A complete theory of Jordan’s algebras was found in Gauss’s unpublished papers. Their first appearance in history seems to be in the early 1930s when the theory burst into maturity from the minds of Pascual Jordan, John von Neumann, and Eugene Wigner in their 1934 paper, On an algebric generalization of the quantum mechanical formalism [1].

Jordan took as his axioms the existence of a bilinear product x ∘ y on a complex vector space satisfying the identities:

A better example of Jordan algebras is obtained by replacing the product of an associative algebra A x y by the Jordan’s product x ∘ y = 1 2 ( x y + y x ) , for all x , y ∈ A . The Jordan algebra obtained will be noted A + . Any Jordan algebra isomorphic to a sub-algebra of such an algebra A + is called a special Jordan algebra. Otherwise, it is called exceptional Jordan algebra. A deep and useful result in the theory of Jordan algebras is the following theorem, due to Shirshov and Cohn ([2], Theorem 3.1.55).

Theorem 1.1Any Jordan algebra generated by two elements(and 1,if unital)is special.

The extension of any notion, in particular generalized inverse, known in the associative case to Jordan’s algebras must take into consideration this gateway from A to A + . For this purpose, we will use the quadratic operator:

For an element a in a Jordan algebra J , we denote by U a the mapping b ↦ a ∘ ( a ∘ b ) − a 2 ∘ b from J to J and note that when J is special, U a ( b ) = a b a . This operator satisfies the fundamental formula U U a ( b ) = U a U b U a for Jordan algebras which was derived from the so-called Macdonald’s theorem.

It is well known that invertibility plays a fundamental role in functional analysis. Hence, the importance of developing the notions of generalized inverse and generalized spectral theory in the Jordan algebras.

In order to give, in the Jordan case, an adequate definition of generalized invertibility, we must remember two prerequisites: First, this definition will allow us to find the one already known in the associative case, since a large class of Jordan algebras come from associative algebras by replacing the initially associative product by that of Jordan. Second, it must simplify the extension of spectral analysis when the algebra in question is endowed with an Jordan-Banach norm. We are also inspired by the works [3] and [4] where the authors used the quadratic operator instead of the multiplication operator.

Definition 2.1Let J be a Jordan algebra and let a be in J . An element c ∈ J is called a generalized inverse of a if the equality a = U a ( c ) holds.

For example, if b is the inverse of a , then b is a generalized inverse of a . If a = a 2 is a projection, then a = U a ( c ) holds with c = a .

In view of the above fundamental formula, this definition emphasizes that, the operator U a satisfies

The above definition generalizes the associative forerunner: Suppose that J is a special Jordan algebra A + . Then the definition 1.1 is formulated here as follows.

which is the well-known generalized inversibility of a in the associative algebra A (see [5]-[8]). Note also that U a ( b ) = a is actually expressed by saying that a isvon Neumann regular.

Proposition 2.1If c is a generalized inverse of a ∈ J then the set of all generalized inverses of a consists of all elements of the form x = c + u − U c U a ( u ) , where u is arbitrary in J .

Proof. Suppose that a = U a ( c ) and x = c + u − U c U a ( u ) for some u ∈ J then

Conversely, if U a ( x ) = a then the operator P = U c U a is a projector, so J = I m ( P ) ⊕ K e r ( P ) with

It follows that the element u = x − c satisfies x = c + u − U c U a ( u ) .

Proposition 2.2Let a be an element of J admitting a generalized inverse c then there exists an element b in J such that

Proof. It is a direct consequence of the fundamental formula.

Throughout the following, B ( J ) denotes the algebra of operators of J .

Proposition 2.3If an element a of J admits a generalized inverse b then U b is a generalized inverse of U a in the associative algebra B ( J )

Proof. Suppose that a , b ∈ J satisfy a = U a ( b ) . Then

Then U b is a generalized inverse of U a in B ( J ) .

In order to state a converse of this proposition, we have:

Proposition 2.4If a ∈ J and T ∈ B ( J ) satisfy U a T U a = U a and T U a T = T then V = U a T (respectively, W = T U a )is a projection of J upon I m ( V ) (respectively, I m ( W ) )in the direction of K e r ( V ) (respectively, K e r ( W ) ).

Proof. Obvious.

Now we need to involve in our development a deep result in the theory of generalized invertibility. This is to unify the different approaches to generalized inverses in the algebraic context, which reads as follows.

Proposition 2.5If a ∈ J and T ∈ B ( J ) satisfy U a T U a = U a and T U a T = T then T is the unique operator of J solution of the system U a X = U a T , X U a = T U a and X U a X = X .

Proof. It is enough to apply ([7], Theorem 1.3(a)) with P = I − T U a and Q = U a T to realize that conclusion in the proposition is fulfilled. We have, P 2 = ( I − W ) 2 = I − 2 W + W 2 = I − W = P and U a ( P ( x ) ) = U a ( x ) − U a T U a ( x ) = 0 for all x ∈ J . Then P ( J ) ⊂ K e r ( U a ) , likewise Q 2 = Q and I m ( Q ) = I m ( U a ) . So, the conditions of ([7], Theorem 1.3(a)) that we want to use for our proof are verified.

Theorem 2.1Let a be not a divisor of zero in J and suppose the operator U a admits ageneralizedinverse T in B ( J ) . Then the following assertions are satisfied:

i) a admits a generalized inverse b = T ( a ) .

ii) T = U T ( a ) .

Proof. Proposition 2.4 allows us to obtain that a = U a T ( u ) + v and a = T U a ( y ) + z , for unique quadruple ( u , v , y , z ) in J with v ∈ K e r ( U a T ) and z ∈ K e r ( T U a ) .

Hence T ( a ) = T ( U a T ( u ) + v ) = T U a T ( u ) + T ( v ) = T ( u ) + T ( v ) and U a T ( a ) = U a T ( u ) + U a T ( v ) = U a T ( u ) + 0 = U a T ( u ) . So, a = U a T ( a ) + v , v ∈ K e r ( U a T ) . On the other hand, T ( a ) = T ( U a T ( a ) + v ) = T U a T ( a ) + T ( v ) = T ( a ) + T ( v ) then T ( v ) = 0 .

Similarly, a = T U a ( a ) + z with U a ( z ) = 0 . Since a is not a divisor of zero then z = 0 and a = T U a ( a ) .

Now, by considering R = 2 I − V − W + T + U a ∈ B ( J ) , we have.

Then

So, W ( v ) = − W ( v ) and W ( v ) = 0 . Since U a = U a T U a = U a W , then

Since a is not a divisor of zero then v = 0 and a = U a ( T ( a ) ) . Then c = T ( a ) is a generalized inverse of a . We claim that

To prove the claim, the equality U a ( T ( a ) ) = a and the fundamental formula give us U a U T ( a ) U a = U a and, multiplying on the right by U T ( a ) , we have U a ( U T ( a ) U a U T ( a ) ) = ( U a U T ( a ) U a ) U T ( a ) = U a U T ( a ) . Since U a is injective, we have U T ( a ) U a U T ( a ) = U T ( a ) , as desired.

Finally, in order to prove assertion (ii), note that it suffices to show that U a U T ( a ) ( x ) = U a T ( x ) , for every x ∈ J . The conclusion follows from the fact that U a is injective. Assume at first that x = U a ( α ) ∈ I m ( U a ) . Then

To conclude the proof, we must show that the same conclusion holds for an arbitrary element x ∈ J . Since U a T and U a U T ( a ) are projectors satisfying

Decomposing x into these two direct sums respectively, x = α + v and x = β + w , we have:

It follows from the first case that U a T ( x ) = U a T ( β ) + U a T ( w ) = β + U a T ( w ) and U a U T ( a ) ( x ) = U a U T ( a ) ( α ) + U a U T ( a ) ( v ) = α + U a U T ( a ) ( v ) . So,

then

So, α − β = U a T ( w ) which leads to w − v = U a T ( w ) and x = α + w − U a T ( w ) . Furthermore, if we set P = U a T and Q = U a U T ( a ) and if we apply the operator Z = 2 I − P − Q + U T ( a ) + T to the two decompositions of x , then we will obtain

and

Then U T ( a ) ( α ) + T ( α ) + Z ( w ) + U T ( a ) ( P ( w ) ) + T ( P ( w ) ) = U T ( a ) ( β ) + T ( β ) + Z ( w ) and U T ( a ) ( α ) + T ( α ) + U T ( a ) ( P ( w ) ) + T ( P ( w ) ) = U T ( a ) ( β ) + T ( β ) . Finally, by applying the linear operator U a to both members of these last equality we derive that

and consequently

Hence, we obtain that

Thus, P ( w ) = − P ( w ) = 0 , and then U a T ( x ) = α = β = U a U T ( a ) ( x ) as desired. Then T = U T ( a ) .

Keeping in mind the arguments and notions above, the following theorem follows easily.

Theorem 2.2Let J be a Jordan algebra over K ( K = ℝ or K = ℂ )and let a be an element in J . We have:

(i) If a possesses a generalized inverse b in J , then the operator U b is a generalized inverse of U a in B ( J ) .

(ii) If a is not a divisor of zero in J and suppose the operator U a admits a generalized inverse T in B ( J ) . Then T ( a ) is a generalized inverse of a in J .

Note that (ii) of the theorem 2.2 shows that generalized inverses of a verifies the well-known equality satisfied by the inverse a − 1 when it exists in the Jordan-algebra J : U a − 1 = U a − 1 (see [2], Theorem 4.1.3).

Proposition 2.6Let J = A + be a special Jordan algebra.For every a ∈ A the following conditions are equivalent:

i) a admits a generalized inverse in A.

ii) a admits a generalized inverse in J.

Proof. Let a ∈ A , by the definition of the operator U a in J , we have

Proposition 2.7If an element a non-divisor of zero in J admits a generalized inverse then the operator U a is left invertible in B ( J ) .

Proof. It follows from Proposition 2.2 that there exists b ∈ J such that a = U a ( b ) and U b ( a ) = b . Which is said in terms of operators that U a U b U a = U a and U b U a U b = U b . So the projector U b U a satisfies J = I m ( U b U a ) ⊕ K e r ( U b U a ) and K e r ( U b U a ) = K e r ( U a ) = { 0 J } . Then we have that U b U a is invertible, that is to say, if we write T U b U a = U b U a T = I d J , from which we deduce that U a is left invertible.

In the following section, J denotes a complex unital Jordan-Banach algebra with unit 1 J . For a ∈ J , σ ( a ) , ϱ ( a ) , and r ( a ) denote the spectrum, the resolvent set and the spectral radius of a, respectively (see [2], Chapiter 4).

Recall that

Note that R ( a , λ ) = ( a − λ 1 J ) − 1 the resolvent of a at point λ is analytic on the open ϱ ( a ) .

Spectrum theory and spectral analysis play a fundamental role in functional analysis. Thus, we would like to define the generalized spectrum where the notion of inverse is replaced by the generalized inverse. Given an element a of J , the natural idea is to define the generalized spectrum of a by considering

But this definition does not preserve the main properties of classical spectral analysis even in associative cases ([5], p. 70). The same defects are obtained by considering the author’s special Jordan algebra. In other words, S ( a ) can be empty for an element a of J and especially the famous spectral mapping theorem is missing. In the following, we extend to Jordan-Banach algebras the notion of generalized spectral analysis defined and studied in associative algebras by numerous authors.

Definition 3.1Let Ω be a part of ℂ ,we will say that a admits a generalized resolvent R g ( a , μ ) in Ω if for all μ ∈ Ω ,there exists R g ( a , μ ) in J such that

U ( a − μ I ) ( R g ( a , μ ) ) = a − μ I and U R g ( a , μ ) ( a − μ I ) = R g ( a , μ )

Definition 3.2An element a of J admits a generalized resolvent R g ( a , λ ) in an open U of ℂ if

U a − λ 1 J ( R g ( a , λ ) ) = a − λ 1 J and U R g ( a , λ ) ( a − λ 1 J ) = R g ( a , λ ) for every λ ∈ U .

In this definition, the condition (which may seem tedious) of the existence of an open U on which a − λ 1 J admits a generalized resolvent is automatically ensured within the framework of classical invertibility. Indeed, the set of invertible elements in J (denoted I n v ( J ) ) is open and the mapping x ↦ x − 1 from I n v ( J ) to J is differentiable at any point a ∈ I n v ( J ) , with derivative equal to the mapping − U a − 1 (See [2], Theorem 4.1.7).

We will also use the following.

Definition 3.3The generalized resolvent set or the regular set of a ∈ J (denoted by R g ( a ) )is the subset of ℂ formed of numbers λ such that a admits an analytical generalized resolvent R g ( a , μ ) in a neighborhood Ω of λ . which means

Example 3.1Let A be the Banach algebra of operators of a Hilbert space H, J is the special Jordan-Banach algebra A⁺. It follows from Proposition2.6 above and ([5], p. 71) that λ is in Rg(a) if and only if

I m ( A − λ I ) is closed and ker ( A − λ I ) n ⊂ I m ( A − λ I )

Definition 3.4The generalized spectrum of an element a of J (relative to J )which will be denoted by σ g ( a ) ,is the complement in ℂ of R g ( a ) .

As the complement of an open, σ g ( a ) is closed. Obviously σ g ( a ) is contained in the compact σ ( a ) , so σ g ( a ) is a compact (since we will show that it is never empty).

As another relation between these two spectra of an element a of J we have the following:

Proposition 3.1Let a in J and ∂ σ ( a ) denotes the boundary of the spectrum σ ( a ) then

Proof. Assume, to derive a contradiction, that the proposition is not true. Then there exists λ 0 ∈ ∂ σ ( a ) ∩ R g ( a ) . So, a admits an analytical generalized resolvent R g ( a , μ ) in a neighborhood Ω of λ 0 which coincides with the analytical resolvent R ( a , μ ) = ( a − μ ) − 1 on the non-empty open set Ω ∩ R ( a ) . So, ( a − λ 0 ) − 1 = R ( a , λ 0 ) exists, which is a contradiction.

Tow easy, but not so straightforward results, are the following.

Corollary 3.1For each a ∈ J ,the generalized spectrum σ g ( a ) of a is not empty.

Corollary 3.2If for a ∈ J its generalized spectrum is at most countable then its two spectrums are equal: σ ( a ) = σ g ( a ) .

Proof. Since σ g ( a ) ⊂ σ ( a ) , it suffices to prove the reciprocal inclusion. Now σ ( a ) is at most countable, then it coincides with its boundary which is contained in σ g ( a ) , as desired.

Proposition 3.2Let a in J and K denotes a connected component of R g ( a ) . Then

Proof. Assume the existence of λ 0 ∈ K ∩ R ( a ) .

So, a admits an analytical generalized resolvent R g ( a , μ ) in a neighborhood Ω of λ 0 which coincides with the analytical resolvent R ( a , μ ) = ( a − μ ) − 1 on the non-empty open set Ω ∩ R ( a ) . So, ( a − λ 0 ) − 1 = R ( a , λ 0 ) exists, which is a contradiction.

Proposition 3.3Let a in J andassumethat σ g ( a ) is at most countable. Then

Proof. The fact that σ g ( a ) is at most countable implies R g ( a ) is connected. it is enough to replace K with R g ( a ) in proposition 3.2 to realize that R g ( a ) = ϱ ( a ) and then σ g ( a ) = σ ( a ) .

Now we define the conorm in a Jordan-Banach algebra, along with some associated results.

Definition 3.5The conorm γ ( a ) of a ∈ J is defined by:

and

Note that the connorm of a in J defined here coincides with the classical connorm of the operator U a in the Banach algebra B ( J ) .

Proposition 3.4If a is invertible in J then γ ( a ) = ‖ U a − 1 ‖ − 1 .

Proof. According to Definition 3.5, we have

Theorem 3.1If an element a of J has a generalized inverse b in J then:

(i) U a ( J ) = { U a ( x ) : x ∈ J } is closed;

(ii) γ ( a ) > 0 :

(iii) if b ∈ J is a generalized inverse of a , then

Proof. Let b ∈ J such that U a ( b ) = a . Then U a = U U a ( b ) = U a U b U a . Put P = U a U b and Q = I − U b U a . Then it is easy to see that P 2 = P and Q 2 = Q . Since U a ( J ) = U a U b U a ( J ) = P U a ( J ) then

and U a ( J ) = P ( J ) is closed. This shows (i).

To prove assertion (ii) it is enough to show that ker ( U a ) = Q ( J ) . Use (i) and [9] (Satz 55.2) to see that (ii) holds.

Now take x ∈ J such that d ( x , ker ( U a ) ) = 1 . Since U a ( x ) = U a U b U a ( x ) , x − U b U a ( x ) ∈ ker ( U a ) , thus

Hence

This gives 1 ‖ U b ‖ ≤ γ ( a ) .

Since ‖ U b ‖ ≤ 3 ‖ b ‖ 2 , it follows that 1 3 ‖ b ‖ 2 ⩽ γ ( a ) , for all generalized inverse b of a .

It follows from the above proposition that γ ( a ) = ‖ U a − 1 ‖ − 1 when a is invertible, this shows that the mapping a → γ ( a ) is not continuous. This justifies the hypothesis of the existence of the limit in the following.

Proposition 3.5Let a in J and λ 0 be a complex number. Assumethat lim λ → λ 0 γ ( a − λ 1 J ) exists. Then the following assertions are equivalent:

(i) λ 0 ∈ R g ( a )

(ii) lim λ → λ 0 γ ( a − λ 1 J ) > 0

Proof. If λ 0 ∈ R g ( a ) , then there exists r 0 > 0 such that the open disc D ( λ 0 , 2 r 0 ) is contained in R g ( a ) . Using the remark γ ( a − λ 1 J ) = d ( λ , σ g ( a ) ) mentioned above, therefore γ ( a − λ 1 J ) ≥ r 0 for all λ ∈ D ( λ 0 , r 0 ) . Hence lim λ → λ 0 γ ( a − λ 1 J ) > 0 .

Conversely, if r = lim λ → λ 0 γ ( a − λ 1 J ) > 0 , then the open disc D ( λ 0 , r 2 ) is contained in R g ( a ) , and the implication (ii) ⇒ (i) is proved.

The author thanks the reviewers for their valuable advice, which contributed to the preparation of this work.