof of Theorem 3.10. Let z = p_{T} − q. We show that T (·)z = 0. Suppose otherwise. Then T (·)z : A ⟶ T (A)z is a nonzero triple homomorphism as T (a^{(3)})z = T (a^{(3)})p_{T} z = (T a)^{(3)}p_{T} z = (T a)^{(3)}z, and z commutes with T (a)^{∗}T (a) because p_{T} and q do. Hence the quotient A/ ker T (·)z is isometrically triple isomorphic to T (A)z. If we identify A with C_{0}(X), then A/ ker T (·)z identifies with C_{0}(Y), where Y is a nonempty closed subset of X and the quotient map is just the restriction map. Pick y ∈ Y. Applying Lemma 3.2 to the isometry C_{0}(Y) ⟶ T (A)z ⊆ B^{∗∗}, we find an extreme point ψ in (B^{∗∗})_{1}^{∗} such that ψ((T f)z) = 1 whenever f ∈ C_{0}(X) satisfies f(y) = ∥f∥ = 1. Let ψ = v^{∗}|ψ| be the polar decomposition with v ∈ B^{∗∗∗∗}. Then |ψ| is a pure state of B^{∗∗} and |ψ|(z) = 1 by Schwarz inequality. Hence

|ψ|(q) = |ψ|(qz) = 0.

We note that |ψ|((T f)^{∗}T f) = 1 since 1 = |ψ|(v^{∗}(T f)z) = |ψ|(v^{∗}T f) ≤ |ψ|((T f)^{∗}T f) ≤ 1. It follows that |ψ| is a pure normal state of B^{∗∗} as it does not vanish on B and a pure state is normal or singular. Therefore ψ is normal on B^{∗∗} since B^{∗} = B^{∗∗∗}z_{0} for some central projection z_{0} in B^{∗∗∗∗} (cf. [19, p. 126]) and we have ψz_{0} = v^{∗}|ψ|z_{0} = v^{∗}|ψ| = ψ. Therefore |ψ| ∈ |Q_{y}| ∩ P (B) because ψ((T f)(111 − z)) = |ψ|(v^{∗}(T f)(111 − z)) = 0 yields ψ(T f) = ψ((T f)z) = 1 for f ∈ A_{y}. It follows that |ψ|(q) = 1, by the definition of q, which gives a contradiction.

(ii) By weak* continuity and Lemma 3.9, we have

T^{∗∗}(111)^{∗}T^{∗∗}(111)ω_{φ} = ω_{φ}, ∀φ ∈ |Q|.

Therefore

T^{∗∗}(111)^{∗}T^{∗∗}(111)q = q

and

p_{T} T^{∗∗}(111)^{∗}T^{∗∗}(111)p_{T} = (T^{∗∗}(111)p_{T})^{∗}(T^{∗∗}(111)p_{T}) = (T^{∗∗}(111)q)^{∗}(T^{∗∗}(111)q) = q.

(iii) Since T (A)z = 0, we have

p_{T} − q = z ≤ 111 − r(T A).

On the other hand, since T (·)(111 − r(T A)) = 0, we have

111 − r(T A) ≤ p_{T} and q(111 − r(T A)) = 0

which gives

p_{T} = q + 111 − r(T A).

The use of dual C*-algebras in Proposition 3.14 hints at the atomic property of B^{∗∗} and a general formulation of the result, without any assumption on B, should relate the atomic part of p_{T} to q, as the following example shows.

Example 3.15. Let A = C_{0}(0, 1] and T : A ⟶ C[−1, 1] be the natural embedding, namely, T f agrees with f on (0, 1] and is zero elsewhere. Then we have p_{T} = 111, r(T A) = ⋁_{f∈A} T (f) = χ_{(0,1]} ∈ C[−1, 1]^{∗∗} and q = z_{at}χ_{(0,1]} is in the atomic part of C[−1, 1]^{∗∗}, where z_{at} is the maximal atomic projection in C[−1, 1]^{∗∗}. We see, in this case, T (·)p_{T} z_{at} = T (·)q and p_{T} z_{at} = q + (111 − r(T A))z_{at}.

4 Isometries into abelian C*-algebras

Every C*-algebra can be embedded into an abelian C*-algebra by a linear isometry. It is therefore natural to consider isometries into abelian C*-algebras. We begin with a description of the structure projection.

Proposition 4.1. Let T : A ⟶ B be a linear isometry between C*-algebras and let B be abelian. Then p_{T} = ⋀_{a∈A} p_{a} where p_{a} is the projection in Corollary 3.12.

Proof. Let p = ⋀_{a∈A} p_{a}. We only need to prove p_{T} ≥ p. For every a ∈ A, we have

T {a, a, a}p = T {a, a, a}p_{a}p = {T a, T a, T a}p_{a}p = {T a, T a, T a}p.

Since B is abelian, T (·)p : A ⟶ B^{∗∗} is a triple homomorphism. Hence p_{T} ≥ p by the maximality of p_{T} in Proposition 2.2.

By a character ρ of a C*-algebra A, we mean an algebra homomorphism ρ : A ⟶ ℂ\{0}. It is clear that the algebra M_{2} does not have a character. Also, a C*-algebra is abelian if, and only if, its pure states are all characters.

Lemma 4.2. Let N be a von Neumann algebra. Then N has a weak* continuous character if, and only if, N contains an abelian summand.

Proof. The sufficiency is obvious. Suppose N has a weak* continuous character ρ. Then N must contain a type I summand N_{I} for otherwise, the ‘Halving Lemma’ implies that N is of the form D⊗M_{2} (cf. [19, Proposition V.1.22]) and the restriction of ρ to 111⊗M_{2} is a character which is impossible. Since N_{I} is of the form ∑_{k} N_{k}⊗B(H_{n_{k}}) where N_{k} is abelian and B(H_{n_{k}}) is a type I_{n_{k}}-factor, N_{I} must contain an abelian summand because the contrary would imply ρ|_{N_{I}} = 0 and ρ = 0.

The above lemma implies that a C*-algebra A has a character if, and only if, A^{∗∗} contains an abelian summand. We show below that this condition is equivalent to the non-triviality of the map T (·)p_{T} if T is a linear isometry from A into an abelian C*-algebra B.

Proposition 4.3. Let T : A ⟶ B be a linear isometry between C*-algebras where B is abelian. Let p_{T} ∈ B^{∗∗} be the structure projection of T. Then

(i) T (·)p_{T} is an isometry if, and only if, A is abelian.

(ii) T (·)p_{T} ≠ 0 if, and only if, A admits a character.

Proof. (i) The necessity is obvious since T (A)p_{T} is an abelian JB*-triple. The sufficiency follows from Theorem 3.10.

For (ii), we first assume that T (·)p_{T} ≠ 0. Then there exists a character ρ of B^{∗∗} which does not vanish on T (A)p_{T}, and hence the composite ρ ∘ (T (·)p_{T}) : A ⟶ ℂ is a non-zero triple homomorphism. Since the closed triple ideals of C*-algebras are algebra ideals, it follows that A⧸ker ρ ∘ (T (·)p_{T}) is a one-dimensional C*-algebra and the natural quotient map ρ̃: A ⟶ A⧸ker ρ ∘ (T (·)p_{T}) is a character of A.

Conversely, let η be a character of A and let B = C_{0}(Y) for some locally compact Hausdorff space Y. Then η is a pure state of A. Since the extreme points in the closed unit ball of T (A)^{∗} can be extended to the extreme points in the closed unit ball of C_{0}(Y)^{∗}, we have η = T^{∗}(λδ_{y}|_{T (A)}) for some y in Y and |λ| = 1 where T^{∗} : T (A)^{∗} ⟶ A^{∗} is an isometry. The support projection p_{δ_{y}} ∈ C_{0}(Y)^{∗∗} of δ_{y} is a minimal projection and we have λT (a^{(3)})p_{δ_{y}} = λT (a^{(3)})(y)p_{δ_{y}} = η(a^{(3)})p_{δ_{y}} = η(a)^{(3)}p_{δ_{y}} = λT (a)^{(3)}p_{δ_{y}} for all a in A. Therefore p_{δ_{y}} ≤ p_{T} by maximality of p_{T}, and thus T (·)p_{T} ≠ 0.

Remark 4.4. Let A, B and T be as in Proposition 4.3. If A has a character, then we actually have

∥T (a)p_{T} ∥ = sup{|η(a)| : η is a character of A},

which gives an alternative proof of the sufficiency in (i). The identity follows from

∥T (a)p_{T} ∥ = sup{|ρ(T (a)p_{T})| : ρ is a character of B^{∗∗}}
= sup{|ρ̃(a)| : ρ is a character of B^{∗∗}}
≤ sup{|η(a)| : η is a character of A},

where ρ̃ is the quotient map A ⟶ A⧸ker ρ ∘ (T (·)p_{T}) and the last term is at most ∥T (a)p_{T} ∥ from the proof of (ii).

The result of Proposition 4.3 does not hold if B is nonabelian. In Example 2.5, we have T (·)p_{T} ≠ 0 for some linear isometry T : M_{2} ⟶ M_{3}. We conclude with the following example.

Example 4.5. There is a linear isometry T : M_{2} ⟶ B(H), where B(H) is the algebra of bounded operators on an infinite dimensional separable Hilbert space H, such that T (·)p_{T} = 0.

To see this, let Y be the closed unit ball of M_{2}^{∗} and j be the canonical linear embedding of M_{2} into C(Y). Take a faithful nondegenerate representation π of C(Y) on a separable Hilbert space H. Then T = π ∘ j is a linear isometry from M_{2} into B(H). By Remark 2.9 and Proposition 4.3, we have T (·)p_{T} = T (·)p_{j} = 0.

References

[1] L. G. Brown, Semicontinuity and multipliers of C*-algebras, Can. J. Math., XL, no. 4 (1988), 865–988.

[2] C-H. Chu, Jordan structures in Banach manifolds, Studies in Adv. Math. 20, Amer. Math. Soc. (2001), 201 -210.

[3] C-H. Chu, T. Dang, B. Russo and B. Ventura, Surjective isometries of real C*- algebras, J. London Math. Soc. 47 (1993), 97–118.

[4] T. Dang, Y. Friedman and B. Russo, Affine geometric proofs of the Banach Stone theorems of Kadison and Kaup, Rocky Mountain J. Math. 20 (1990), 409–428.

[5] E.G. Effros, Order ideals in a C*-algebra and its dual, Duke Math. J. 30 (1963) 391-417.

[6] L.A. Harris, Bounded symmetric homogeneous domains in infinite dimensional spaces, in Proceedings on infinite holomorphy, (Lecture Notes in Math. 364 Springer-Verlag, Berlin 1974) 13-40.

[7] L.A. Harris, A generalization of C*-algebras, Proc. London Math. Soc. 42 (1981) 331–361.

[8] W. Holsztynski, Continuous mappings induced by isometries of spaces of continuous functions, Studia Math. 26 (1966), 133-136.

[9] J-S. Jeang and N-C. Wong, Weighted composition operators of C_{0}(X)’s, J. Math. Anal. Appl. 201 (1996), 981–993.

[10] R. V. Kadison, Isometries of operator algebras, Ann. of Math. 54 (1951), 325– 338.

[11] W. Kaup, A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces, Math. Z. 138 (1983), 503–529.

[12] K. Morita, Analytical characterization of displacements in general Poincaré space, Proc. Imper. Acad. Tokyo 17 (1941) 489-494.

[13] A. L. T. Paterson and A. M. Sinclair, Characterization of isometries between C*-algebras, J. London Math. Soc., 5 (1972), 755–761.

[14] G. K. Pedersen, C*-algebras and their automorphism groups, London Mathematical Society Monographs 14, Academic Press, London-New York, 1979.

[15] R.T. Prosser, On the ideal structure of operator algebras, Memoir Amer. Math. Soc. 45 (1963).

[16] A. Rodriguez Palacios, Isometries and Jordan-isomorphisms onto C*-algebras, J. Operator Theory, 40 (1988), 71–85.

[17] A. Rodriguez Palacios, Jordan structures in analysis, Proceedings of Oberwolfach conference (1992) on Jordan algebras, Walter de Gruyter, Berlin (1994) 97-186.

[18] B. Russo, Structures of JB*-triples, Proceedings of Oberwolfach conference (1992) on Jordan algebras, Walter de Gruyter, Berlin (1994) 208-280.

[19] M. Takesaki, Theory of operator algebras I, Springer-Verlag, Berlin, 1979.

[20] H. Upmeier, Symmetric Banach manifolds and Jordan C*-algebras, North-Holland, Amsterdam, 1985.

Goldsmiths College University of London London SE14 6NW, England maa01chc@gold.ac.uk

Department of Applied Mathematics National Sun Yat-sen University Kaohsiung 80424, Taiwan, R.O.C. wong@math.nsysu.edu.tw