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«HEREDITY IN FUNDAMENTAL LEFT COMPLEMENTED ALGEBRAS Marina Haralampidou and Konstantinos Tzironis Abstract. In the present paper, we introduce the ...»

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Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)

Volume 11 (2016), 93 – 106

HEREDITY IN FUNDAMENTAL LEFT

COMPLEMENTED ALGEBRAS

Marina Haralampidou and Konstantinos Tzironis

Abstract. In the present paper, we introduce the notion of a fundamental complemented linear

space, through continuous projections. This notion is hereditary. Relative to this, we prove that if a certain topological algebra is fundamental, then a concrete subspace is fundamental too. For a fundamental complemented linear space, we define the notion of continuity of the complementor. In some cases, we employ a generalized notion of complementation, that of (left) precomplementation.

In our main result, the continuity of the complementor for a certain fundamental complemented (topological) algebra is inherited to the induced vector complementor of the underlying linear space of a certain right ideal. Weakly fundamental algebras are also considered in the context of locally convex ones.

1 Introduction and Preliminaries In 1970, F.A. Alexander dealt with representation theorems in the context of Banach complemented algebras [1]. For this, and among others, continuity of the complementor is assumed. A respective representation theory, in the non-normed case, is faced in [11]. An appropriate context to work in this theory, is that of fundamental (pre)complemented algebras, studied in [6] and [18]. Here, the genetic property has to do with the existence of certain continuous linear maps (Definition 1). Fundamentality is also considered in the context of topological linear spaces (Definition 17). For certain topological algebras being also fundamental, the latter property is inherited to appropriate subspaces (Theorem 19); this information is used to obtain our main result (Theorem 20). Besides, an issue relative to the representation theory of non-normed complemented topological algebras is again the continuity of the complementor, which is faced via the “fundamental property”. The latter continuity is inherited to the induced vector complementor of the underlying linear space of a certain right ideal, say, R [ibid.]. In fact, the continuity of a complementor turns 2010 Mathematics Subject Classification: 46H05; 46H10 Keywords: Fundamental complemented algebra, complemented linear space, fundamental complemented (topological) linear space, vector complementor, weakly fundamental algebra, axially closed element.

****************************************************************************** http://www.utgjiu.ro/math/sma 94 M. Haralampidou and K. Tzironis out to be a key property of the aforementioned representation theory, and this was the motivation for the present work. Under conditions, a complemented algebra E admits a faithful continuous representation T in an inner product space. The question that arises then is, when every complete subalgebra F of the topological algebra of all continuous linear operators on R as in Theorem 20, that contains the image T (E), and certain projections, relative to closed subspaces of R, gains “complementation” and “fundamentality”. Indeed, this is true in the context of certain fundamental complemented algebras [10]. A further application of our main Theorem 20 assures a kind of the continuity of the complementor on F [ibid.]. A quick reference to weakly fundamental algebras is given in Section 2.

All algebras, employed below, are taken over the field C of complexes. A topological algebra E is an algebra which is a topological vector space and the ring multiplication is separately continuous (see e.g., [13]). If, in particular, the topology is defined by a family (pα )α∈A of seminorms (resp. submultiplicative seminorms), then E is named a locally convex (in particular, locally m-convex) algebra. We use the notation (E, (pα )α∈A ). We also employ the notation S for the (topological) closure of a subset S of a topological algebra E.

We denote by Al (S) (resp. Ar (S)) the left (right) annihilator of a (non empty) subset S of an algebra E, being a left (resp. right) ideal of E. If S is a left (resp.

right) ideal of E, then the ideals Al (S) and Ar (S) are two-sided. If Al (E) = {0} (resp. Ar (E) = {0}) we say that E is a left (resp. right ) preannihilator algebra. For a right preannihilator algebra, it is also used the term proper algebra. E is named preannihilator if it is both left and right preannihilator.

For a topological algebra E, Ll (E) ≡ Ll (resp. Lr (E) ≡ Lr, L(E) ≡ L) stands for the set of all closed left (right, two-sided) ideals of E. If for I ∈ L the relation I 2 = {0} implies I = {0}, then E is called topologically semiprime, while E is topologically simple, if it has no proper closed two-sided ideals.

The next notion was introduced in [6, p. 3723, Definition 2.1].

A topological algebra E is called left complemented, if there exists a mapping ⊥ : Ll −→ Ll : I ↦→ I ⊥, such that

–  –  –

⊥, as before, is called a left complementor on E. In what follows, we denote by (E, ⊥) a left complemented algebra with a left complementor ⊥.

****************************************************************************** Surveys in Mathematics and its Applications 11 (2016), 93 – 106 http://www.utgjiu.ro/math/sma Heredity in fundamental left complemented algebras A right complemented algebra is defined analogously and we talk about a right complementor. A left and right complemented algebra is simply called a complemented algebra.





A topological algebra E is named left precomplemented, if for every I ∈ Ll, there exists I ′ ∈ Ll such that E = I ⊕ I ′. Similarly, a right precomplemented algebra is defined. A left and right precomplemented algebra is called a precomplemented algebra (see [6, p. 3725, Definition 2.7]).

An idempotent element 0 ̸= e = e2 is called minimal, if the algebra eEe is a division one. On the other hand, if e is a minimal (idempotent) element, then Ee is a minimal left ideal and eE is a minimal right ideal of E (see [15, p. 45, Lemma 2.1.8]).

If E is a left precomplemented algebra and I, I ′ ∈ Ll with E = I ⊕ I ′, then there exists a linear map T : E → E with T 2 = T (projection) such that ImT = I and ker T = I ′. Indeed, for x ∈ E, there are unique y ∈ I and z ∈ I ′ with x = y+z. Then the mapping T (x) = y is well defined, linear and unique with the aforementioned properties.

For the sake of completeness, we refer some notions, which were introduced in [18, Definitions 2.5, 2.8, 2.9 and 2.10].

Definition 1. A left precomplemented algebra E is called fundamental if, for any I ∈ Ll, and a (pre)complement of I, say I ′ ∈ Ll (viz. E = I ⊕ I ′ ) there is a continuous linear mapping T = T (I, I ′ ) : E → E such that T 2 = T, Im T = I and ker T = I ′.

A fundamental right precomplemented algebra is defined analogously. A fundamental left and right precomplemented algebra is simply called a fundamental precomplemented algebra.

A left complemented algebra is named fundamental, if it is fundamental as left precomplemented. Similarly, for a right complemented algebra, and for a complemented one.

In the locally convex case, if E is the topological direct sum of I and I ′ (in the sense of [16, p. 90]), then E is named a weakly fundamental algebra.

EXAMPLE. Every left precomplemented Banach algebra is fundamental (see [18, Proposition 2.6]).

Definition 2. Let (E, ⊥) be a fundamental left complemented algebra. A net (Iδ )δ∈∆ of minimal closed left ideals (of E) is ⊥-convergent to I0 ∈ Ll, if

–  –  –

on any minimal right ideal of E.

Definition 3. An element x in a topological algebra E is said to be axially closed if the left ideal Ex is minimal closed.

–  –  –

In particular, a subset of E is named axially closed if each of its elements is axially closed.

Examples 4. (1) Any primitive idempotent element x in a precomplemented algebra E is axially closed (see [4, p. 964, Theorem 2.1]).

Recall that x is primitive if it can not be expressed as the sum of two orthogonal idempotents; namely, of some non-zero idempotents y, z ∈ E with yz = zy = 0.

(2) Any primitive idempotent element in a topologically semiprime algebra, in which, moreover, every left ideal contains a minimal left ideal, (in short (Dl )-algebra) is axially closed (see [5, p. 154, Theorem 3.9]).

(3) Any non-unital commutative semisimple topological algebra E with discrete space of maximal regular ideals is a (Dl )-algebra (see [9, p. 148, Examples 3.8, (1)]).

So, since E is also topologically semiprime, all its primitive idempotents are axially closed.

(4) Any semisimple finite-dimensional topological algebra is a (Dl )-algebra [ibid.

Examples 3.8, (2)], and thus its primitive idempotents are axially closed (see also (3)).

For more examples of (Dl )-algebras see [ibid.].

Definition 5. Let (E, ⊥) be a fundamental left complemented algebra. The mapping ⊥ is called continuous whenever for each convergent, axially closed net (aδ )δ∈∆ with aδ −→ a0 ∈ E, a0 ̸= 0, and such that Ea0 ∈ Ll, the net (Eaδ )δ∈∆ is ⊥-convergent δ in Ea0. Namely,

–  –  –

Proof. Since Al (I) is a closed two-sided ideal, either Al (I) = E or Al (I) = {0}. If the first case holds, EI = {0}. In particular, I 2 = {0} and by [5, p. 149, Theorem 2.1], I = {0}, a contradiction.

2 Weakly fundamental (pre)complemented algebras In this section, we get realizations of the notion “weakly fundamental” for locally convex algebras (see Definition 1). The same results are obviously applied in locally convex spaces when the notion of (pre)complemented linear spaces is considered in ****************************************************************************** Surveys in Mathematics and its Applications 11 (2016), 93 – 106 http://www.utgjiu.ro/math/sma Heredity in fundamental left complemented algebras the fashion of Definition 15. In what follows, “topological direct sum” is taken in the sense of [16, p. 90].

We start with the following useful result.

Theorem 7. Let E be a locally convex algebra. Consider the assertions:

(1) For every I ∈ Ll, there exists a continuous linear mapping T : E → E with T 2 = T, Im T = I and E = I ⊕ ker T.

(2) E is a left precomplemented algebra.

Then (1) ⇒ (2). Besides, (2) ⇒ (1), if in particular,

E is the topological direct sum of I and I ′ (: the complement of I in E). (2.1)

Proof. It is enough to show that (2) ⇒ (1). Consider I ∈ Ll and its complement I ′ ∈ Ll. Since both of them are locally convex spaces, the assumption that E = I ⊕ I ′ is the topological direct sum of I and I ′ is meaningful. Thus, the projections PI : E → I and PI ′ : E → I ′ are continuous (see [ibid. p. 90, Proposition 21 and p.

95, Proposition 29]). The comments preceding Definition 1 complete the proof.

Concerning the previous result, we note that E need not be the topological direct sum of I and I ′ when they carry the induced topologies (see the comments before Proposition 29 in [16, p. 95]). This reveals the necessity of the assumption (2.1).

Corollary 8. Every left complemented locally convex algebra, satisfying (2.1), is weakly fundamental.

In the next, by a locally C ∗ -algebra we mean an involutive complete locally (m)convex algebra (E, (pα )α∈A ), such that pα (x∗ x) = pα (x)2 for all x ∈ E and α ∈ A.

We also remind that a topological algebra E is said to be an annihilator algebra, if it is preannihilator with Ar (I) ̸= {0} for every I ∈ Ll, I ̸= E and Al (J) ̸= {0} for every J ∈ Lr, J ̸= E (see [5]).

Theorem 9. Every annihilator locally C ∗ -algebra, satisfying (2.

1), is weakly fundamental.

Proof. Let E be a topological algebra as in the statement. Then, by [8, p. 226, Theorem 3.1], E is left complemented. The assertion now follows from Corollary 8.

The next example is, under (2.1), a realization of the previous theorem.

Example 10. [6, p. 3724, Example 2.4]. Let X be a discrete (completely regular, k-space). Consider the locally m-convex algebra Cc (X) of all C-valued continuous functions on X in the topology of compact convergence, defined by the family of seminorms (pK )K, where K runs over all compact subsets of X, where

–  –  –

(cf., for instance [13, p. 19, Example 3.1]). Cc (X) is complete, since X is a k-space (see [12, p. 230 and p. 231, Theorem 12]), and it is a locally C ∗ -algebra under the involution given by f ∗ (x) = f (x), f ∈ Cc (X), x ∈ X. Since a locally C ∗ -algebra is semisimple, Cc (X) is topologically semiprime and hence preannihilator (cf. [5, p.

150, Lemma 2.3]). Besides, by [9, p. 144, Examples 2.2, (4)], if I is a proper closed ideal in Cc (X), there exists some closed maximal ideal M with I ⊂ M. Hence, A(M ) ⊆ A(I), where A ≡ Al = Ar. Since X is a completely regular space,

–  –  –

(cf. [13, p. 223, Corollary 1.3; see also its proof]). On the other hand, the closed subset {x}c corresponds to the proper closed ideal of Cc (X) given by

–  –  –

(Of course, {x}c is a non empty proper subset of X and hence I{x}c is a non trivial proper ideal of Cc (X); see [ibid., p. 221, Lemma 1.5, and p. 222, Remark 1.2]).

Obviously, I{x}c ⊆ A(Mx ) and hence A(I) ̸= {0}. Thus, Cc (X) is an annihilator algebra.

3 Complemented subspaces in (left) complemented algebras In this section, we pave the way to succeed vector complementors topically, in the sense that minimal closed one-sided ideals in certain left complemented algebras to be complemented as topological vector spaces (Theorem 14). Continuity of the vector complementor in question, is faced in the context of Theorem 20. So, we start with the next result, that generalizes Lemma 10 in [17, p. 658].

Proposition 11. Let E be a topologically simple left precomplemented algebra and I ∈ Ll (E), {0} ̸= R ∈ Lr (E). Put S = I ∩ R. If, for some x ∈ E, rx ∈ S for every r ∈ R, then x ∈ I. Namely {x ∈ E : Rx ⊆ S} ⊆ I.



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