TRACE AND DETERMINANT IN JORDAN-BANACH ALGEBRAS

Using an appropriate definition of the multiplicity of a spectral value, we introduce a new definition of the trace and determinant of elements with finite spectrum in Jordan-Banach algebras. We first extend a result obtained by J. Zemánek in the associative case, on the connectedness of projections which are close to each other spectrally (Theorem 2.3). Secondly we show that the rank of the Riesz projection associated to a finite-rank element a and a finite subset of its spectrum is equal to the sum of the multiplicities of these spectral values (Theorem 2.6). Then we turn to the study of properties such as linearity and continuity of the trace and multiplicativity of the determinant.


Introduction
Determinants of infinite matrices were investigated for the first time by the astronomer G. W. Hill in his studies on lunar theory and his ideas were put into a rigorous form by Henri Poincaré in 1886.Ten years later H. von Koch refined and generalized Poincaré's results.In 1903, I. Fredholm developed a determinant theory for integral operators.Unlike von Koch, Fredholm studied eigenvalues and looked at the analyticity of Det(I + λM ).Fredholm's determinant theory is certainly one of the milestones in the history of functional analysis.In the early fifties A. F. Ruston, T. Lezański and A. Grothendieck almost simultaneously defined determinants for nuclear or integral operators on a Banach space.In the seventies, A. Pietsch developed an axiomatic approach to the determinant of elements of certain operator ideals.In 1978, J. Puhl [17] studied the trace on the socle and nuclear elements of a semisimple Banach algebra, basing his difficult arguments on the standard trace defined for finite-rank operators.In [5] it was shown that the trace and determinant on the socle of a Banach algebra can be developed in a purely spectral and analytic way, that is to say internally, without using operators on the algebra.
The essential ingredient in all the arguments is the fact that the spectrum is an analytic multifunction.So this point of view gives us the possibility to extend almost all the results of [5] to more general situations where the spectrum is also analytic, for instance the case of Jordan-Banach algebras.For a finite-dimensional Jordan algebra A the trace is defined on A using the trace of the left multiplication operator L x defined on A by L x a = xa.In this paper we shall define the trace and determinant for elements of the socle of an arbitrary Jordan-Banach algebra in a purely spectral and internal way.At this point it is worth mentioning that purely algebraic proofs of some of our results have been given by O. Loos in [12].
We recall that a complex Jordan algebra A is non-associative and the product satisfies the identities ab = ba and (ab)a 2 = a(ba 2 ) for all a, b in A (see [21] for the theory of Jordan algebras).A unital Jordan-Banach algebra is a Jordan algebra with a complete norm satisfying xy ≤ x y , for x, y ∈ A and 1 = 1.Equivalently, we could define a unital Jordan-Banach algebra as a Banach space A over C with a continuous quadratic map U from A to the bounded linear operators on A, written as x → U x and a distinguished element 1 ∈ A satisfying in particular the following identities: The U operators can be recovered by U x y = 2x(xy) − x 2 y.The linear Jordan product xy is defined by 2xy = U x,y 1, and powers inductively by x 0 = 1, x n+1 = xx n .An element x ∈ A is said to be invertible if there exists y ∈ A such that xy = 1 and x 2 y = x.This is equivalent to the operator U x having inverse U y .For an element x of a Jordan-Banach algebra A, the spectrum of x is by definition the non-empty set of complex numbers λ such that λ − x is not invertible.
The main tool in all the arguments is the following theorem which is an immediate consequence of the fact that λ −→ Sp(f (λ)) is an analytic multifunction, where f is an analytic function from a domain D ⊂ C into a Jordan-Banach algebra, and of Theorem 7.1.7 of [1].In the following we use the symbol #S to denote the cardinality of the set S. Theorem 1.1.Let A be a Jordan-Banach algebra with nonzero socle.Suppose that f is an analytic function from a domain D of C into the socle of A. Then there exist an integer N and a closed discrete subset F of D such that

The multiplicity of a spectral value
Throughout this paper we shall assume that the Jordan-Banach algebra A is semisimple with an identity.Here semisimple means that the Jacobson radical vanishes.We recall that an idempotent p is minimal if U p A = Cp, in which case U p A is a minimal inner ideal.By definition the socle of A, denoted by Soc A, is the sum of all minimal inner ideals of A. By Osborn-Racine theorem [16] the socle of A is an ideal and it is the sum of simple ideals generated by minimal projections.By a theorem of Aupetit-Baribeau [4] the socle of a scattered Jordan-Banach algebra is not reduced to zero.It is easy to see that the elements of the socle are algebraic, so they have finite spectrum.Consequently by Newburgh's theorem the spectrum function is continuous on the socle (see [1,Corollary 3.4.5],and [15]).
For a semisimple Jordan-Banach algebra, it is known that the socle coincides with the largest von Neumann regular ideal, as shown by A. Fernández López and A. Rodríguez Palacios in [8].Given a semisimple Jordan-Banach algebra, A. Fernández López proved that its socle is an algebraic ideal and conversely if I is any algebraic ideal then every element of I can be written as the sum of an element of the socle and a nilpotent element.This result was improved by M. Benslimane, O. Jaa and A. Kaïdi [6] who proved that every element of a spectrally finite ideal can be written as the sum of an element of the socle and an element whose square is zero.Using recent results of O. Loos [11] and A. Rodríguez Palacios [18], it can be proved that a spectrum-finite ideal of A is included in the socle.
The rank of a ∈ A is defined in [2] as the maximum of the number of points of Sp U x a\{0}, when x ∈ A, if it exists, otherwise the rank is infinite.It is also proved that the set of finite-rank elements coincides with the socle.This rank is equal to the algebraic rank defined by O. Loos [10], as this was shown in [9].We recall that a ∈ Soc A is said to be a maximal-finite-rank element if #(Sp a\{0}) = rank(a).Let a ∈ A be a finite-rank element.By the definition of the rank, the set consequently U x a is a maximal-finite-rank element when x belongs to E(a).In the associative case, maximal finite-rank elements have a simple structure, they are linear combinations of orthogonal projections ([5, Theorem 2.8]).What can be said in the Jordan-Banach case?

Theorem 2.1 (Density of maximal finite-rank elements). Let a be a finite-rank element of A. Then E(a) is a dense open subset of A. Consequently the set of maximal finite-rank elements is dense in the socle.
Proof: The proof is almost identical to the proof of Theorem 2.2 in [5], replacing everywhere xa by U x a. Theorem 2.2.Let a be a finite-rank element of A. Let Γ be an oriented regular contour, not intersecting Sp a, and denote by 0 , 1 respectively its interior and its exterior.By upper semicontinuity of the spectrum there exists an open ball B in A, centred at the identity, such that Sp and let D be the convex domain of C containing 0 and 1 such that which has finite-rank, and applying the Localisation Principle ([1, Theorem 7.1.5])to the analytic multifunctions λ −→ Sp f (λ) ∩ i , by the Scarcity Theorem we conclude that there exist two integers N 0 , N 1 and two closed discrete subsets To prove the theorem we have to show that 0, 1 , from the definition of the rank, where m = rank(a), so this is absurd.If 0 / ∈ Sp U y a by upper semicontinuity of the spectrum we can choose λ / and this is also absurd.By a similar argument the case 1 ∈ F 0 ∪ F 1 also gives a contradiction, so the theorem is proved.
The number #(Sp(U x a) ∩ 0 ) which is independent of x by Theorem 2.2, is denoted by m(Γ, a) and is called the multiplicity of a associated to Γ.It is independent of Γ if the corresponding spectral points of a in 0 , 1 do not change.For α in Sp a we define m(α, a), the multiplicity of a at α, as m(Γ, a) where Γ is a small circle centred at α and isolating α from the rest of the spectrum.We have m(α, a) ≥ 1.It is not difficult to see that If a is a maximal finite-rank element then necessarily we must have m(α, a) = 1 for every α ∈ Sp a, because 1 ∈ B ∩ E(a).
In particular if we take for Γ a contour surrounding all the spectrum of a we obtain ( In fact we shall see that the multiplicity of a at α = 0 is equal to the rank of the Riesz projection associated to a and α.
Let α ∈ C and Γ be a small curve isolating α from the rest of the spectrum of a.By definition the Riesz projection is The Holomorphic Functional Calculus implies that the p(α, a) are orthogonal projections for different α and their sum is one.
For α = 0, the identity (λ − a) where Γ is a small circle isolating α from the rest of Sp a ∪ {0}.
In the case of Banach algebras, J. Zemánek proved in [20] that two projections p, q such that ||p − q|| < 1 are analytically connected.In particular they have the same rank by Lemma 2.5 of [5].We now extend this result to Jordan-Banach algebras.
Theorem 2.3.Let p, q be two projections of A such that ρ(p − q) < 1, where ρ denotes the spectral radius.Then there exists s ∈ A such that s 2 = 1, q = U s p and p = U s q.In particular rank(p) = rank(q).Proof: a) Let L be the non-closed subalgebra generated by 1, p, q.By the Shirshov-Cohn Theorem [21] this algebra has a faithful representation as a subalgebra of a special Jordan algebra B + , where B is an associative algebra with identity.We denote by x • y the associative product.Then taking a = p+q−1 we have p 2 .Consequently we have p • y = y • p and q • y = y • q for all y in the Jordan algebra C((p − q) 2 ) generated by 1 and ).All of this proves that the following Jordan identities are true in A: for t = U y a, y ∈ C((p − q) 2 ).By continuity these identities certainly extend for y in the closure of C((p − q) 2 ).b) Now suppose that ρ(p − q) < 1.By the Holomorphic Functional Calculus we can define (1 − (p − q) 2 ) − 1 4 by a convergent series containing only terms with even powers of p − q.Consequently this element is in the closure of C((p − q) 2 ).Taking y = (1 − (p − q) 2 ) − 1  4 and s = U y a we conclude from (5) that s 2 = 1, U s p = q and p = U s −1 q = U s q.Consequently p and q have same rank by [2, Theorem 3.5].
Corollary 2.4.Let p, q be two projections of A. If p, q are in the same connected component of the set of projections there exists a sequence s 1 , . . ., s n of elements of A such that s 2 i = 1 (i = 1, . . ., n), p = U s1 . . .U sn q and q = U sn . . .U s1 p.In particular p and q have the same rank.
Proof: Since p and q lie in the same connected component, we know by a standard topological result (see for example [7,Proposition 19.2]) that there exist a chain of points r 1 , . . ., r n+1 such that p = r 1 , q = r n+1 and r i − r i+1 < 1 (i = 1, . . ., n).By n successive applications of Theorem 2.3, we obtain the result.
We now prove the fundamental result of this paper.Theorem 2.6.Let a be a finite-rank element and α 1 , . . ., α n some nonzero elements of its spectrum.If p denotes the Riesz projection associated to a and {α 1 , . . ., α n } (which is the sum of the Riesz projections associated to a and the different α i ) then we have Proof: a) From Theorem 2.2 and the definition of multiplicity there exists 0 > 0 such that for 0 < < 0 , where 0 is a domain limited by a regular contour Γ separating α 1 , . . ., α n from the rest of the spectrum of a and 0. Consequently by the Holomorphic Functional Calculus we have where q denotes the Riesz projection associated to U x a and Γ, that is the sum of the Riesz projections associated to U x a and the different α 1 , . . ., α n .The proof of this is almost identical to the proof of Theorem 3.3.4 in [1].Consequently by [2,Corollary 3.6].If we take > 0 small enough, then we can suppose that p − q < 1, so by Theorem 2.3, we have b) We now prove a small preliminary result.If b has finite-rank, α = 0 is in the spectrum of b and if q is the Riesz projection associated to b and α, then U q A ⊂ U b A. This comes immediately from the fact that we have q where u is in a maximal associative Jordan algebra containing 1 and b, so q = q 2 = (bu c) We now prove the converse inequality of (6).By Theorem 2.2 we may choose x arbitrarily near to 1 such that We know that Sp b contains m points in 0 and r − m nonzero points outside.Denote by q 1 , . . ., q r−m the r−m Riesz projections associated to b and to the r − m non-zero points outside of 0 .These projections are orthogonal and we have U qi A ⊂ U b A by part b).Moreover these q i are orthogonal to q, the Riesz projection associated to b and Γ. Suppose rank(q) > m, then by Lemma 2.5 there exists y ∈ A such that #(Sp U q y\{0}) > m.
By the Holomorphic Functional Calculus applied to U q y we can construct m + 1 orthogonal non-zero projections p 1 , . . ., p m+1 which are in U q A, consequently in U b A, by part b).Every p i is orthogonal to every q j because p i ∈ U q A, q j ∈ U qj A, q is orthogonal to q j , so we apply Proposition 6 of [4].Hence U b A contains r + 1 nonzero orthogonal projections q 1 , . . ., q r−m , p 1 , . . ., p m+1 , but this violates Lemma 2.5 applied to b.So rank(q) ≤ m.But for x near to 1, p and q satisfy the conditions of Theorem 2.3, that is rank(p) = rank(q) Corollary 2.7.If p is a non-zero projection of the socle and p = 1 then m(0, p) = 1 and m(1, p) = rank(p).

The trace and the determinant
Using the previous arguments with some results from [5] we obtain: 3) Exactly the same as in [5,Theorem 3.5].
Corollary 3.3.Let p 1 , . . ., p n be orthogonal finite-rank projections and let α 1 , . . ., α n be non-zero complex numbers.Then Proof: We may suppose that all the projections are non-zero.By [2, Theorem 3.9], we have rank(α by [2,Corollary 3.6] and the fact that the algebra generated by the p i is associative.So without loss of generality we may suppose that The following lemma is essential in the proof of the next theorem.We include a detailed proof here for the sake of completeness.Lemma 3.4.Let f (λ, µ) be a complex-valued function of two complex variables which is separately entire in λ, µ and such that f (λ, µ) = 0 for all λ, µ ∈ C. Suppose moreover that there exists two positive constants A, B such that |f (λ, µ)| ≤ e A|λ|+B|µ| .
Remark 3.6.Property 4) is not true in general, even for a finitedimensional Jordan-Banach algebra.To see this, one has just to consider the Jordan algebra of matrices M 2 (C) with the usual Jordan product x • y = 1 2 (xy + yx).Now take the following two matrices

Lemma 2 . 5 .
Let a be a finite-rank element.Then rank(a 2 ) = max x∈A #(Sp U a x\{0}) ≤ rank(a).Consequently the quadratic ideal U a A contains at most rank(a) orthogonal non-zero projections.If p is a projection then rank(p) = max x∈A #(Sp U p x\{0}).Proof: By the Shifting Principle [21, p. 315] we have Sp U a x 2 \{0} = Sp U x a 2 \{0}, for a, x ∈ A. Consequently, it follows from the definition of the rank that, rank(a 2 ) = max x∈A #(Sp U a x 2 \{0}).
If a ∈ Soc A we define the trace of a by Tr(a) = λ∈Sp a λm(λ, a), (7) and the determinant of 1 + a by Det(1 + a) = λ∈Sp a

Theorem 3 . 1 .
Let f be an analytic function from a domain D of C into the socle of A. Then Tr(f (λ)) and Det(1 + f (λ)) are holomorphic on D.
In this case we set p = p 1 + • • • + p n which is a non-zero projection.By Corollary 2.7 and Theorem 3.2 1) we have rank