On continuous Peirce decompositions, Schur multipliers and the perturbation of triple functional calculus

Jonathan Arazy ^{1}

Department of Mathematics University of Haifa Haifa 31905 Israel

jarazy@mathcs2.haifa.ac.il

Wilhelm Kaup ^{1}

Mathematisches Institut Universität Tübingen Auf der Morgenstelle 10 D-72076 Tübingen Germany

kaup@uni-tuebingen.de

1. Introduction

We illustrate our results with a simple matrix example: Let E: = C^{n×n} be the algebra of all complex n × n-matrices z = (z_{jk}) and denote by H ⊂ E the IR-linear subspace of all hermitian matrices z (i.e. satisfying z_{jk} = \overline{z}_{kj} for all j, k). Then E = H ⊕ iH and every diagonal matrix in H has real entries. Now fix a function f: IR → IR. Then to every z ∈ H there is associated a matrix f(z) ∈ H in a well known way: In case d ∈ H is diagonal, f(d) is the diagonal matrix obtained from d by replacing all diagonal entries by their f-images. In case z ∈ H is arbitrary, write z = udu^{∗} for a unitary matrix u ∈ E and a diagonal matrix d ∈ H, then f(z): = uf(d)u^{∗} does not depend on the choice of u, d for z. Another way of saying this is that z and f(z) have the same eigen vectors in C^{n} but the corresponding eigen values are f-images of each other.

By the above functional calculus, the scalar function f induces a matrix function f: H → H that we also denote by f_{H} for better distinction. It is well known that, for instance, f_{H} has continuous derivatives if the function f has the same property. Then for every a ∈ H the derivative df_{H}(a) as operator on H can be described in the following way: For every j, k denote by H_{jk} the subspace of all z ∈ H with z_{lm} = 0 if {j, k} ≠ {l, m}. If we assume without loss of generality that a is a diagonal matrix with diagonal entries λ_{j}: = a_{jj}, the derivative df_{H}(a) is multiplication by the divided difference m_{f}(λ_{j}, λ_{k}) on every H_{jk}, where

m_{f}(x, y) = \frac{f(x) − f(y)}{x − y} if x ≠ y  and  = f′(x) otherwise.

In particular, the derivative df_{H}(a) is a Schur multiplier on H, i.e. a linear operator L on H such that L(z) = (h_{jk}z_{jk}) for a (necessarily symmetric) matrix h ∈ IR^{n×n}. The space of all Schur multipliers on H clearly is a commutative real algebra. It is easily seen that a linear subspace F ⊂ H is invariant under all f_{H}, f: IR → IR arbitrary, if and only if F ⊂ E is closed under the Jordan product x ∘ y: = (xy + yx)/2. This indicates that functional calculus and Jordan theory are related.

Besides the (hermitian operator) functional calculus there is another calculus that applies to all of E, but then f: IR → IR has to be an odd function: For every real diagonal matrix d define f(d) as above. For z ∈ E arbitrary write z = udv for unitary matrices u, v and a real diagonal matrix d, then put f(z): = uf(d)v, which does not depend on the choice of u, d, v for z, since by assumption f is odd. The induced map f = f_{E}: E → E also is odd and coincides with f_{H} on H. While f_{H} relies on the eigen values of its arguments, these are the singular values in case of f_{E}.

1 Supported by a grant from the German-Israeli Foundation (GIF), I-0415-023.06/95.

For instance, if f is defined by f(0) = 0 and f(t) = 1/t for t ≠ 0 then for every matrix z ∈ E the matrix f(z)^{∗} is the Penrose pseudo-inverse of z. Furthermore, a (complex) linear subspace F ⊂ E is invariant under all transformations f_{E}, f odd, if and only if F is closed under the Jordan triple product {xyz}: = (xy^{∗}z + zy^{∗}x)/2 on E, compare the more general Remark 3.1.

Again, the derivatives of f_{E} have a nice description in case the odd function f is of class C^{1}: For every 1 ≤ j ≤ k ≤ n put E^{j,k}: = H_{jk} and E^{j,−k}: = iH_{jk}, so that E = ⨁_{j≤|k|} E^{j,k}. Then, if a is the real diagonal matrix with diagonal entries λ_{j} as above and if we put λ_{−k}: = −λ_{k} for all k, then df_{H}(a) is multiplication by m_{f}(λ_{j}, λ_{k}) on E^{j,k} for all 1 ≤ j ≤ |k| ≤ n.

Both calculi extend to infinite dimensions where they have been studied by various authors, compare f.i. [1], [3], [4], [6], [15], [17], [19]. For the first calculus, H is the space of all bounded self-adjoint operators on an arbitrary complex Hilbert space H or – more generally – H can be an arbitrary unital JB-algebra in the sense of [9]. In contrast to the finite dimensional case the operator function f_{H} is not differentiable in general even if the scalar function f is of class C^{1}, see [6] for an explicit counter example. The second calculus extends to a large class of complex Banach spaces, the so called JB*-triples. These are precisely those complex Banach spaces E for which the open unit ball D is homogeneous under biholomorphic transformations, or what amounts to the same, for which D is a bounded symmetric domain. All these Banach spaces come with a natural Jordan triple product, and there are much more examples than the algebra L(H) of all bounded linear operators on a complex Hilbert space H. Since the function f_{E}: E → E is given in terms of the triple product we speak of the triple functional calculus in this case.

In this paper we study for odd C^{1}-functions f: IR → C differentiability of f_{E} at a given point a in the JB*-triple E. As main tool we define a Peirce spectrum Σ(a) ∈ IR^{2} and with it the notion of Schur multiplier on E. Every Schur multiplier has a symbol (that is, a certain continuous function on the compact space Σ(a)) and is uniquely determined by its symbol. We establish various cases where the derivative df_{E}(a) exists and is a Schur multiplier with the divided difference m_{f} as symbol. Finally, in Proposition 7.10 we show that the class of all odd C^{1}-functions on IR giving a C^{1} triple functional calculus on every JB*-triple coincides with the class of all odd C^{1}-functions giving a C^{1} hermitian operator functional calculus on every complex Hilbert space.

In all previous papers on Schur multipliers and operator differentiability the authors use spectral resolution of the element a ∈ E and this restricts the theory to the case where E has a predual, i.e. is a JBW*-triple. Our results are consistent with those of [1], although the two papers have slightly different points of view: In [1] the authors use divided differences and divided sums, which act as ’Schur multipliers’ on the symmetric and the anti-symmetric parts respectively. Here we use only divided differences, but they are defined on a bigger space – the Peirce spectrum Σ(a).

In [12] the notion of a real JB*-triple has been introduced and studied. Most of our results carry over to these more general structures.

Notation. For complex Banach spaces E, F we denote by L(E, F) the Banach space of all bounded linear operators E → F and by E^{∗} = L(E, C) the dual of E. Furthermore, L(E): = L(E, E) is the Banach algebra of all continuous endomorphisms and H(E) ⊂ L(E) is the IRlinear subspace of all hermitian operators, i.e. of all operators λ with ∥ exp(itλ)∥ = 1 for every t ∈ IR. The group of all invertible operators in L(E) is denoted by GL(E). The identity of GL(E) will be denoted by 11_{E} or simply by 11 if the reference to E is clear. The spectrum of an operator L ∈ L(E) will be denoted by σ(L) ⊂ C, and σ′(L): = σ(L) ∪ {0} is the extended spectrum of L. For every set S and every compact topological space Ω let ℓ^{∞}(S, E) and C(Ω, E) be the Banach spaces of all bounded maps S → E and of all continuous maps Ω → E, where for S = Ω = ∅ these spaces are put to be {0}. In case E = C we simply write ℓ^{∞}(S) and C(Ω). By L_{IR}(E) we denote the complex Banach algebra of all bounded IR-linear operators E → E. Every L ∈ L_{IR}(E) has a unique representation L = L^{+} + L^{−} with L^{+} complex linear and L^{−} conjugate linear, i.e. L^{ε}(iz) = εiL^{ε}(z) for ε = ±. Finally, we denote by E_{IR} the real Banach space underlying E.

2. JB*-triples and the triple spectrum

2.1 Definition A complex Banach space E together with a continuous map

E × E × E → E, (x, y, z) ↦ {xyz}

is called a JB*-triple (compare [14]), if for all a, b, x, y, z ∈ E the following conditions hold

(i) {xyz} is symmetric bilinear in the outer variables (x, z) and conjugate linear in the inner variable y.

(ii) {ab{xyz}} = {{abx}yz} − {x{bay}z} + {xy{abz}}.

(iii) The operator on E defined by w ↦ {aaw} is hermitian, has spectrum ≥ 0 and norm ∥a∥^{2}.

For JB*-triples E, F a linear map L: E → F is called a (triple) morphism if

(2.2)  L{xyz} = {(Lx)(Ly)(Lz)} for all x, y, z ∈ E

is satisfied. In case, such an L is only IR-linear, we speak of a real morphism. It is known that every real morphism is continuous, induces an isometry E/ Ker(L) → F and, in particular, has as image a closed real subtriple of F.

Not every complex Banach space allows a triple product making it to a JB*-triple. However, if such a triple product exists on the Banach space E, it is unique. For instance, for every pair H, K of complex Hilbert spaces, the Banach space L(H, K) is a JB*-triple and the corresponding triple product is given by {xyz} = (xy^{∗}z + zy^{∗}x)/2. Also, for every compact topological space T, the Banach space C(T) of all continuous functions f: T → C endowed with the sup-norm and the triple product \{fgh\} = f\overline{g}h is a JB*-triple. In case, Λ is a compact subset of a topological vector space with Λ = −Λ, the closed linear subspace

(2.3)  C^{−}(Λ): = {f ∈ C(Λ) : f(−λ) = −f(λ) for all λ ∈ Λ}

is a JB*-subtriple of C(Λ).

Let E always be a JB*-triple in the following. For every a, b ∈ E the operators D(a, b) ∈ L(E) and Q(a, b) ∈ L_{IR}(E) are defined by z ↦ {abz} and z ↦ {azb} respectively. Also,

B(a, b): = 11 − 2D(a, b) + Q(a)Q(b)

is the Bergman operator. Furthermore, define D(a): = D(a, a), Q(a): = Q(a, a), B(a): = B(a, a) and P (a): = Q(a)^{2}. Finally, for subsets A, B ⊂ E put

D(A, B): = {D(a, b) : a ∈ A, b ∈ B}, D(A): = {D(a) : a ∈ A}

and the same for Q and B in place of D.

The elements a, b ∈ E are called (triple) orthogonal and we write a ⊥ b if D(a, b) = 0 (or equivalently, if D(b, a) = 0 ) holds, compare also Corollary 3.8. The element e ∈ E is called a tripotent if it satisfies {eee} = e. Every tripotent e ∈ E induces a Peirce decomposition

(2.4)  E = E_{1}(e) ⊕ E_{1/2}(e) ⊕ E_{0}(e)

where every Peirce space E_{k}(e) is the k-eigenspace of the operator D(e) in E. In particular, E_{1}(e) and E_{0}(e) are orthogonal subtriples of E. Since for all u ⊥ v in E the operators D(u), D(v) commute, there is a joint Peirce decomposition with respect to every finite family e_{1}, e_{2}, …, e_{r} of pairwise orthogonal tripotents in E, more precisely

(2.5)  E = ⨁_{0≤i≤j≤r} E_{i,j}

with

(2.6)  E_{i,j}: = {z ∈ E : 2D(e_{k})z = (δ_{ik} + δ_{kj})z for all 1 ≤ k ≤ r} .

Fix an arbitrary element a ∈ E in the following and denote by C = C_{a} ⊂ E the smallest closed (complex) subtriple containing a and by R = R_{a} the smallest closed real subtriple containing a. Then C = R ⊕ iR and there exists a unique compact subset Λ = −Λ ⊂ IR with 0 ∈ Λ together with a (surjective) triple isomorphism

ψ: C → C^{−}(Λ) satisfying ψ(a) = id_{Λ} ,

compare [16]. We call Λ the extended spectrum of a and denote it by Sp′(a). The closure of Λ\{0} in Λ is called the spectrum of a and is denoted by Sp(a). This can also be characterized as (comp. [16])

Sp(a) = \sqrt{σ(D(a)|_{C})} = \{t ∈ C : a ∉ (D(a) − t^{2}11)E\}

where σ denotes the usual spectrum of a Banach space operator and where \sqrt{T}: = \{s ∈ C : s^{2} ∈ T } is the set of square roots of a subset T ⊂ C. Besides the spectrum Sp(a) we also consider

SP(a): = \sqrt{σ(D^{(}a^{)})}

which we call the big spectrum of a. It contains Sp(a) and is contained in the real interval [−∥a∥, ∥a∥]. Actually, it can be shown that it is contained in the set of all ±\sqrt{(x^{2} + y^{2})/2} with x, y running in the extended spectrum Sp′(a) = Λ, compare [14], Theorem 3.3.

The element a ∈ E is called algebraic if there exists a decomposition

(2.7)  a = λ_{1}e_{1} + ⋯ λ_{r}e_{r}

(called spectral decomposition) with e_{1}, e_{2}, …, e_{r} a finite family of pairwise orthogonal tripotents in E and complex coefficients λ_{k}. For the algebraic element a ∈ E the decomposition (2.7) always can be chosen in such a way that every e_{k} is non-zero and also that

(2.8)  λ_{1} > λ_{2} > … > λ_{r} > 0

holds. Under this additional assumption the corresponding spectral decomposition is unique. Furthermore, if P_{i,j} ∈ L(E) is the canonical projection onto the Peirce space E_{i,j} we have with λ_{0}: = 0

D(a) = ∑_{0≤i≤j≤r} \frac{1}{2}(λ_{i}\overline{λ}_{i} + λ_{j}\overline{λ}_{j}) P_{i,j}

Sp(a) = { ± |λ_{1}|, …, ±|λ_{r}|} and
SP(a) = \{ ± \sqrt{(λ_{i}\overline{λ}_{i} + λ_{j}\overline{λ}_{j})/2} : i, j ≥ 0 with E_{i,j} ≠ 0\} .

To avoid the trivial situation Sp′(a) = {0}, i.e. Sp(a) = ∅, we always assume a ≠ 0 in the following. For every f ∈ C^{−}(Sp^{(}a^{)}) = C^{−}(Λ) we denote the element ψ^{−1}(f) ∈ C also by f(a). In particular, a^{3} = {aaa} arises for the cube function f(s) = s^{3} and C is the closed linear span of all odd powers a^{n}, n ∈ 2IN + 1. It can be shown that D(C, C) is a closed linear subspace of L(E) and that D(f(a), g(a)) only depends on the even function h = f\overline{g} ∈ C(Λ), more precisely (comp. [16]),

D(C, C) → C(Λ)
D(f(a), g(a)) ↦ f\overline{g}

defines a linear isometry with image

C_{0}^{+}(Λ): = {h ∈ C(Λ) : h even and h(0) = 0} .

Although this space has a natural algebra structure, the space D(C, C) is not a subalgebra of L(E) in general. It is clear that the spaces C^{−}(Λ) and C_{0}^{+}(Λ) both are isometrically isomorphic to C_{0}(Λ^{+}) where Λ^{+}: = {λ ∈ Λ : λ > 0}. Also, D(R, R) is a closed real subspace of D(C, C) and D(C) is a closed convex cone in D(R, R).

Denote by A (resp. B) the smallest closed real (resp. complex) subalgebra of L_{IR}(E) that contains D(a), Q(a) and the identity operator 11. Then B is a unital complex Banach algebra, and B = B^{+} ⊕ B^{−} is a topologically direct sum where B^{+} = B ∩ L(E) is the subalgebra of all complex linear and B^{−} is the subspace of all conjugate linear operators in B. From

D(a)Q(a) = 2Q(a^{3}, a) − Q(a)D(a) and Q(a^{3}, a) = 2D(a)Q(a) − Q(a)D(a)

we derive that A is commutative, whereas B itself is not commutative. Also, B^{+} is the closed unital complex subalgebra of L(E) generated by the operators D(a), P (a) and 1, and B^{−} is the closure of B^{+}Q(a) in B. In particular, also B^{+} is commutative. For the element â: = |a|a ∈ R (that is f(a) for the odd function f(s) = |s|s on Λ) we have P (a) = 2D(a)^{2} − D(â), i.e. B^{+} can be generated as unital complex Banach algebra also by the hermitian operators D(a), D(â). From

D(a^{n+2}, a^{m}) = 2D(a^{n}, a)D(a^{m}, a) − Q(a)Q(a^{n}, a^{m})
Q(a^{n+2}, a^{m}) = 2D(a^{n}, a)Q(a^{m}, a) − Q(a)D(a^{n}, a^{m})

for all n, m ∈ 2IN + 1 it is clear that D(C, C) and Q(C, C) are contained in B.

Every operator L ∈ L(E_{IR}) has a complex linear extension to the formal complexification E⊕iE of E. We remark that this complexification again has the structure of a JB*-triple. Indeed, let \overline{E} be the JB*-triple obtained from E by changing the complex structure to its conjugate, i.e. if we denote by \overline{x} ∈ \overline{E} the corresponding element of x ∈ E then scalar multiplication is given by α\overline{x} = \overline{^{(}\overline{α}x^{)}} for all α ∈ C and x ∈ E. Then E ⊕ \overline{E} is a JB*-triple and its real form \{(x, \overline{x}) : x ∈ E\} can be identified with E_{IR} in the obvious way. Now every L ∈ B^{+} extends to the complex linear operator (x, \overline{y}) ↦ (Lx, \overline{Ly}) and every L ∈ B^{−} extends to the complex linear operator (x, \overline{y}) ↦ (Ly, \overline{Lx}) of E ⊕ \overline{E}.

3. An extension of the odd continuous functional calculus

Let E always be a JB*-triple in the following. In section 2 we already mentioned briefly the odd continuous functional calculus on E: This is a way to define an element f(a) ∈ E for every a ∈ E and every odd continuous function f: Sp(a) → C. An equivalent way to do this is as follows: In case, f ∈ C[t] is an odd polynomial write f(t) = g(t^{2})t for a polynomial g ∈ C[t] and put f(a) = g(D^{(}a^{)})a = g(Q^{(}a^{)})a. For f arbitrary, approximate f uniformly on Sp(a) by a sequence (f_{n}) of odd polynomials and put f(a): = lim f_{n}(a), compare [15]. Then, in particular, ∥f(a)∥ = ∥f∥: = sup |f(Sp^{(}a^{)})| holds.

3.1 Remark For every closed (complex) linear subspace F ⊂ E the following conditions are equivalent.

(i) F is a JB*-subtriple of E. (ii) f(F) ⊂ F for every f ∈ C^{−}(IR). (iii) f(F) = F for f(s) ≡ s^{3}. Proof. (i) ⟹ (ii). For every a ∈ F we have f(a) ∈ C_{a} ⊂ F if F is a subtriple. (ii) ⟹ (iii). For f(s) ≡ s^{3} and g(s) ≡ s^{1/3} we have g(a) ∈ F and f(g^{(}a^{)}) = a for all a ∈ F. (iii) ⟹ (i). f(a) = {aaa} ∈ F for all a ∈ F implies by polarization immediately (i).

In the following we want to extend this functional calculus to arbitrary bounded odd functions f: Sp(a) → C. The price to pay is that then the element f(a) in general will no longer be in E but only in the bidual E^{∗∗}. We always consider E in the natural way as a closed linear subspace of E^{∗∗}. It is known [5] that E^{∗∗} is again a JB*-triple containing E as a JB*-subtriple. In particular, the bidual E^{∗∗} is a JBW*-triple – by definition a JB*-triple F is called a JBW*- triple if F = (F_{∗})^{∗} holds for a suitable complex Banach space F_{∗}, called a predual of F. Then, if F_{∗} is considered as a linear subspace of the dual F^{∗} in a natural way, it is well known that the predual F_{∗} is uniquely determined for every JBW*-triple F. Furthermore, it is known that the triple product on F is separately w^{∗}-continuous for w^{∗}: = σ(F, F_{∗}), compare [2]. In particular, for every tripotent e ∈ F and every k the corresponding Peirce-k-space F_{k}(e) is w^{∗}-closed in F.

To start with, let F be an arbitrary JBW*-triple in the following. The following statement seems to be well known, for completeness we present it with a proof:

3.2 Lemma Let A ⊂ F be a bounded subset of mutually orthogonal elements. Then

u = ∑_{a∈A} a ∈ F

exists as w^{∗}-convergent sum and ∥u∥ = sup_{a∈A} ∥a∥. Proof. We may assume that sup_{a∈A} ∥a∥ = 1 holds. Let φ ∈ E^{∗} be an arbitrary linear form on E with ∥φ∥ = 1. For every a ∈ A choose a complex number t_{a} with |t_{a}| = 1 and φ(t_{a}a) ≥ 0. For every finite subset B ⊂ A we then have

∑_{a∈B} |φ(a)| = ∑_{a∈B} φ(t_{a}a) ≤ ∥∑_{a∈B} t_{a}a∥ ≤ 1

since also the family (t_{a}a) consists of mutually orthogonal elements in F. As a consequence,

u(φ): = ∑_{a∈A} φ(a), φ ∈ F_{∗},

defines an element u ∈ F with ∥u∥ ≤ 1. For every a ∈ A the elements a and u−a are orthogonal in F, i.e. ∥u∥ ≥ ∥a∥ and hence ∥u∥ = 1.

3.3 Lemma Let A and u be as in Lemma 3.2. Then Sp(u) is the smallest compact subset of IR containing all Sp(a), a ∈ A. Moreover, for every odd continuous function f: Sp(u) → C we have

(3.4)  f(u) = ∑_{a∈A} f(a) .

Proof. For every a ∈ A the elements a and b: = u − a are orthogonal. This implies Sp(u) = Sp(a) ∪ Sp(b) and hence Ω ⊂ Sp(u) where Ω is the smallest compact subset of IR containing all Sp(a), a ∈ A. Therefore ∥f(a)∥ ≤ ∥f∥ for all a ∈ A and v: = ∑_{a∈A} f(a) exists in F by 3.2 since also (f^{(}a^{)})_{a∈A} is an orthogonal family. For every odd polynomial p ∈ C[t] the equation p(u) = ∑_{a∈A} p(a) is easily seen by induction on the degree of p using the separate w^{∗}-continuity of the triple product. This implies

f(u) − v = g(u) − ∑_{a∈A} g(a) for g: = f − p ∈ C^{−}(Sp^{(}u^{)})

and hence

∥f(u) − v∥ ≤ 2∥g∥

as a consequence of Lemma 3.2. Since for every ε > 0 we can choose p with 2∥g∥ < ε we get f(u) = v as desired. Now consider a number t ∈ IR\Ω. Then there exists an odd continuous function f: IR → IR with f(t) = 0 and f(ω) = ω for all ω ∈ Ω. This implies f(u) = u by (3.4) and hence t ∉ Sp(u) if t ≠ 0. As a consequence we get Sp′(u) = Ω ∪ {0} and finally Sp(u) = Ω.

3.5 Corollary Let Λ be a set and let (e_{λ})_{λ∈Λ} be a family of pairwise orthogonal tripotents in the JBW*-triple F. Let furthermore (c_{λ})_{λ∈Λ} be a bounded family in IR. Then

u = ∑_{λ∈Λ} c_{λ}e_{λ}

exists as w^{∗}-convergent sum in F and Sp(u) is the closure of the set {±c_{λ} : c_{λ}e_{λ} ≠ 0} in IR. For every odd continuous function f: Sp(u) → C furthermore

f(u) = ∑_{λ∈Λ} f(c_{λ})e_{λ}

holds.

Let always F: = E^{∗∗} be the bidual of E for the rest of the section and consider E in the natural way as linear subspace of E^{∗∗}. Fix an element a ∈ E and put Λ^{+}: = {λ ∈ Sp(a) : λ > 0}. 3.6 Proposition For a ∈ E given and Λ = Sp′(a) there exists a unique orthogonal family (e_{λ})_{λ∈Λ^{+}} of non-zero tripotents in E^{∗∗} such that

(3.7)  a = ∑_{λ∈Λ^{+}} λe_{λ}

holds as w^{∗}-convergent sum in E^{∗∗}. Proof. We may assume a ≠ 0 and denote by C = C_{a} the closed subtriple of E generated by a. We identify C with C_{0}(Λ^{+}) in such a way that a = id_{Λ^{+}}. Also, we consider C^{∗∗} in the natural way as linear subspace of the bidual E^{∗∗}. Fix λ ∈ Λ^{+} and choose for every n ∈ IN a function f_{λ,n} ∈ C with f_{λ,n}(λ) = 1 = ∥f_{λ,n}∥ such that lim_{n→∞} f_{λ,n}(t) = 0 for all t ∈ Λ^{+}\{λ}. For every φ ∈ E^{∗} then μ: = φ|_{C} ∈ C^{∗} is a finite Borel measure on Λ^{+} and by Lebesgue’s Bounded Convergence Theorem

e_{λ}(φ): = lim_{n→∞} φ(f_{λ,n}) = lim_{n→∞} ∫_{Λ^{+}}f_{λ,n}dμ = μ({λ}) =: φ(e_{λ})

defines an element e_{λ} ∈ C^{∗∗} ⊂ E^{∗∗} that does not depend on the choice of the sequence (f_{λ,n})_{n∈IN}. This implies

δ_{λω} e_{λ} = {e_{λ} f_{ω,j} f_{ω,k}} = {e_{λ} e_{ω} f_{ω,k}} = {e_{λ} e_{ω} e_{ω}}

for all λ, ω ∈ Λ^{+} and all j, k ∈ IN. In particular, (e_{λ})_{λ∈Λ^{+}} is an orthogonal family of non-zero tripotents in C^{∗∗}. By Lemma 3.2 the element u: = ∑ λe_{λ} ∈ C^{∗∗} is well defined and we have to show a = u. For this it is enough to show φ(a) = φ(u) for all positive φ ∈ C^{∗}: Fix n ∈ IN, and for 1 ≤ k ≤ n put

Λ_{n,k}: = \{λ ∈ Λ : \frac{k − 1}{n} < λ ≤ \frac{k}{n}\} and e_{n,k}: = ∑_{λ∈Λ_{n,k}} e_{λ} .

Then we get from the positivity of φ

∑_{k=1}^{n} \frac{k − 1}{n} φ(e_{n,k}) ≤ ∑_{λ∈Λ^{+}} λφ(e_{λ}) = φ(u) ≤ ∑_{k=1}^{n} \frac{k}{n} φ(e_{n,k}) .

From φ(e_{n,k}) = μ(Λ_{n,k}), μ: = φ|_{C} we hence get

∑_{k=1}^{n} \frac{k − 1}{n} μ(Λ_{n,k}) ≤ φ(u) ≤ ∑_{k=1}^{n} \frac{k}{n} μ(Λ_{n,k}) .

Letting n → ∞, we obtain

φ(u) = ∫_{Λ^{+}} λ dμ(λ) = μ(a) = φ(a)

since a = id |_{Λ^{+}}. This proves the existence of the representation (3.7). For the uniqueness proof assume that also a = ∑_{α∈Λ^{+}} αẽ_{α} holds for an orthogonal family (ẽ_{α}) of tripotents in E^{∗∗}. Then, for fixed λ ∈ Λ^{+} and f_{λ,n} ∈ C chosen as above, we have

f_{λ,n}(a) = ∑_{α∈Λ^{+}} f_{λ,n}(α)ẽ_{α}

by Corollary 3.5. Thus, for every positive φ ∈ C^{∗} and μ the corresponding measure on Λ^{+}, we see that with n → ∞ the φ(f_{λ,n}) = ∑_{λ∈Λ^{+}} f_{λ,n}(α)φ(ẽ_{α}) converge to φ(ẽ_{λ}). By the same argument we have convergence to φ(e_{λ}). This shows φ(e_{λ}) = φ(ẽ_{λ}) for all positive φ ∈ C^{∗} and hence e_{λ} = ẽ_{λ} for all λ.

For λ ∈ Λ^{+} the tripotent e_{λ} ∈ E^{∗∗} in general is not contained in E. However, if λ is an isolated point of Λ^{+}, then e_{λ} ∈ E. More generally, for every compact open subset K ⊂ Λ^{+} the tripotent e_{K}: = ∑_{λ∈K} e_{λ} is in E. Equation (3.7) may be used to define f(a): = ∑_{λ∈Λ^{+}} f(λ)e_{λ} ∈ E^{∗∗} for every a ∈ E and every bounded odd function Sp(a) → C. In particular, e: = ∑_{λ∈Λ^{+}} e_{λ} is the support tripotent of a in E^{∗∗}, which is in E if 0 ∉ Sp(a).

3.8 Corollary The elements a, b ∈ E are orthogonal if and only if {aab} = 0. Proof. Suppose {aab} = 0. Fix λ ∈ Λ^{+} and put e: = e_{λ}, v: = a − λe and write b = b_{1} + b_{1/2} + b_{0} with b_{k} ∈ F_{k}(e) and F = E^{∗∗}. Then e ⊥ v and

0 = \{aab\} = λ^{2}\{eeb\} + \{vvb\} = λ^{2}b_{1} + (\frac{1}{2}λ^{2}b_{1/2} + \{vvb_{1/2}\}) + \{vvb_{0}\}

implies b_{1} = 0 and D(v)b_{1/2} = −\frac{1}{2}λ^{2}b_{1/2}. Since D(v) has non-negative spectrum we also conclude b_{1/2} = 0, that is e_{λ} ⊥ b for all λ ∈ Λ^{+}. For every z ∈ E finally we get

{abz} = ∑_{λ∈Λ^{+}} λ{e_{λ}bz} = 0

and hence a ⊥ b. The converse implication is trivial.

4. The Peirce spectrum

Let the commutative real Banach algebra A ⊂ L_{IR}(E), E a given JB*-triple and a ∈ E a given element, be as in section 2 and consider its spectrum spec(A), that is the compact space of all non-zero multiplicative IR-linear functionals ξ: A → C. Since A is generated as unital algebra by the two operators D(a) and Q(a) we can realize spec(A) as a subset K ⊂ C^{2} in the following way: Define κ: spec(A) → C^{2} by ξ ↦ (ξ(D(a), ξ(Q(a))) and let K be the image. Then κ: spec(A) → K is a homeomorphism and the Gelfand transform gives a canonical algebra homomorphism A → C(K). We conjecture that K is contained in the sector {(x, y) ∈ IR^{2} : x ≥ |y|}.

4.1 Lemma K is invariant under the reflection ρ : (λ, μ) ↦ (λ, −μ). Also (x, y) ∈ K implies that x, y^{2} are real and that x ≥ 0 holds. Proof. For every ξ ∈ spec(A) define the linear form ξ̃ on A by ξ̃(L) = L if L is complex linear and by ξ̃(L) = −L if L is conjugate linear. Then also ξ̃ is multiplicative, i.e. K is ρ-invariant. y^{2} ∈ IR follows from the fact that P (a) = Q(a)^{2} = 2D(a)^{2} − D(â) is a real polynomial of hermitian operators.

Now suppose that Σ is a compact topological space and that η: Σ → K is a continuous map. This induces an algebra homomorphism η^{∗}: C(K) → C(Σ) and together with the Gelfand transform of A a homomorphism φ: A → C(Σ). In the following we present a space Σ together with a map η in such a way that φ will be injective.

Let F: = E^{∗∗} be the bidual of E and let (e_{λ})_{λ∈Λ^{+}} be the orthogonal family of non-zero tripotents in F satisfying (3.7) where Λ = Sp′(a). Extend λ ↦ e_{λ} in the unique way to an odd mapping Λ → F. Define in analogy to (2.6) for every λ, μ ∈ Λ with λ·μ ≥ 0 the Peirce space

(4.2)  F_{λ,μ} = {z ∈ F : 2D(e_{ν})z = (δ_{λν} + δ_{νμ})z for all ν ∈ Λ, ν ≠ 0} ,

which is a w^{∗}-closed subtriple of F (in general not contained in E). Also define for all 0 ≤ λ ≤ μ mutual orthogonal projections P_{λ,μ} from F onto F_{λ,μ} by

(4.3)  P_{λ,μ} = \{ \begin{matrix}(2 − δ_{λμ})Q(e_{λ})Q(e_{μ}) & if λ > 0 \\ 4(D(e_{μ}) − D(e_{μ})^{2})B(e − e_{μ}) & if μ > λ = 0 \\ B(e) & if μ = 0 , \end{matrix}

where e = ∑_{λ>0} e_{λ} is the support tripotent of a. The algebraic sum

(4.4)  G: = ⨁_{0≤λ≤μ} F_{λ,μ}

is direct in F but not norm dense in general. However, G is w^{∗}-dense in F, compare [11] Lemma 3.15. In case λ · μ > 0, the operator Q(e) with e: = ∑_{λ>0} e_{λ} the support tripotent, splits F_{λ,μ} into a real (+1) – and a real (−1)–eigenspace that we denote by F^{λ,μ} and F^{λ,−μ} respectively. Also, we put F^{λ,μ}: = F_{λ,μ} if λ·μ = 0. Then every F^{λ,μ} is a real subtriple of F and F^{λ,−μ} = iF^{λ,μ} holds for all λ, μ. Put

Θ: = \{(λ, μ) ∈ Λ^{2} : F^{λ,μ} ≠ 0\} , Σ = Σ(a): = \overline{Θ} and Σ′: = Σ ∪ \{(0, 0)\}

and call Σ, Σ′, the Peirce spectrum and the extended Peirce spectrum of a ∈ E respectively. Also denote by W the group generated by the following two reflections of IR^{2}

(λ, μ) ↦ (μ, λ) and (λ, μ) ↦ (−μ, −λ) .

Notice that, in case A ⊂ E is a subtriple with a ∈ A, then Sp(a) is the same whether computed with respect to A or to E. In case of the Peirce spectrum Σ this is not the case. Therefore, we better should write Σ_{E} = Σ_{E}(a) if the reference to E is not obvious.

4.5 Proposition Σ is a W -invariant compact subset of Λ^{2} ⊂ IR^{2} and in addition is also invariant under the reflection ρ : (λ, μ) ↦ (λ, −μ). There is a unique continuous algebra homomorphism φ: A → C(Σ, IR) such that

d(λ, μ) = \frac{λ^{2} + μ^{2}}{2} and q(λ, μ) = λμ

holds for d: = φ(D^{(}a^{)}) and q: = φ(Q^{(}a^{)}). The closure of φ(A) in C(Σ, IR) is the subalgebra

C(Σ, IR)^{W}: = {h ∈ C(Σ, IR) : h ∘ γ = h ∀ γ ∈ W } ≅ C(Σ/W)

of all real-valued W -invariant continuous functions on Σ. The homomorphism φ is injective. In particular, the operator L ∈ A is complex linear (conjugate linear) if and only if h ∘ ρ = h (h ∘ ρ = −h) holds for the φ-image h: = φ(L) ∈ C(Σ). Proof. Let F = E^{∗∗}, Λ, K, Θ and F^{λ,μ} be as above. Then Θ is W -invariant as well as ρinvariant. As closure of Θ then also Σ has the same property. Define the W -invariant continuous map η: Σ → IR^{2} by (λ, μ) ↦ ((λ^{2} + μ^{2})/2, λμ). Denote by G ⊂ F the smallest norm-closed linear subspace containing all F^{λ,μ}. Then G is w^{∗}-dense in F and for every L ∈ A the operator L^{∗∗} ∈ L_{IR}(F) is a w^{∗}-continuous extension of L leaving G invariant. In particular, every L ∈ A is uniquely determined by the restriction L^{∗∗}|_{G} ∈ L_{IR}(G). Since for every (λ, μ) ∈ Θ all elements of F^{λ,μ} are eigenvectors of L ∈ A we conclude that η(Θ) is contained in K. Therefore η: Σ → K is a continuous mapping and the composition of the Gelfand transform A → C(K) with η^{∗} gives the homomorphism φ. For every L ∈ A and h: = φ(L) then

(4.6)  L^{∗∗}(z) = h(λ, μ) z for all (λ, μ) ∈ Θ, z ∈ F^{λ,μ}

shows that φ is injective. Since the functions d, q ∈ C(Σ) separate the W -orbits in Σ we get from the Stone-Weierstraß Approximation Theorem that φ has dense image in C(Σ, IR)^{W}. Finally, L ↦ L̃: = L^{+} − L^{−} defines a real algebra isomorphism of L_{IR}(E) that leaves the subalgebra A invariant. The identity φ(L̃) = φ(L) ∘ ρ proves the remaining statements.

4.7 Proposition The extended triple spectrum Λ = Sp′(a) can be realized as the diagonal of the extended Peirce spectrum Σ′, more precisely, for Δ: = {(λ, λ) : λ ∈ IR}

Σ′ ∩ Δ = {(λ, λ) : λ ∈ Sp′(a)} .

In case D(a) is injective the same holds for the non-extended spectra, i.e.

Σ ∩ Δ = {(λ, λ) : λ ∈ Sp(a)} .

Proof. e_{λ} ∈ F^{λ,λ} holds for every λ > 0, i.e. (λ, λ) ∈ Σ′ for all λ ∈ Λ, proving the first equation. Now suppose that 0 ∈ Sp(a). Then 0 = lim λ_{n} for a sequence (λ_{n}) in Λ^{+} and consequently (0, 0) ∈ Σ, since Σ is compact. Conversely, suppose 0 ∉ Sp(a) and (0, 0) ∈ Σ. Then (0, 0) is isolated in Σ, and σ(D^{(}a^{)}) ⊂ {(λ^{2} + μ^{2})/2 : λ, μ ∈ Sp′(a)} implies that 0 is isolated in the extended spectrum σ′(D^{(}a^{)}). Since 0 is not an eigenvalue of D(a) by assumption, we get 0 ∉ σ(D^{(}a^{)}) and hence that D^{(}a^{)}^{∗∗} is injective. On the other hand, by the separate w^{∗}-continuity of the triple product on F, we have for all b ∈ F^{0,0}

D(a)^{∗∗}b = {aab} = ∑_{λ>0} λ{e_{λ}( ∑_{μ>0} μe_{μ})b} = ∑_{λ>0} λ^{2}{e_{λ}e_{λ}b} = 0 ,

i.e. F^{0,0} = {0} and thus (0, 0) ∉ Θ, a contradiction, since Σ is the closure of Θ and (0, 0) is isolated in Σ.

Many properties of the element a ∈ E are determined by the corresponding family (e_{λ}) in F = E^{∗∗}. We state as an example, where as before P (a) = Q(a)^{2}:

4.8 Proposition The following conditions are equivalent.

(i) D(a) and P (a) commute with every operator D(b), b ∈ E. (ii) Every D(e_{λ}) commutes with every operator D(b), b ∈ F. (iii) λ = μ for every (λ, μ) ∈ Σ(a). If these conditions are satisfied then also Q(a) commutes with every D(b), b ∈ E. Proof. (i) ⟹ (iii). Suppose that (i) holds. Since every F_{λ,μ} is a joint eigen space of D(a) and P (a), it is invariant under D(b) for every b ∈ E and hence also for every b ∈ F as a consequence of the separate continuity of the triple product. Now fix λ ∈ Λ^{+} and μ ∈ Sp′(a) with λ ≠ μ. For every x ∈ F_{λ,μ} ⊂ F_{1/2}(e_{λ}) and b: = e_{λ} + x then D(b)e_{λ} ∈ F_{λ,λ} = F_{1}(e_{λ}) implies 0 = D(e_{λ})x = \frac{1}{2}x, i.e. F_{λ,μ} = 0. (iii) ⟹ (ii). Suppose that (iii) holds and fix λ ∈ Λ^{+}. Since F_{1/2}(e_{λ}) is the w^{∗}-closed span of all F_{λ,μ} with λ ≠ μ, we have F = F_{1}(e_{λ}) ⊕ F_{0}(e_{λ}). For every b = b_{1} + b_{0} ∈ F_{1}(e_{λ}) ⊕ F_{0}(e_{λ}) then D(e_{λ})D(b) = D(b_{1}) = D(b)D(e_{λ}) implies (ii). (ii) ⟹ (i). Suppose that (ii) holds. As before we see that then also (iii) and hence P (a) = D(a)^{2} holds. For every λ ∈ Λ^{+} the operator D(e_{λ}, a) = λD(e_{λ}) commutes with every D(b), b ∈ E, and hence also D(a) commutes with every D(b).

4.9 Proposition Let L: E → Ẽ be a morphism of JB*-triples and put ã: = L(a). Then (i) Sp(ã) ⊂ Sp(a), (ii) Σ(ã) ⊂ Σ(a) if L is surjective. Proof. ad (i): L extends in a unique way to a w^{∗}-continuous operator F → F̃ of the corresponding biduals. By the separate w^{∗}-continuity of the triple product on F, F̃ this extension again is a morphism of JB*-triples. Write a in the form (3.7) and put ẽ_{λ}: = L(e_{λ}). Then (ẽ_{λ})_{λ∈Λ^{+}} is an orthogonal family of tripotents in F̃ and ã = ∑ λẽ_{λ} holds as w^{∗}-convergent sum in F̃. From Lemma 3.3 and the uniqueness property in Proposition 3.6 we derive Λ̃^{+} = {λ ∈ Λ^{+} : ẽ_{λ} ≠ 0}, proving the first claim. ad (ii): Suppose, L is surjective. Then the w^{∗}-continuous extension of L to the biduals has dense image and hence is surjective, too. For every λ ≥ μ ≥ 0 from Sp′(ã) the operator L maps F_{λ,μ} into F̃_{λ,μ}. It is enough to show that always F̃_{λ,μ} = L(F_{λ,μ}) holds. Therefore, fix an arbitrary x ∈ F̃_{λ,μ} and pick an element y ∈ L^{−1}(x). Then x = L(z) for z: = P_{λ,μ}(z) ∈ F_{λ,μ}.

Let us call an operator L ∈ L_{IR}(E) a (refined) Schur multiplier (relative to a ∈ E) if there exists a function h: Θ → C satisfying (4.6), compare [8] for this notion in the finite dimensional case. Then L is uniquely determined by h and we write also M_{h} instead of L. The function h clearly is W -invariant and will be called the symbol of M_{h}. The space M of all Schur multipliers (relative to a) is a complex subalgebra of L_{IR}(E) containing B. The subalgebra M^{+}: = M∩L(E) of all complex linear Schur multipliers is commutative and consists of all M_{h} with h ∘ ρ = h. Also, the IR-linear subalgebra N ⊂ M of all real Schur multipliers M_{h} (i.e. h is real-valued) is commutative and contains A. In particular, we have N ∩ iN = 0 and hence A ∩ iA = 0.

The multiplier algebra M is not commutative. It is easy to see that the symbol h of the multiplier product M_{h} = M_{h_{1}}M_{h_{2}} is given by

(4.10)  h(λ, μ) = (au − b̃v) + i(ãv + bu)

for all (λ, μ) ∈ Σ, where

h_{1}(λ, μ) = a + ib, , h_{1}(λ, −μ) = ã + ib̃ h_{2}(λ, μ) = u + iv

are the decompositions into real and imaginary parts. In particular, h = h_{1}h_{2} if h_{2} is real.

Instead of considering the symbol h of a Schur multiplier L as a complex-valued function we could instead also use matrix valued functions on Θ: Consider the algebra IM: = IR^{2×2} of all real 2 × 2-matrices. Then (_{b d}^{ac}) ↦ (_{−c a}^{d −b}) defines an involutory inner automorphism τ of IM and the subalgebra Fix(τ) is isomorphic to C after identifying (_{b a}^{a−b}) ∈ Fix(τ) with a + ib ∈ C. For every (λ, μ) ∈ Θ define H(λ, μ): = (_{b u}^{a−v}) ∈ IM if h(λ, μ) = a + ib and h(λ, −μ) = u + iv. Then the function H: Θ → IM satisfies τ ∘ H = H ∘ ρ where the reflection ρ is defined as before by (λ, μ) ↦ (λ, −μ). In particular, h(λ, μ) = H(λ, μ) ∈ C ⊂ IM if λ · μ = 0. In case λ · μ > 0 every matrix (_{b d}^{ac}) ∈ IM acts on F_{λ,μ} = F^{λ,μ} ⊕ iF^{λ,μ} by x + iy ↦ (ax + cy) + i(bx + dy). In this sense, the Schur multiplier L acts on every Peirce space F_{λ,μ}, λ · μ =≥ 0, as the matrix H(λ, μ), and we call H the matrix symbol of L and write also M_{H} in place of L. For the matrix symbol, formula (4.10) now gets the more natural form

M_{H_{1}}M_{H_{2}} = M_{H_{1}H_{2}} .

5. Some particular functions

For fixed a ∈ E let the Peirce spectrum Σ and the algebras A ⊂ B ⊂ M be as in section 4. Notice that as a consequence of (4.6) for every f, g ∈ C^{−}(Λ, IR) the operators

D(f(a), g(a)),  Q(f(a), g(a))  and  B(f(a), g(a))

are in A and have symbols

\frac{f(λ)g(λ) + f(μ)g(μ)}{2},  \frac{f(λ)g(μ) + f(μ)g(λ)}{2}  and  (1 − f(λ)g(λ))(1 − f(μ)g(μ)) .

For every c ∈ E put

(5.1)  X(c): = D(c, a) + Q(c, a)

in the following. Then X(c) depends complex linearly on c and if c = f(a), f ∈ C^{−}(Λ, IR), then X(c) is in A and has symbol

(5.2)  \frac{1}{2}(f(λ) + f(μ))(λ + μ)

on Σ (multiplied sums). Later we will use also various types of divided differences, more precisely: For every odd C^{1}-function f: IR → C denote by m_{f} or f^{[1]} the unique continuous function on IR^{2} satisfying

m_{f}(λ, μ) · (λ − μ) = f(λ) − f(μ)

for all λ, μ, that is m_{f}(λ, μ) = ∫_{0}^{1} f′(sλ + (1 − s)μ) ds and, in particular, m_{f}(λ, λ) = f′(λ). Then clearly m_{f} in a unique way can be written as a sum m_{f} = m_{f}^{+} + m_{f}^{−} of continuous functions where m_{f}^{+}(λ, μ) is even in μ and m_{f}^{−}(λ, μ) is odd in μ. These functions satisfy

m_{f}^{+}(λ, μ) · (λ^{2} − μ^{2}) = λf(λ) − μf(μ) and m_{f}^{−}(λ, μ) · (λ^{2} − μ^{2}) = μf(λ) − λf(μ)

for all λ, μ. The divided difference m_{f} plays a crucial in the study of the operator differentiability of f, compare f.i. [1], and we may ask: When is m_{f} (more precisely m_{f}|_{Σ} ) the symbol of an operator L ∈ B and what is the significance of such an L? Clearly, then m_{f}^{+} is the symbol of the complex linear part L^{+} of L.

Now put

Θ^{+}: = {(λ, μ) ∈ Θ : λ, μ ≥ 0} and Ω: = {(λ, μ) ∈ Θ^{+} : {F_{λ,λ}F_{λ,μ}F_{μ,μ}} ≠ 0}.

5.3 Lemma Let L = M_{h} ∈ M^{+} be a complex linear Schur multiplier with symbol h. A necessary condition for L to be hermitian is

(5.4)  h(λ, μ) = \frac{h(λ, λ) + h(μ, μ)}{2} ∈ IR for all (λ, μ) ∈ Ω .

In case Ω = Θ^{+} (5.4) is also a sufficient condition. Proof. L is hermitian if and only if

(5.5)  L{uvw} = {(Lu)vw} − {u(Lv)w} + {uv(Lw)}

holds for all u, v, w ∈ E. Since L^{∗∗} is hermitian on E^{∗∗} and is the multiplication by h(λ, μ) on every F_{λ,μ} we derive from

0 ≠ {F_{λ,λ}F_{λ,μ}F_{μ,μ}} ⊂ F_{λ,μ}

that (5.4) is necessary for L to be hermitian. Conversely, suppose (5.4) and Ω = Θ^{+} holds. For the validity of (5.5) we have only to check triple products {uvw} ∈ F_{κ,ν} with u ∈ F_{κ,λ}, v ∈ F_{λ,μ}, w ∈ F_{μ,ν}, since the linear span of all F_{λ,μ} is w^{∗}-dense in F. But this is obvious.

The analogue for M_{h} to be the exponential of an hermitian operator is obtained when replacing condition (5.4) by its multiplicative analogue

(5.6)  h(λ, μ) = \sqrt{h(λ, λ)h(μ, μ)} > 0 for all (λ, μ) ∈ Ω .

This is true for instance for every operator B(c) with c ∈ C of norm < 1 and for every B(c, −c). Exponentials of hermitian operators have the remarkable property that their norm coincides with their spectral radius.

6. Functional calculus and derivatives

Let E, a, C, Λ, A, B, Σ, φ be as before and suppose throughout that E ≠ 0. Notice that for V = −V ⊂ IR open the set U: = {z ∈ E : Sp(z) ⊂ V } may not be open in E and also that for f ∈ C^{−}(V) the map f: U → E, z ↦ f(z), may not be continuous. As an example of this phenomenon take V: = IR\{0} and f(t) = 1/t. Then f is not continuous at 0 ∈ U. If E has infinite rank, 0 is not an interior point of U.

For 𝕋: = {t ∈ C : |t| = 1} denote by C^{𝕋}(C) the space of all 𝕋-equivariant continuous functions f: C → C, i.e. satisfying f(tz) = tf(z) for all z ∈ C and t ∈ 𝕋. Clearly C^{𝕋}(C) is a semigroup with respect to composition and f ↦ f|_{IR} induces an isomorphism of complex vector spaces C^{𝕋}(C) ≈ C^{−}(IR). Every f ∈ C^{−}(IR) induces by the triple functional calculus a mapping E → E that we denote by f_{E}. In case of E = C the mapping f_{C} is nothing but the 𝕋-equivariant extension of f to C. For all f, g ∈ C^{−}(IR) the formula (f_{C} ∘g)_{E} = f_{E} ∘g_{E} holds for all JB*-triples E.

In the following f: IR → C is always an odd continuous function. For simplicity we also write f (the corresponding boldface letter to f) for the induced vector function f_{E} if no confusion is likely. It is clear that f^{(}a^{)} ∈ E only depends on the restriction of f to the spectrum Sp^{(}a^{)} ⊂ IR of a. The question arises, when does there exist the (real) Fréchet derivative

df^{(}a^{)} ∈ L_{IR}(E) ,

(or the Gateaux derivative) or at least, for a given b ∈ E, the directional derivative

df(a; b): = lim_{t→0} \frac{f(a + tb) − f^{(}a^{)}}{t} ∈ E .

For instance, for all a ⊥ b ≠ 0 in E the directional derivative df(a; b) ∈ E exists if and only if f is differentiable in 0 and then df(a; b) = f′(0)b. For every real triple morphism L: E → F of JB*- triples, compare (2.2), f_{F} (L^{(}a^{)}) = L(f_{E}^{(}a^{)}) is obvious and with it also df_{F} (L^{(}a^{)})∘L = L∘df_{E}^{(}a^{)} if both derivatives exist.

Let us consider a simple example in more detail, that is, let E = H be a complex Hilbert space of complex dimension > 1 with inner product (z|w). Then, for a ∈ E fixed with r: = ∥a∥ > 0, we have

Sp(a) = \{±r\},  SP(a) = \{±r/\sqrt{2}, ±r\},  Σ = \{(εr, ηr) ≠ (0, 0) : ε, η = −1, 0, 1\}
E^{r,r} = IRa,  E^{r,0} = {z ∈ E : (z|a) = 0},  E^{r,−r} = iIRa .

It is easy to see that the derivative df(a) exists if and only if the scalar function f has a derivative at r = ∥a∥ (that is, in Sp(a)). In that case, f(z) = g((z|z))z for z ≠ 0 and g defined by g(s): = s^{−1/2}f(s^{1/2}) for all s > 0. This implies df(a; b) = g′(r^{2})((a|b) + (b|a))a + g(r^{2})b by elementary calculus and hence that df(a) is the Schur multiplier with symbol h defined by

h(r, r) = 2r^{2}g′(r^{2}) + g(r^{2}) = f′(r)  and  h(r, 0) = h(r, −r) = g(r^{2}) = f(r)/r ,

that is, h = m_{f} is the divided difference of f on Σ. Finally, we have for the derivative

df(a) = (g(r^{2}) − r^{2}g′(r^{2}))11 + g′(r^{2})(Q(a) + 2D(a)) ∈ B .

In case E contains an element with uncountable spectrum, the map f: E → E can be Fréchet differentiable everywhere only if the scalar function f is of class C^{1} (compare the simple proof for the analogue algebra situation in [19]). Therefore, let us assume for the rest of the section that the odd function f: IR → C is of class C^{1}. Then in particular the divided difference m_{f} is a continuous function on IR^{2}. Also, there is a unique continuous function g on {t ∈ IR : t ≥ 0} with f(t) = g(t^{2})t for all t ≥ 0. We start with polynomials.

6.1 Proposition Let f ∈ C[t] be an odd polynomial. Then f: E → E is a real-analytic polynomial map of same degree as f. Furthermore, the Fréchet derivative df(a) is in B and has symbol m_{f} (more precisely, the restriction of m_{f} to Σ). Proof. We may assume without loss of generality that f(t) = t^{2n+1}. Then

df^{(}a^{)} = D^{(}a^{)}^{n} + ∑_{j=1}^{n} D^{(}a^{)}^{n−j}X(a^{2j−1}) ∈ A

is easily verified, where the operator function X is defined in (5.1). For n = 0 our claim is obvious, so we may suppose n > 0. For p(t): = t^{2n−1} and all z ∈ E then

f(z) = D(z)p(z) and hence df^{(}a^{)} = X(p^{(}a^{)}) + D^{(}a^{)} dp^{(}a^{)} .

By induction hypothesis and (5.2) we get for the symbol h of df^{(}a^{)}

2h(λ, μ)(λ − μ) = (λ^{2} − μ^{2})(g(λ) + g(μ)) + (λ^{2} + μ^{2})(g(λ) − g(μ)) = 2(f(λ) − f(μ))

for all λ, μ and hence h = m_{f}.

6.2 Corollary Suppose there exists a neighbourhood U of a in E together with a sequence of odd polynomials f_{n} ∈ C[t] satisfying

(i) f(a) = lim_{n→∞} f_{n}(a), (ii) the sequence (df_{n}) converges uniformly on U to a map Φ: U → L_{IR}(E).

Then f has Fréchet derivative df(a) = Φ(a) ∈ B at a with symbol m_{f}. Proof. We may assume that U is bounded and convex in E. From elementary calculus we know that f is of class C^{1} on U with Fréchet derivative df(a) = Φ(a) and that f = lim_{n→∞}(f_{n}) uniformly on U. This implies

(∗)  f(λ) = lim_{n→∞} f_{n}(λ) and f′(λ) = lim_{n→∞} f′_{n}(λ) for all λ ∈ Λ^{+} .

Since B is closed in L(E) we have df(a) ∈ B. Let h ∈ C^{−}(Σ) be the symbol of df(a). We only have to show that h and m_{f} coincide on Σ: The restriction of df_{n}(a)^{∗∗}, n ∈ IN, to every Peirce space F^{λ,μ} ⊂ F: = E^{∗∗} is multiplication by the scalar m_{f_{n}}(λ, μ), i.e. h(λ, μ) = lim_{n→∞} m_{f_{n}}(λ, μ) for all (λ, μ) ∈ Σ. Because of (∗) and the continuity of h, m_{f} it therefore remains to show h(0, 0) = f′(0) in case that (0, 0) is an isolated point of Σ. But then 0 ∉ Sp(a) and the support tripotent e = ∑_{λ∈Λ^{+}} e_{λ} is contained in E. From E_{0}(e)^{∗∗} = F_{0}(e) = F^{(0,0)} ≠ {0} we get E_{0}(e) ≠ {0}. Therefore there exists a vector v ≠ 0 with a ⊥ v and df(a)v = f′(0)v implies h(0, 0) = f′(0).

As a further application of Proposition 6.1 we can generalize the above Hilbert space example (i.e. the rank 1 case) to arbitrary JB*-triples of finite rank. By definition, E has finite rank if every element of E is contained in a subtriple if finite dimension, or equivalently, if every element in E is algebraic. In particular, every JB*-triple of finite dimension has finite rank. The JB*-triples of finite rank are fully classified, compare [13].

6.3 Proposition Suppose that the JB*-triple E has finite rank and that f: IR → C is an odd C^{1}-function. Then f: E → E also is of class C^{1} and has Fréchet derivative df(a) ∈ B at a ∈ E with symbol m_{f}. Proof. We may assume that f is real-valued. Fix for a the decomposition (2.7) satisfying (2.8). E = F for F: = E^{∗∗} by [13], and there exists an equivalent Hilbert norm |||·||| on E_{IR} (a generalization of the Hilbert-Schmidt norm for matrices) for which all Peirce spaces F^{λ,μ} (compare section 4) are pairwise orthogonal. For every L ∈ L_{IR}(E) denote by |||L||| the corresponding operator norm. Fix a compact interval I ⊂ IR containing Sp′(a) in its interior. Then U: = {z ∈ E : Sp(z) ⊂ I} is a neighbourhood of a in E. To every integer n > 0 there is an odd polynomial p_{n} ∈ IR[t] with

|f(t) − p_{n}(t)| + |m_{f}(s, t) − m_{p_{n}}(s, t)| < 1/n for all s, t ∈ I .

This implies that the sequence (p_{n}) converges uniformly on U to f and, as a consequence of Proposition 6.1, that

(∗)  ||| dp_{n}(a) − dp_{m}(a) ||| < 2/n for all m ≥ n ,

i.e. (dp_{n}^{(}a^{)}) is a Cauchy sequence in L_{IR}(E). As a consequence, df(a) = lim dp_{n}(a) ∈ B exists and has symbol m_{f}, compare Corollary 6.2. Since all the limits are uniform in U the map f is of class C^{1} on U and hence on E, since a is arbitrary.

6.4 Corollary Let E be a JB*-triple of finite rank and let S be an arbitrary set. Then for every odd f ∈ C^{1}(IR) the induced map f: ℓ^{∞}(S, E) → ℓ^{∞}(S, E) has Fréchet derivatives everywhere. Proof. We use the notation of the proof to 6.3. Fix a = (a_{s})_{s∈S} ∈ ℓ^{∞}(S, E) and put |||a|||: = sup |||a_{s}|||. Then also (∗) holds and gives the result.

Next consider the special situation where the odd function f: IR → C is real-analytic in a neighbourhood of SP^{(}a^{)}. Then also f: E → E is real-analytic in a neighbourhood of a ∈ E. Indeed, there exists in C an open neighbourhood U of σ(D^{(}a^{)}) together with a holomorphic function g on U such that f(t) = g(t^{2})t for all t ∈ SP^{(}a^{)}. For all z in a suitable neighbourhood Ω of a ∈ E then

σ(D^{(}z^{)}) ⊂ U and f(z) = g(D^{(}z^{)}) z

holds where the operator g(D^{(}z^{)}) ∈ L(E) is given by the holomorphic functional calculus and hence depends in a real-analytic way on z ∈ Ω. To be a little more specific, we may assume that there is a compact subset K ⊂ U with piece-wise smooth boundary ∂K such that σ(D^{(}z^{)}) is in the interior of K for all z ∈ Ω. Cauchy’s integral formula then gives

(6.5)  f(z) = \frac{1}{2πi} ∫_{∂K}g(s)(s − D(z))^{−1}z ds

for all z ∈ Ω. Differentiation of (6.5) gives with X as defined in (5.1)

(6.6)  df^{(}a^{)} = g(D^{(}a^{)}) + \frac{1}{2πi} ∫_{∂K}g(s)(s − D^{(}a^{)})^{−1}X(a_{s}) ds

where

a_{s}: = (s − D^{(}a^{)})^{−1}a for all s ∈ C\σ(D^{(}a^{)}) .

Notice that the spectrum of D^{(}a^{)} is real and hence has connected complement in C. As a consequence, every (s − D^{(}a^{)})^{−1} is contained in the algebra B, and hence, the derivative df^{(}a^{)} also is contained in B and has symbol m_{f}.

We may ask whether for f Fréchet differentiable the derivative df(a) is in the algebra B and has symbol m_{f}. Let us point out that in B the Schur multipliers with divided difference as symbol form a small linear subspace – more precisely, for M_{h} ∈ B a necessary condition is

(λ − μ)h(λ, μ) = (λ − ν)h(λ, ν) + (ν − μ)h(ν, μ) for all (λ, μ), (λ, ν), (ν, μ) ∈ Σ .

7. Connection to hermitian operator calculus

As before, we consider for given f ∈ C^{−}(IR) and JB*-triple E the induced map f_{E}: E → E that we also denote by f for short. Let us assume that there is a continuous scalar function g(s), s ≥ 0, with f(t) = g(t^{2})t for all t ∈ IR. In section 6 we used that for f real-analytic every f(z), z ∈ E, can be given by

(7.1)  f(z) = g(D^{(}z^{)})z ,

where the operator g(D^{(}z^{)}) ∈ L(E) is given by the holomorphic functional calculus and hence every directional derivative df(z)y, y ∈ E, is given in terms of dg(D^{(}z^{)}) ∈ L(L(E)) by

(7.2)  df(z)y = g(D^{(}z^{)})y + (dg(D^{(}z^{)})(D^{(}z, y^{)} + D^{(}y, z^{)}))z .

Formula (7.1) suggests that it would be desirable to give a meaning to g(x) ∈ L(E) for every x ∈ H(E), or at least to all operators of the form x = D(z), z ∈ E. In the following we discuss several instances where this is possible, thereby restricting either the function g or the JB*-triple E to special classes. Recall that for every complex Hilbert space H, every hermitian operator x ∈ H: = H(H) and every (complex-valued) locally bounded g ∈ L^{∞}(IR) the operator g(x) ∈ L(H) is well defined: If e is the spectral measure of x on IR then

x = ∫_{IR} λ de(λ) and g(x) = ∫_{IR} g(λ) de(λ) .

The induced mapping g: H → L(H) will also be denoted by g_{H}. In case g is continuous and odd on IR, the map g_{H} is nothing but the restriction to H of the map g_{E} given by the triple functional calculus on the JB*-triple E: = L(H).

Case I: Suppose that g: IR → IR is defined in terms of a Fourier transform, i.e.

(7.3)  g(t): = ∫_{IR} e^{ist} dμ(s) ,

where μ is a complex-valued finite Borel measure on IR (compare also [19]). Assume in addition that μ has first moment, that is

(7.4)  ∫_{IR} |s| d|μ|(s) < ∞ .

The Weyl calculus associates the operator g(L): = ∫_{IR} exp(isL) dμ(s) ∈ L(E) to every hermitian operator L ∈ H(E). It is not difficult to see that g is of class C^{1} on IR and also that formula (7.1) holds for the associated odd function f. We claim that in this sense also formula (7.2) holds:

7.5 Proposition Let A be a unital complex Banach algebra and let x be an hermitian element of A (i.e. ∥ exp(itx)∥ = 1 for all t ∈ IR). Then

(7.6)  g(x): = ∫_{IR} e^{isx} dμ(s) ∈ A

is well defined. Furthermore, if H ⊂ A is the IR-linear subspace of all hermitian elements in A, then x ↦ g(x) defines a Fréchet differentiable map g: H → A with

(7.7)  dg(x)(y) = ∫_{IR} (∫_{0}^{1} e^{i(1−u)sx}ye^{iusx} du) is dμ(s) .

Proof. Since x is hermitian and since the Radon-Nikodym derivative dμ/d|μ| is unimodular we see that the integral in (7.6) is absolutely convergent and satisfies ∥g(x)∥ ≤ |μ|(IR). To prove (7.7) let x, y ∈ H and use Lemma 9.1 to obtain

g(x + y) = ∫_{IR} e^{is(x+y)} dμ(s) = g(x) + ∫_{IR} (∫_{0}^{1} e^{i(1−u)x}isye^{ius(x+y)} du) dμ(s)
= g(x) + L(y) + R(y) with
L(y): = ∫_{IR} (∫_{0}^{1} e^{i(1−u)x}ye^{iusx} du) is dμ(s) and

R(y): = ∫_{IR} (∫_{0}^{1} e^{i(1−u)x}y(e^{ius(x+y)} − e^{iusx}) du) is dμ(s) .

The linear operator L satisfies ∥L∥ ≤ ∫_{IR} |s| d|μ|(s) < ∞ and hence is bounded. The remainder R(y) satisfies

∥R(y)∥ ≤ ∥y∥ε(y) for ε(y): = ∫_{IR} ∫_{0}^{1} ∥e^{ius(x+y)} − e^{iusx}∥ |s|du d|μ|(s) .

Since ∥e^{ius(x+y)} −e^{iusx}∥ ≤ 2 and lim_{y→0} ∥e^{ius(x+y)} −e^{iusx}∥ = 0 pointwise for (u, s) ∈ [0, 1]×IR, an application of Lebesgue’s Dominated Convergence Theorem yields lim_{y→0} ε(y) = 0, i.e. L = dg(x) is the Fréchet derivative of g at x.

We apply Proposition 7.5 to the special case where A = L(E) and x = D(z) for z ∈ E.

7.8 Proposition Under the assumptions (7.3) and (7.4) the map f: E → E has Fréchet derivative everywhere and df: E → L_{IR}(E) is continuous. Furthermore,

(7.9)

f(a) = g(D^{(}a^{)})a = ∫_{IR} e^{isD(a)}a dμ(s) and

df(a) = ∫_{IR} e^{isD(a)} dμ(s) + ∫_{IR} ( ∫_{0}^{1}e^{i(1−u)sD(a)} X(e^{iusD(a)}a) du)isdμ(s) ∈ B ,

where X(c) = D(c, a) + Q(c, a) for all c ∈ E. Proof. By Proposition 7.5 g(D(z)) = ∫_{IR} e^{isD(z)} dμ(s) ∈ L_{IR}(E) is well defined and f(z) = g(D^{(}z^{)})z is easily verified for all z ∈ E. This implies again by 7.5 that df(a) exists and that

\begin{aligned}df(a; b) & = lim_{t→∞} \frac{f(a + tb) − f(a)}{t} = g(D^{(}a^{)})b + (dg(D^{(}a^{)})(D(a, b) + D(b, a)))a \\ & = g(D^{(}a^{)})b + ∫_{IR} ( ∫_{0}^{1} e^{i(1−u)sD(a)} (D(a, b) + D(b, a)) e^{iusD(a)}a du)is dμ(s) \end{aligned}

for all b ∈ E proving (7.9). Since dg(D^{(}a^{)}) depends in a continuous way on a, the same is true for df(a).

Case II: E has finite rank and g ∈ C(IR) is arbitrary. Then there exists an equivalent Hilbert norm |||·||| on E for which every D(z), z ∈ E, is hermitian. In this sense g(D^{(}z^{)}) ∈ L(E) is well defined by the hermitian operator calculus on complex Hilbert spaces, and also (7.1) holds in this situation. More generally, for every set S and every z ∈ ℓ^{∞}(S, E) the operator D(z) can be defined pointwise in the following natural way: (D^{(}z^{)}w)(s) = D(z^{(}s^{)})(w^{(}s^{)}) for all w ∈ ℓ^{∞}(S, E) and all s ∈ S. By 6.4 f is of class C^{1} on ℓ^{∞}(S, E) if the scalar function f on IR has the same property.

Case III: E = L(H, K) for complex Hilbert spaces H, K, and g ∈ C(IR) is arbitrary. Denote for given z ∈ E by e and f the spectral measures of the hermitian operators (z^{∗}z)^{1/2} ∈ H(H) and (zz^{∗})^{1/2} ∈ H(K) respectively. Then the operator D(z) ∈ H(E) is given by

D(z) : w ⟼ ∫_{0}^{∞}∫_{0}^{∞} \frac{λ^{2} + μ^{2}}{2} df(μ)w de(λ)

and we define g(D^{(}z^{)}) ∈ L(E) by

g(D^{(}z^{)}) : w ⟼ ∫_{0}^{∞}∫_{0}^{∞}g(\frac{λ^{2} + μ^{2}}{2} ) df(μ)w de(λ) .

For the scalar function f(t) = g(t^{2})t we have for all z ∈ E

f(z) = g(D^{(}z^{)})(z) = g(zz^{∗})z = zg(z^{∗}z) .

In particular, df(a) ∈ L_{IR}(E) exists if one of the derivatives

dg(aa^{∗}) ∈ L_{IR}(H(H), L(H))  or  dg(a^{∗}a) ∈ L_{IR}(H(K), L(K))

exists. Actually, we can do better. Let H̃: = H ◯ ⊥ K be the orthogonal Hilberst space sum of H, K and set H̃: = H(H̃). Write every operator in L(H̃) as operator matrix (_{b d}^{ac}) with a ∈ L(H), b ∈ L(H, K) and so on. For every z ∈ E and w: = z^{∗} put z̃: = (_{z 0}^{0w}) ∈ H̃. Then z ↦ z̃ defines an isometric real morphism E ↪ L(H̃) of JB*-triples and for every z ∈ E

Sp′(z) = σ′(z̃) = \sqrt{σ′(z^{∗}z)} = \sqrt{σ′(zz^{∗})}

can easily be seen. For every f ∈ C^{−}(IR) and every z ∈ E we have f̃_{E}(z) = f_{H}(z̃). As a consequence we get together with Corollary 6.4.

7.10 Proposition For every odd C^{1}-function f: IR → C the following conditions are equivalent. (i) f_{E}: E → E is of class C^{1} for every JB*-triple E. (ii) f_{H}: H → L(H) is of class C^{1} for every complex Hilbert space H and H: = H(H). Proof. (i) ⟹ (ii). For every complex Hilbert space H consider the JB*-triple E: = L(H). Then f_{H} is the restriction of f_{E} to H. (ii) ⟹ (i). By the Gelfand-Naimark Theorem [7] the JB*-triple E can be realized as a JB*- subtriple of L(H) ⊕ ℓ^{∞}(S, F), where H is a complex Hilbert space, S is a set and F is the irreducible exceptional JB*-triple of rank 3 and dimension 27. Therefore we only have to consider the special cases E = L(H) and E = ℓ^{∞}(S, F). The latter is contained in Corollary 6.4 while the first follows from the above discussion, i.e. the triple functional calculus on L(H) can be obtained by restricting the hermitian operator calculus on H(H ◯ ⊥ H) to the image of L(H) under the map z ↦ z̃.

8. The associated Lie algebra

Let E be a JB*-triple and denote by

Der(E): = {δ ∈ L(E) : δ{xyz} = {(δx)yz} + {x(δy)z} + {xy(δz)}}

the space of all triple derivations of E. Then 𝔨: = i Der(E) is a real Banach Lie algebra with respect to the commutator product [δ_{1}, δ_{2}] = δ_{1} ∘ δ_{2} − δ_{2} ∘ δ_{1}, and 𝔨 can be embedded into a bigger (real) Banach Lie algebra 𝔤 defined in the following way: Put 𝔭: = E and extend the Lie product from 𝔨 to 𝔤: = 𝔨 ⊕ 𝔭 by

[a, b] = 2(D(b, a) − D(a, b)) ∈ 𝔨 and [a, δ] = −[δ, a] = δ^{(}a^{)} ∈ 𝔭

for all a, b ∈ 𝔭 and all δ ∈ 𝔨. It is known that 𝔤 can be identified with the Lie algebra of all complete holomorphic vector fields on the open unit ball D ⊂ E if we identify every (δ, a) ∈ 𝔨⊕𝔭 with the polynomial vector field f(z) = a + δ(z) − {zaz} and define the bracket h: = [f, g] of any two holomorphic vector fields on D by h(z) = df^{(}z^{)}(g^{(}z^{)}) − dg^{(}z^{)}(f^{(}z^{)}).

The linear subspace 𝔭 = E of 𝔤 is a Lie triple system with respect to the Lie triple product

[abc]: = [[a, b], c] = 2({abc} − {bac}) .

Lie triple and Jordan triple product determine each other uniquely since 2{abc} is the part of [abc] that is C-linear in a.

Now fix an element a ≠ 0 in E = 𝔭. The spectrum σ = σ(ad^{(}a^{)}) ⊂ C of ad^{(}a^{)} by definition is the spectrum of the complex linear extension of ad^{(}a^{)} to the complexification 𝔤^{C}: = 𝔤 ⊕ i𝔤. Then σ is symmetric to the real axis and also symmetric to the origin in C since (δ+c) ↦ (δ−c), δ ∈ 𝔨 and c ∈ 𝔭, obviously is a Lie algebra automorphism of 𝔤. By the same reason every eigenspace 𝔤_{t}: = {x ∈ 𝔤 : [a, x] = tx}, t ∈ IR, is isomorphic to 𝔤_{−t}. Also, 𝔤_{0} = Ker(ad^{(}a^{)}) contains the element a, i.e. 0 is contained in σ. Clearly, [𝔤_{s}, 𝔤_{t}] ⊂ 𝔤_{s+t} holds for all s, t ∈ IR.

8.1 Lemma For fixed a ∈ E define on E the operator Y: = 2(D^{(}a^{)} − Q^{(}a^{)}). Then

σ(ad^{(}a^{)}) = \sqrt{σ(Y)} .

Moreover, if we denote by 𝔭_{s} ⊂ 𝔭 = E the s-eigenspace of Y we have for every real t ≠ 0 and s: = t^{2}

𝔤_{t} = { t^{−1}[a, b] + b : b ∈ 𝔭_{s}} .

Proof. Denote by α ∈ L(𝔨, 𝔭) and β ∈ L(𝔭, 𝔨) the restrictions of ad^{(}a^{)} to 𝔨 and 𝔭. Then ad^{(}a^{)}^{2} = βα ⊕ αβ and

σ(ad^{(}a^{)})^{2} = σ(ad^{(}a^{)}^{2}) = σ(βα) ∪ σ(αβ) .

But αβ = Y and 0 ∈ σ(Y), i.e. σ(βα) ⊂ σ(αβ) and hence σ(ad^{(}a^{)}) = σ(Y)^{1/2}. Now fix δ ∈ 𝔨 and b ∈ 𝔭. Then (δ + b) is in 𝔤_{t} if and only if tδ = [a, b] and t^{2}b = [a, tδ] = Y (b).

With the notation of section 2 the operator Y = 2(D^{(}a^{)} − Q^{(}a^{)}) ∈ A has symbol (λ − μ)^{2} on Σ. This implies

(8.2)  { ± (λ − μ) : (λ, μ) ∈ Σ} ⊂ σ(ad^{(}a^{)}) .

8.3 Problem Is σ(ad^{(}a^{)}) real or does even hold equality in 8.2? The answer is affirmative if the Gelfand map φ: A → C(Σ, IR) is surjective (this holds for instance if a is algebraic in E).

Suppose for the rest of the section that a is a tripotent in E and that E_{k}(a), k = 1, 1/2, 0, are the corresponding Peirce spaces. Also, let E^{k}(a) for k ∈ IR be the k-eigenspace of the conjugate linear operator Q(a). Then σ(Y) ⊂ {0, 1, 4} and

𝔭_{0} = E^{1}^{(}a^{)} ⊕ E_{0}^{(}a^{)}, 𝔭_{1} = E_{1/2}^{(}a^{)}, 𝔭_{4} = E^{−1}^{(}a^{)}

and hence

𝔤 = 𝔤_{−2} ⊕ 𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1} ⊕ 𝔤_{2} .

For simplicity let us also assume that the tripotent a is maximal, i.e. E_{0}^{(}a^{)} = 0. Consider in 𝔤 the closed linear subspace

𝔰: = 𝔭_{0} ⊕ 𝔤_{1} ⊕ 𝔤_{2} .

Then 𝔰 is the graph of the linear operator T ∈ L(𝔭, 𝔨) defined by T (x + y + z) = [a, y + z/2] for all (x, y, z) ∈ 𝔭_{0} × 𝔭_{1} × 𝔭_{4}. Clearly 𝔤_{1} ⊕ 𝔤_{2} is a nilpotent Lie algebra and 𝔞 ⊕ 𝔤_{1} ⊕ 𝔤_{2} is a solvable Lie algebra for every abelian subalgebra 𝔞 ⊂ 𝔭 containing a, compare [8] for a more detailed discussion in the finite dimensional setting.

9. Appendix

In the proof of Proposition 7.5 we used the following result. 9.1 Lemma Let x, y be arbitrary elements in the unital real (or complex) Banach algebra A. Then

(9.2)  e^{x+y} − e^{x} = ∫_{0}^{1} e^{(1−u)x}ye^{u(x+y)} du .

Proof.

e^{x+y} − e^{x} = ∑_{n=1}^{∞} \frac{1}{n!}((x + y)^{n} − x^{n}) = ∑_{n=1}^{∞} \frac{1}{n!} ∑_{j=0}^{n−1} x^{j}y(x + y)^{n−1−j}

= ∑_{j,k=0}^{∞} \frac{x^{j}y(x + y)^{k}}{j! k!} ∫_{0}^{1}(1 − u)^{j}u^{k} du

= ∫_{0}^{1} ( ∑_{j=0}^{∞} \frac{((1 − u)x)^{j}}{j!} )y( ∑_{k=0}^{∞} \frac{(u(x + y))^{k}}{k!} ) du .

If we write exp(x) instead of e^{x} for the exponential function, then formula (9.2) gives

(9.3)  d exp(x)y = ∫_{0}^{1}e^{(1−u)x}y e^{ux} du

for the derivative of exp at x in the direction y. Now, for x ∈ A fixed, formula (9.2) immediately gives by iteration the full power series expansion of e^{x+y} about x. Indeed, put for all n > 0 and all y ∈ A

p_{n}(y) = ∫_{Q_{n}} e^{(1−u_{1})x}y e^{u_{1}(1−u_{2})x}y ⋯ e^{u_{1}u_{2}···u_{n−1}(1−u_{n})x}y e^{u_{1}u_{2}···u_{n}x} du

where Q_{n} ⊂ IR^{n} is the unit cube and u = (u_{1}, …, u_{n}). Then, with p_{0}(y): = e^{x}, every p_{n}: A → A is homogeneous of degree n and

(9.4)  e^{x+y} = ∑_{n=0}^{∞} p_{n}(y) for all y ∈ A .

Formula (9.3) can also be written in operator form. For this let for every c ∈ A the left and right translation operators L(c) and R(c) be defined by z ↦ cz and z ↦ zc. Also, put as usual ad(c): = L(c) − R(c). Then by (9.3)

d exp(x)y = e^{x}∫_{0}^{1}e^{−ux}y e^{ux} du = e^{x}∫_{0}^{1}exp(−u ad^{(}x^{)})y du

and hence

(9.5)  d exp(x) = L(e^{x}) ∘ ∫_{0}^{1}exp(−u ad^{(}x^{)}) du = L(e^{x}) ∘ \frac{1 − exp(− ad^{(}x^{)})}{ad^{(}x^{)}} .

The last identity for arbitrary Lie groups is well known and occurs for instance as Theorem II.1.7 in [10].

If we replace A by its opposite algebra A^{op} (i.e. by reversing the order of multiplication) all powers and hence also the exponential function remain unchanged. In particular, formula (9.5) for A^{op} gives for every x in the algebra A:

(9.6)  d exp(x) = R(e^{x}) ∘ ∫_{0}^{1}exp(u ad^{(}x^{)}) du = R(e^{x}) ∘ \frac{exp(ad^{(}x^{)}) − 1}{ad^{(}x^{)}} .

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