#### Abstract

We initiate a study of quasi-Jordan normed algebras. It is demonstrated that any quasi-Jordan Banach algebra with a norm unit can be given an equivalent norm making the algebra isometrically isomorphic to a closed right ideal of a unital split quasi-Jordan Banach algebra; the set of invertible elements may not be open; the spectrum of any element is nonempty, but it may be neither bounded nor closed and hence not compact. Some characterizations of the unbounded spectrum of an element in a split quasi-Jordan Banach algebra with certain examples are given in the end.

#### 1. Introduction

Looking back at the development of modern mathematics, we see that early formal study of algebra was mostly commutative and associative. With an abstract study of functions and matrices, it became noncommutative but still associative; then introduction of nonassociative structures, such as Lie structures due to Sophus Lie [1] and Jordan structures due to Jordan [2], has led us to the mathematics which at present is noncommutative as well as nonassociative.

There is a strong relationship between Lie algebras and Jordan algebras [3]. Jordan structures have been extensively studied by a large number of mathematicians: P. Jordan, von Neumann, E. Wigner, N. Jacobson, K. McCrimmon, R. Braun, M. Koecher, E. Neher, and O. Loos, to name but a few. A vast literature containing many important results on Jordan algebras has been developed (cf. [3, 4]). After the mid-1960s, people began studying Jordan structures from the point of view of functional analysis. Interesting theories of Jordan Banach algebras, -algebras, -algebras, and -triples have been developed, which closely resemble that of -algebras and have found surprisingly important applications in a wide range of mathematical disciplines including analysis, physics, and biology (cf. [5–7]).

A Jordan algebra is a nonassociative algebra with the product satisfying and the Jordan identity: , where . Any associative algebra becomes a Jordan algebra with the same linear space structure and the Jordan product ; it becomes a Lie algebra under the skew-symmetric product , so called the Lie bracket (cf. [4]). For any Jordan algebra , there is a Lie algebra such that is a linear subspace of and the product of can be expressed in terms of the Lie bracket in . Moreover, the universal enveloping algebra of a Lie algebra has the structure of an associative algebra; see the original work due to Kantor, Koecher, and Tits appearing in [8–10].

Loday introduced a generalization of Lie algebras, called the Leibniz algebras [11, 12], and successfully demonstrated that the relationship between Lie algebras and associative algebras can be translated into an analogous relationship between Leibniz algebras and the so-called dialgebras (cf. [13]): a* dialgebra* over a field is a -module equipped with two bilinear associative maps satisfying ; ; and . The maps and are called the left product and the right product, respectively. Any dialgebra () becomes a Leibniz algebra under the Leibniz bracket , and the universal enveloping algebra of a Leibniz algebra has the structure of a dialgebra; for details, see [12, 13].

Recently in [14], Velásquez and Felipe introduced the notion of quasi-Jordan algebras, which have relation with the Leibniz algebras similar to the existing relationship between the Jordan algebras and the Lie algebras [15]. The quasi-Jordan algebras are a generalization of Jordan algebras where the commutative law is replaced by a quasicommutative identity and a special form of the Jordan identity is retained. These facts indicate the significance of studying the quasi-Jordan algebras; within a few years time, many mathematicians, including M. K. Kinyon, M. R. Bremner, L. A. Peresi, J. Sanchez-Ortega, and V. Voronin, have got their interests in this new area. In [16], Felipe made an attempt to study dialgebras from the functional analytic point of view.

A Jordan Banach algebra is a real or complex Jordan algebra with a complete norm satisfying ; basics of Jordan Banach algebras may be seen in [7]. In this paper, we initiate a study of the* quasi-Jordan normed algebras*. The class of complete quasi-Jordan normed algebras, called* quasi-Jordan Banach algebras*, properly includes all Jordan Banach algebras and hence all -algebras (cf. [7]). This study may provide a better mathematical foundation for some important areas such as quantum mechanics. We are interested specially in extending, as much as possible, the theory of Jordan Banach algebras to the general setting of quasi-Jordan Banach algebras. We investigate the notions of invertibility and spectrum of elements in the setting of unital quasi-Jordan Banach algebras. Among other results, we demonstrate that any quasi-Jordan Banach algebra with a norm unit can be given an equivalent norm that makes the algebra isometrically isomorphic to a norm closed right ideal of a unital split quasi-Jordan Banach algebra. We show that the set of invertible elements in a unital quasi-Jordan Banach algebra generally is not open and that the spectrum of any element is nonempty but it may be neither bounded nor closed, hence not compact. Moreover, if the spectrum of an element in a complex unital split quasi-Jordan Banach algebra (see below) is unbounded then it coincides with the whole complex plane and vice versa. Some examples are also given in the end.

#### 2. Quasi-Jordan Banach Algebras

We begin by recalling some basics of the quasi-Jordan algebra theory from [14, 17]. A* quasi-Jordan algebra* is a vector space over a field of characteristic equipped with a bilinear map , called the* quasi-Jordan product*, satisfying * (right commutativity)* and * (right Jordan identity)*, for all . Here, and for ; in the sequel, we will see that may not coincide with . Every quasi-Jordan algebra includes two important sets: * the linear span of the elements* and , respectively, called the* annihilator* and the* zero part* of . It follows from the right commutativity that and that the quasi-Jordan algebra is a Jordan algebra if and only if .

*Example 1. *Let be a dialgebra over a field of characteristic . One can define another product by for all , which satisfies the identities ; ; and ; however, this quasi-Jordan product generally is not commutative. Therefore, is a quasi-Jordan algebra, which may not be a Jordan algebra (cf. [14]). This quasi-Jordan algebra is denoted by , called a* plus quasi-Jordan algebra*. Any quasi-Jordan algebra which is a homomorphic image of a plus quasi-Jordan algebra is called* special*.

We define a* quasi-Jordan normed algebra* as a quasi-Jordan algebra over the field of complex numbers endowed with a norm satisfying , for all . Thus, the quasi-Jordan product “” in a normed quasi-Jordan algebra is continuous. A quasi-Jordan normed algebra is called a* quasi-Jordan Banach algebra * if it is complete as a normed space.

If there is an element in a quasi-Jordan algebra satisfying for all , called the left unit, then becomes commutative and hence a Jordan algebra. Due to this fact, we will consider only the* right unit* elements; henceforth, unit in a quasi-Jordan algebra would mean a right unit. An element in a quasi-Jordan algebra is called a* unit* if , for all . A quasi-Jordan algebra may have many (right) units (cf. [15, 17]). If the dialgebra has a* bar-unit * (i.e., ) then , for all , and hence is a unit of the plus quasi-Jordan algebra .

For any quasi-Jordan algebra with unit , we know from [17] that and are two-sided ideals of , , and , where denotes the set of all (right) units in .

One can always attach a (two-sided) unit to any Jordan algebra by following the standard unitization process. This unitization process no longer works for quasi-Jordan algebras (cf. [17, pages 210-211]). Adding a unit to a quasi-Jordan algebra is yet an open problem. In giving a partial solution to this unitization problem, Velásquez and Felipe [17] introduced the following special class of quasi-Jordan algebras, called split quasi-Jordan algebras: let be a quasi-Jordan algebra and let be an ideal in such that . We say that is a* split quasi-Jordan algebra* (more precisely, is* split over* ) if there exists a subalgebra of such that , the direct sum of and . It is easily seen that such a subalgebra is a Jordan algebra. One can attach a unit to any split quasi-Jordan algebra; details of such a unitization process are given in [17]. If the algebra has a unit then . Thus, a quasi-Jordan algebra with unit is a* split quasi-Jordan algebra* if and only if for some subalgebra of ; the algebra is called the* Jordan part* of . In such a case, each element has a unique representation with and , respectively, called the* Jordan part* and the* zero part* of . Moreover, if is split quasi-Jordan algebra with unit , then there exists a unique element which acts as a unit of the quasi-Jordan algebra and at the same time a unit of the Jordan algebra .

Proposition 2. *Let be a unital split quasi-Jordan Banach algebra with unit . Then the Jordan part is a norm closed subalgebra of and hence a unital Jordan Banach algebra.*

*Proof. *From the above discussion, it is clear that the Jordan part is a unital Jordan normed algebra. Next, let be any fixed Cauchy sequence in . Then, the same is Cauchy sequence also in the quasi-Jordan Banach algebra since is a normed subspace of . Hence, for some .

Now, since each and since is the unit of the Jordan algebra under the (restricted) product “,” we get for all . So, by the continuity of the product, . Hence, by the uniqueness of the limit of any convergent sequence in a normed space, . However, . Thus, the required result follows.

We know from [17] that for any quasi-Jordan algebra , equipped with the sum , scalar multiplication , and product is a split quasi-Jordan algebra and the map is an embedding of into , where stands for the usual right multiplication operator on . It is easily seen that is the Jordan part of and the zero part . Moreover, the embedding preserves the units: clearly, for all so that is a unit in whenever is a unit in . In fact, is a unit in , for all ; it may be noted here that is the only unit in the Jordan part of .

From [17], we also know that , with product “” defined by , for all , is a quasi-Jordan algebra. Moreover, is the direct product of the quasi-Jordan algebras and .

Indeed, the split quasi-Jordan algebra is a quasi-Jordan Banach algebra with unit whenever is a quasi-Jordan Banach algebra with a norm unit , and that is a closed unital quasi-Jordan normed subalgebra of . To justify this claim, we need the following result.

Proposition 3. *Suppose is a quasi-Jordan Banach algebra with a norm unit . Then the algebra as above is a quasi-Jordan Banach algebra with unit of norm .*

*Proof. *Clearly, each is a bounded linear operator with . Hence, the usual operator norm is a norm on the quasi-Jordan algebra with the quasi-Jordan product . Further, we observe that

Alternately, by exploiting the right commutativity of the quasi-Jordan product in , we get

Thus, together with the operator norm is a quasi-Jordan normed algebra. Moreover, for any , we have and ; that is, is a norm (right) unit in .

Suppose is any fixed Cauchy sequence in . Then, for any fixed , as , and so is a Cauchy sequence in . But, is complete. Hence, the sequence for any is convergent in . In particular, the sequence converges to some . Moreover, the Cauchy sequence converges to in the operator norm because as , for all . Thus, the quasi-Jordan normed algebra is complete.

Now, we show that the corresponding algebra is a unital split quasi-Jordan Banach algebra.

Proposition 4. *Let be a quasi-Jordan Banach algebra with a norm unit and let be as above. Then is the direct product of the quasi-Jordan algebras and . Moreover, equipped with the product norm is a split quasi-Jordan Banach algebra with unit , and is a closed right ideal in in the norm topology.*

*Proof. *For the first part, see [17]. Clearly, is a norm, and it satisfies
for all . Keeping in view Proposition 3, we deduce that being the product of complete spaces and is complete in the product norm. Thus, is a split quasi-Jordan Banach algebra with (right) unit .

Clearly, is a subspace of with for all , , so that is a quasi-Jordan normed subalgebra with the unit included. Further, let be any fixed Cauchy sequence in . Then as , so that is a Cauchy sequence in (the complete space) and hence it converges to some . Now, by using the fact for all (since is a norm unit in ; see above), we get the convergence of arbitrarily fixed Cauchy sequence to since as . Thus, being complete is a closed right ideal in in the norm topology.

Next, we observe the isometry between and .

Proposition 5. *Let be a quasi-Jordan Banach algebra with norm unit. Then there exists an equivalent norm that makes isometrically isomorphic to a quasi-Jordan closed subalgebra of the split quasi-Jordan Banach algebra .*

*Proof. *Clearly, defines a norm on the quasi-Jordan algebra . It follows that is a quasi-Jordan Banach algebra. Moreover, is isomorphic to the subalgebra of under the isomorphism , as seen above. Further, we observe that is an isometry since . Finally, we note that the two norms and on are equivalent since .

#### 3. Invertible Elements

As in [17], an element in a quasi-Jordan algebra is called* invertible with respect to a unit* if there exists such that and ; such an element is called an* inverse of * * with respect to* . Let denote the element . Then, has an inverse with respect to and .

We know from the above discussion that the embedding of a quasi-Jordan algebra into the split quasi-Jordan algebra preserves the units. The embedding also preserves the corresponding invertible elements: if is an inverse of with respect to a unit in , then and . Hence, Thus, is an inverse of , with respect to the unit , in .

Let be a unital split quasi-Jordan algebra. Then, = , for all . Thus, for all .

In this section, we demonstrate that the set of invertible elements, with respect to a fixed unit, in a quasi-Jordan Banach algebra, may not be open. For this, we proceed as follows.

Proposition 6. *Let be a unital split quasi-Jordan algebra with unit . If is invertible with respect to , then so is for all .*

*Proof. *Let be an inverse of in with respect to . We show that is an inverse of with respect to . Observe that
Hence, by the uniqueness of the representation as sum of Jordan and zero parts, we get , , , and . Therefore, = and = because and .

Proposition 7. *Let be a quasi-Jordan normed algebra with a unit . Let be an open set and . Then for all .*

*Proof. *Suppose and . If then . Next, suppose . Since is an open set, there exists such that whenever . Hence, with .

Let and be inverses of and with respect to , respectively. Then
Hence, by setting with , we see that
since . Thus, is an inverse of with respect to the unit .

Corollary 8. *Under the hypothesis of Proposition 7, implies for all .*

Proposition 9. *Let be a split quasi-Jordan normed algebra and let be a unit in such that the set is open. Then for all .*

*Proof. *Of course, the element is the unit of the Jordan algebra . Then, for any fixed element , there exists such that is invertible in ; otherwise, we would get the negation of the well-known fact that spectrum of an element of a unital Jordan algebra is bounded. That is, there exists such that and . However, . Hence, is an inverse of in the quasi-Jordan algebra with respect to the unit . By Corollary 8, is also invertible for any . This in turn gives the existence of satisfying and . Multiplying the first equation from the right by , the second equation by , and using the right Jordan identity, we get
Hence,
so that
This last equation reduces to since and is the unit of . Hence, for all and . Now, for any , the last equation with and gives . Thus, = for all .

Corollary 10. *If a unital split quasi-Jordan algebra has an element with then the set of invertible elements, with respect to the unit of the Jordan part, is not open.*

In the sequel, we will show the existence of a unital split quasi-Jordan Banach algebra with elements such that . Thus, the above result establishes that the set of invertible elements, with respect to a fixed unit, in a quasi-Jordan Banach algebra may not be open.

#### 4. The Spectrum of Elements in a Unital Quasi-Jordan Algebra

As usual, we define the spectrum of an element in a unital quasi-Jordan algebra , denoted by , to be the collection of all complex numbers for which is not invertible. Thus, . Here, the subscript indicates that the invariability depends on the choice of unit , which generally is not unique.

Proposition 11. *Let be a unital quasi-Jordan algebra. Then for all .*

*Proof. *Let . Then, for any , is an inverse of with respect to the unit because and + .

Proposition 12. *Let be a quasi-Jordan algebra with unit . Then for all .*

*Proof. *For any fixed and nonzero scalar , the vector satisfies and + . So is an inverse of with respect to . This means for all . However, the zero vector is not invertible. Thus, .

Proposition 13. *Let be a split quasi-Jordan algebra with unit . Then for all idempotents (i.e., ).*

*Proof. *Let be any fixed idempotent in . Since , has a unique representation with and . Clearly, . Then, by uniqueness of the representation in the split quasi-Jordan algebra , and ; this means is an idempotent in the Jordan algebra . Hence, . Thus, is invertible in with the unique inverse , for all .

We show that is an inverse of in with respect to the unit ; for this, we note that , , = , and = = = = = = = . Hence, = = = = = = = = = and = = = = = = = = = = = .

As mentioned in Section 2, if is a quasi-Jordan algebra with a unit then the set coincides with the set of all units in .

Proposition 14. *Let be a unital split quasi-Jordan algebra with unit and invertible with respect to some . Then is invertible, with respect to the unit , in the Jordan algebra .*

*Proof. *Clearly, and . Since , we have for some . Hence, the invertibility of in , with respect to the unit , gives the existence of such that and . So, by the uniqueness of the representations in the split algebra , we get and . Thus, is the inverse of in the Jordan algebra .

Next, we observe that the spectrum of in a unital split quasi-Jordan algebra with respect to any unit includes the spectrum of in the Jordan part .

Corollary 15. *Let be a unital split quasi-Jordan algebra with unit , and let . Then for all .*

*Proof. *Let with , and let . Then, is invertible in with respect to the unit . Hence, its Jordan part is invertible in the Jordan algebra by Proposition 14. Thus, .

It is well known that the spectrum of any element in a unital Jordan Banach algebra is nonempty (cf. [7]). This together with Proposition 2 and Corollary 15 gives the following result.

Corollary 16. *The spectrum of any element in a unital split quasi-Jordan Banach algebra is nonempty.*

The next result extends Corollary 16 to any quasi-Jordan Banach algebra with a norm unit.

Proposition 17. *The spectrum of any element in a quasi-Jordan Banach algebra with a norm unit is nonempty.*

*Proof. *Let be a unital quasi-Jordan Banach algebra with a norm unit . From Section 2, we know that the map embeds into the unital split quasi-Jordan Banach algebra , equipped with the sum , scalar multiplication , and product , and the image is a norm closed right ideal isomorphic to with norm unit . Moreover, it is seen in Section 3 that the embedding also preserves the corresponding invertible elements; that is, is an inverse of , with respect to the unit , in whenever is an inverse of , with respect to a unit , in . Hence, by Corollary 16, it follows that , for all .

Proposition 18. *Let be a unital split quasi-Jordan algebra with unit and let . Then .*

*Proof. *By Corollary 15, . For the reverse inclusion, let , then is invertible in ; that is, there exists such that and . However, . Hence, is an inverse of in with respect to the unit ; that is, . Thus, .

Proposition 19. *Let be a unital quasi-Jordan normed algebra, and let be a unit in for which is open. Then , for all .*

*Proof. *By Corollary 8, is invertible if and only if is invertible, for all . Thus, if and only if for all .

Corollary 20. *Let be a unital split quasi-Jordan normed algebra, and let be a unit in such that is open. Then for all . Further, if the unit then .*

Lemma 21. *Let be a unital split quasi-Jordan algebra with a unit , and let be invertible with respect to . Then .*

*Proof. *As is invertible, there exists such that and . Hence, by the uniqueness of the representation in a split quasi-Jordan algebra, we obtain
Thus,

Proposition 22. *Let be a unital split quasi-Jordan algebra with a unit ; satisfies . Then,*(1),(2).

*Proof. *(1) Let . Then is invertible with respect to . By Lemma 21, we have
since the zero part of is . However,

Therefore, (14) becomes
which after simplification reduces to the required equation .

(2) Since , we have
by the part . But, since is in the Jordan algebra . Therefore, .

*Remark 23. *In any quasi-Jordan algebra, if an element satisfies , then for all positive integers . For this, suppose satisfies and for any fixed . Then (by the right Jordan identity) .

Proposition 24. *Let be a unital quasi-Jordan Banach algebra with unit , and let satisfy . Then whenever .*

*Proof. *First note that gives for all . Hence, the infinite geometric series converges absolutely to some element . We show that the geometric series sum is an inverse of , with respect to the unit . For any fixed positive integer , let . Then, the sequence of partial sums converges to . By setting , we get
Thus, by allowing , we obtain since .

Next, by Remark 23, we have
Taking the limit as , we get

Proposition 25. *Let be a unital split quasi-Jordan Banach algebra with unit . If with , then for all .*

*Proof. *If with , then, the noninvertibility of means the noninvertibility of , with respect to the unit . However, by Proposition 24, must be invertible with respect to the unit , whenever . It follows that for all .

#### 5. Unbounded and Nonclosed Spectrum

In this section, we show that the spectrum of an element in a split quasi-Jordan Banach algebra may be neither bounded nor closed, and hence not compact. The following result gives a couple of characterizations of the unbounded spectrum of an element in a split quasi-Jordan Banach algebra.

Proposition 26. *Let be an element of a unital split quasi-Jordan Banach algebra with a unit . Then, the following statements are equivalent.*(1)*.*(2)*.*(3)*, for all **.*

*Proof. *(12): See Proposition 22.

(23): Suppose . Then,

From (21), we get
Hence, , for all by Proposition 25.

(31): Immediate.

*Remark 27. *There do exist unital split quasi-Jordan algebras containing elements that have the spectrum, with respect to the unit of the Jordan part, equal to the whole of , and hence unbounded. To justify this claim, we proceed as follows.

Let be a unital associative algebra and let be an -bimodule. Let be an -bimodule map (i.e., an additive map satisfying and , for all , ). Then, one can put a dialgebra structure on as follows: and (cf. [13, Example 2.2(d)]). Hence, is a quasi-Jordan algebra under the quasi-Jordan product “” given by . Further, for any , we observe that
However, the right hand sides of the above equations (23) may not be equal; see the following example (Example 28). For such elements , we have . Hence, by Proposition 26, the spectrum of is unbounded whenever is a unital split quasi-Jordan Banach algebra.

*Example 28. *Let be the collection of matrices with entries from the field , and let be the algebra of all matrices of the form with . Then, it is easily seen that is an -bimodule. Next, we define by . Of course, is an additive map satisfying
Hence, is an -bimodule map. Thus, by Remark 27, is a quasi-Jordan algebra with the quasi-Jordan product as below:

Indeed, , where is a subalgebra of and