Nuclear elements in Banach Jordan pairs

Ottmar Loos Institut für Mathematik, Universität Innsbruck Technikerstr. 25/7, A-6020 Innsbruck, Austria

Abstract We introduce nuclear elements in Banach Jordan pairs, generalizing the nuclear elements in Banach algebras. We also define a notion of metric and bounded approximation property in Banach Jordan pairs and show that the trace form Tr introduced in [3] may be extended to the nuclear elements, assuming the bounded approximation property.

1 Nuclear elements

Let V = (V^{+}, V^{−}) be a complex Banach Jordan pair. Denoting the norm by |x|, it thus satisfies the inequality |{xyz}| ≤ 2|x||y||z| and hence |Q_{x}y| = |\frac{1}{2}\{xyx\}| ≤ |x|^{2}|y|. Let Soc V = (S^{+}, S^{−}) be the socle of V. We recall from [3, sec. 1] that the socle is the span of the reduced elements, where an element u ∈ V^{±} is reduced if Q_{u}V^{∓} ⊂ Cu. Thus a nonzero element is reduced if it is either trivial (Q_{u} = 0) or of rank 1 (Q_{u}V^{∓} = Cu). The socle is an ideal of V, stable under structural transformations. An element x ∈ V^{+} is called nuclear if there exists a sequence (u_{n}) of reduced elements such that

∑_{1}^{∞} |u_{n}| < ∞  (1)

and

x = ∑_{1}^{∞} u_{n}.  (2)

Note that the series (2) is convergent since V^{+} is complete. Let N^{+} = N^{+}(V) be the set of nuclear elements in V^{+} and define N^{−} ⊂ V^{−} analogously. The nuclear norm ν(x) of a nuclear element x is the infimum of the sums (1) taken over all representations of x as in (2). It is easy to see that N^{±} is a vector space containing S^{±}, and ν is a norm on N^{±} satisfying ν(u) = |u| for reduced u and ν(x) ≥ |x| for all nuclear x.

The following propositions are extensions to the Jordan Banach setting of well-known results for operators.

Proposition 1. (a) N^{±} is complete in the nuclear norm.

(b) If x ∈ N^{±} is represented as in (2) then the series converges to x in the nuclear norm. Hence N^{±} may be identified with the completion of S^{±} with respect to the nuclear norm.

Proof. (a) By a standard result, it suffices to show: If (x_{n}) is a sequence in N^{±} with the property that ∑_{1}^{∞} ν(x_{n}) < ∞ then the series ∑_{1}^{∞} x_{n} converges in the nuclear norm to an element x ∈ N^{±}. For each n, choose representations x_{n} = ∑_{j} u_{nj} where u_{nj} is reduced and ∑_{j} |u_{nj}| ≤ ν(x_{n}) + 2^{−n}. Then

∑_{j,n} |u_{nj}| ≤ ∑_{n} (ν(x_{n}) + 2^{−n}) = ( ∑_{n} ν(x_{n})) + 1 < ∞.

Hence the double series x = ∑_{n,j} u_{nj} converges absolutely in V^{±}, and therefore x ∈ N^{±}. It remains to show that x = lim_{n→∞} ∑_{1}^{n} x_{i} in the nuclear norm. Now

x − ∑_{i=1}^{n} x_{i} = ∑_{k>n} x_{k} = ∑_{k>n} ∑_{j} u_{kj}

is a representation as in (2), and therefore

ν(x − ∑_{i=1}^{n} x_{i}) ≤ ∑_{k>n} ∑_{j} |u_{kj}| ≤ ∑_{k>n} (ν(x_{k}) + 2^{−k}) → 0

for n → ∞, because the numerical series ∑_{1}^{∞} (ν(x_{n}) + 2^{−n}) converges.

(b) Let x = ∑_{1}^{∞} u_{n} ∈ N^{±} be represented as a series of reduced elements with ∑_{1}^{∞} |u_{n}| < ∞. Then ν(x_{n}) = |u_{n}| so by completeness of N^{±}, the series converges to x in the nuclear norm.

Proposition 2. N (V) = (N^{+}, N^{−}) is an ideal of V containing the socle. If (φ, ψ) : V ⇀ ↽ W is a continuous structural transformation between Banach Jordan pairs then φ(N^{+}(V)) ⊂ N^{+}(W) and

ν(φ(x)) ≤ ∥φ∥ν(x),  (3)

for all x ∈ N^{+}(V), and similarly for ψ: W^{−} → V^{−}. Here ∥φ∥ denotes the operator norm of φ. In particular,

ν(Q_{a}y) ≤ |a|^{2}ν(y),
ν(B(a, b)x) ≤ (1 + |a||b|)^{2}ν(x),  (4)

for (a, b) ∈ V^{σ} × V^{−σ} and (x, y) ∈ N^{σ} × N^{−σ}, σ ∈ {+, −}.

Proof. Let (φ, ψ) be a continuous structural transformation, let ε > 0 and let x = ∑ u_{n} ∈ N^{+} as in (2) with ν(x) + ε ≥ ∑_{1}^{∞} |u_{n}|. Then φ maps reduced elements to reduced elements [4, Th. 1.3], and by continuity,

∑_{1}^{∞} |φ(u_{n})| ≤ ∑_{1}^{∞} ∥φ∥ · |u_{n}| ≤ ∥φ∥(ν(x) + ε).

Hence φ(x) = ∑_{1}^{∞} φ(u_{n}) ∈ N^{+}(W), and (3) follows for ε → 0. By applying this to to the inner structural transformations (B(a, b), B(b, a)) : V ⇀ ↽ V and (Q_{a}, Q_{a}) : V^{op} → V we obtain (4) and (4). From structural stability it follows that N (V) is an outer ideal and therefore an ideal since the characteristic of the base field is not 2.

Remark. In [6, Th. 5.4], Puhl proves that, for an associative Banach algebra and a nuclear element x, the operator Q_{x} is a nuclear operator. The corresponding statement for Banach Jordan pairs (or even algebras) is false. As an example, let V = (X, X) be the Jordan pair determined by a continuous nonzero quadratic form q on an infinite dimensional Banach space X with Q_{x}y = b(x, y)x − q(x)y, where b(x, y) = q(x + y) − q(x) − q(y) is the associated bilinear form. Then V = Soc V and for an element x with q(x) ≠ 0 the operator Q_{x} is invertible. Since X is infinite dimensional, Q_{x} is not compact and therefore not nuclear.

2 Approximation properties and extension of the trace form

Let V be a Banach Jordan pair. We say V has the metric approximation property (= m.a.p.) for V^{+} if for every s ∈ S^{+} and every ε > 0 there exist elements (a, b) ∈ Soc V such that

B(a, b)s = 0 and |a||b| ≤ 1 + ε.  (5)

The m.a.p. for V^{−} is defined in the same way. We say V has the m.a.p. if it has the m.a.p. for both V^{+} and V^{−}.

V is said to have the bounded approximation property (= b.a.p.) for V^{+}, if there exists a constant C > 0 with the following property: For all s ∈ S^{+} there exist (a, b) ∈ Soc V such that

B(a, b)s = 0 and |a| |b| ≤ C.  (6)

The b.a.p. for V^{−} and for all of V is defined as in case of the m.a.p. Clearly the m.a.p. implies the b.a.p.

We remark that the requirement (a, b) ∈ Soc V may be relaxed to a ∈ S^{+} or b ∈ S^{−}. Indeed, suppose for all s ∈ S^{+} there exists (a, b) in S^{+} × V^{−} or in V^{+} × S^{−} such that (6) holds. Then also B(a, Q_{b}a)s = B(Q_{a}b, b)s = B(a, −b)B(a, b)s = 0 by the identity JP24 of [2], and Q_{b}a resp. Q_{a}b is in the socle since the socle is an ideal. Also, |a| |Q_{b}a| ≤ |a| |b|^{2} |a| ≤ C^{2}. Similar remarks apply to the m.a.p.

Example. Let X be a Banach space and let V^{±} = L(X) (bounded linear operators) with Q_{x}y = xyx (composition of linear maps). It is well known [5] that if X has the m.a.p. (in the usual sense) then for every ε > 0 and every finite rank operator s ∈ S^{+} there exists a finite rank operator b ∈ L(X) such that |b| ≤ 1 + ε and bs = s. By the remark made above, this implies that V has the m.a.p. in the Jordan sense, because B(1, b)s = (1 − b)s(1 − b) = 0. Also, if the dual X^{∗} has the m.a.p. then there exists a finite rank operator a ∈ L(X) with |a| ≤ 1 + ε and sa = a. Again this implies the m.a.p. in the Jordan sense because of B(a, 1)s = (1 − a)s(1 − a) = 0. Analogous statements hold for the b.a.p.

Lemma. (a) Let u ∈ V^{+} be reduced and let y ∈ V^{−}. Then

| Tr(u, y)| ≤ |u||y|.  (7)

Similarly,

| Tr(x, v)| ≤ |x||v|  (8)

for all reduced elements v ∈ V^{−} and all x ∈ V^{+}.

(b) For fixed s ∈ S^{+}, the linear form y ↦ Tr(s, y) on V^{−} is continuous. Similarly, for fixed t ∈ S^{−} the linear form x ↦ Tr(x, t) on V^{+} is continuous.

Proof. (a) Since u is reduced, Q_{u}y = Tr(u, y)u whence | Tr(u, y)||u| = |Q_{u}y| ≤ |u|^{2}|y|, showing (7). The second formula follows in the same way.

(b) Write s = ∑_{1}^{n} u_{i} as a finite sum of reduced elements. Then by bilinearity of the trace form and (7),

| Tr(s, y)| ≤ ∑_{1}^{n} | Tr(u_{i}, y)| ≤ ( ∑_{1}^{n} |u_{i}|) · |y|.

The proof of the second statement is analogous.

Extension Theorem. (a) Let V be a Banach Jordan pair and suppose V has the bounded approximation property for V^{+}. Then the trace form Tr : S^{+} × V^{−} → C may be extended by continuity to a bilinear form, again denoted Tr : N^{+} × V^{−} → C, which satisfies the inequality

| Tr(x, y)| ≤ ν(x) · |y|  (9)

for all x ∈ N^{+} and y ∈ V^{−}.

(b) Let (φ, ψ) : V ⇀ ↽ W be a continuous structural transformation between Banach Jordan pairs which satisfy the b.a.p. for V^{+} and W^{+}. Then

Tr(φ(x), y) = Tr(x, ψ(y))  (10)

for all x ∈ N^{+}(V) and all y ∈ W^{−}.

Proof. (a) Let s ∈ S^{+}, y ∈ V^{−}, and ε > 0. Choose reduced elements u_{n} such that s = ∑ u_{n} and ∑ |u_{n}| ≤ ν(s) + ε. Also choose (a, b) as in (6). Then by the properties of the trace form [3, Th. 2.8],

0 = Tr(B(a, b)s, y) = Tr(s, B(b, a)y) = Tr(s, y) − Tr(s, {bay} + Q_{b}Q_{a}y)

and therefore Tr(s, y) = Tr(s, t) where t = {bay} − Q_{b}Q_{a}y ∈ S^{−} satisfies the estimate

|t| ≤ |{bay}| + |Q_{b}Q_{a}y| ≤ (2|b| |a| + |b|^{2} |a|^{2})|y| ≤ (2C + C^{2}) |y| = C′ |y|, (11)

independent of s.

By part (b) of the Lemma, the linear form x ↦ Tr(x, t) on V^{+} is continuous. Hence we obtain

| Tr(s, y)| = | Tr(s, t)| = | ∑_{n} Tr(u_{n}, t)|
≤ ∑_{n} | Tr(u_{n}, t)| ≤ ∑_{n} |u_{n}| |t|  (by (7))
= ( ∑_{n} |u_{n}|) |t| ≤ (ν(s) + ε) |t|
≤ C′(ν(s) + ε) |y|.  (by (11))

Since ε > 0 was arbitrary, this implies | Tr(s, y)| ≤ C′ν(s) |y|. Now N^{+} is the completion of S^{+} with respect to the nuclear norm by Proposition 1(a). Hence the linear form s ↦ Tr(s, y) on S^{+} extends uniquely to a continuous linear form Φ_{y}: N^{+} → C. We prove (9). Let x ∈ N^{+} and ε > 0, and write x = ∑ u_{n} where the u_{n} are reduced and ∑_{n} |u_{n}| ≤ ν(x) + ε. Then by continuity of Φ_{y} and since x = ∑_{n} u_{n} in the nuclear norm by Proposition 1(b)

|Φ_{y}(x)| = | ∑_{n} Tr(u_{n}, y)|
≤ ∑ | Tr(u_{n}, y)| ≤ ∑ |u_{n}| |y|  (by (7))
≤ (ν(x) + ε) |y|,

so that (9) follows for ε → 0. Finally, linearity of Φ_{y} in y is an easy consequence of bilinearity of the trace form on S^{+} × V^{−} and continuity.

(b) By [3, Th. 2.8(b)], we have Tr(φ(s), y) = Tr(s, ψ(y)) for all s ∈ Soc V^{+} and y ∈ W^{−}. Since φ: N^{+}(V) → N^{+}(W) is continuous in the nuclear norm by (3), (10) follows by continuity.

3 Some open questions

1. Optimal constants in the b.a.p. If V satisfies the b.a.p. with a constant C then it does so for any C′ ≥ C. Hence it is of interest to find lower bounds for the possible values of C. A rather coarse such lower bound is

C ≥ \sqrt{2} − 1.  (12)

Indeed, let 0 ≠ s ∈ Soc V^{+} and (a, b) ∈ Soc V with B(a, b)s = 0. Then s = {abs} − Q_{a}Q_{b}s and hence |s| ≤ |{abs}| + |Q_{a}Q_{b}s| ≤ (2|a||b| + |a|^{2}|b|^{2})|s| ≤ (2C +C^{2})|s|. This implies 1 ≤ 2C +C^{2} or 2 ≤ (1+C)^{2}, so that (12) follows. On the other hand, the lower bound for C in case of operators is 1. It is therefore natural to conjecture the same in the Jordan case.

2. Approximation properties for Jordan algebras. We say a Banach Jordan algebra J has the b.a.p. if there exist a constant C > 0 such that for every s ∈ Soc J there exists an element a ∈ Soc J with U_{1−a}s = 0 and |a| ≤ C. Note here that U_{1−a} = Id − V_{a} + U_{a} makes sense even if J does not have a unit element. Define the m.a.p. for J similarly. Then it is easy to see that one of the approximation properties for J implies the corresponding property for the Jordan pair (J, J). What about the reverse?

3. Banach spaces and Jordan approximation properties. Suppose that X is a Banach space and J = L(X), considered as a Jordan algebra. As in the Example in Section 2, the metric or bounded approximation property implies the same for J. Conversely, suppose J satisfies the m.a.p. or b.a.p. What does this mean in terms of the geometry of X? The same question can be asked for a pair (X, Y) of Banach spaces and the associated Jordan pair (L(X, Y), L(Y, X)).

4. Weaker conditions for extending the trace. The assumption of b.a.p. in the Extension Theorem may be too strong. For a semi-prime associative Banach algebra A, Puhl [6, 5.10] introduces a notion of quasi-approximation property (=q.a.p.) in terms of the approximation property of the Banach spaces Ap and pA where p is a minimal idempotent. He then shows [6, 5.11, 5.12] that for A commutative or a C^{∗}-algebra the q.a.p. holds, and that it is sufficient for extending the trace to the nuclear elements. In the Jordan case, the analogue of a minimal idempotent is a division idempotent e = (e_{+}, e_{−}). Since the Peirce 2-spaces are V_{2}^{±}(e) = Ce_{±}, the role of Ap and pA is essentially played by the Peirce 1-spaces V_{1}^{±}(e). It might be possible to use a similar idea in the Jordan case, involving approximation properties of the Banach spaces V_{1}^{±}(e).

5. Vanishing of the trace on nilpotent pairs. In [6, 5.12], Puhl also proves, under the assumption of the q.a.p., that the trace vanishes on nilpotent elements. In the Jordan case, it is an open problem whether the extended trace form Tr on N^{+}×V^{−} vanishes on nilpotent pairs. For the trace form on S^{+}×V^{−} this is true by [3, 2.8(f)].

6. The Lidskij trace formula. The famous Lidskij trace formula for operators expresses the trace of a nuclear operator on a Hilbert space as the sum of its spectral values counted with multiplicities. This was extended to nuclear elements of C^{∗}-algebras by Puhl in [6, 6.6]. It should therefore be possible to prove an analogue for JB^{∗}-triples. Note that for Jordan pairs, the trace Tr(x, y) for x in the socle and y arbitrary is indeed the (finite) sum of the spectral values of (x, y) counted with multiplicities, see [3, 3.8] and also [1] for the case of Banach Jordan algebras. A successful extension to the nuclear elements will most likely depend upon the solution of the problems in 4 and 5.

References

[1] B. Aupetit, A. Maouche, and H. Du T. Mouton, Trace and determinant in Jordan-Banach algebras (to appear).

[2] O. Loos, Jordan pairs, Lecture Notes in Math., vol. 460, Springer-Verlag, 1975.

[3] O. Loos, Trace and norm for reduced elements of Jordan pairs, Comm. Algebra 25 (1997), 3011–3042.

[4] K. McCrimmon, Reduced elements in Jordan triple systems, J. Algebra 97 (1985), 540–564.

[5] A. Pietsch, Operator ideals, North-Holland, 1980.

[6] J. Puhl, The trace of finite and nuclear elements in Banach algebras, Czechoslovak Math. J. 28 (1978), 656–676.