MATHEMATICS

S. N. SLUGIN

A COMPLEX PARTIALLY ORDERED SPACE

AND MODULES OVER IT

(Presented by Academician S. L. Sobolev on 28 V 1962)

In the paper \((^1)\), modules over a \(K\)-space with unit \((^{2,3})\) \(Z\) were constructed, generalizing the concepts of real Banach and Hilbert spaces. Here the concept of a complex partially ordered space \(V=Z\times Z\) is introduced, and modules over \(V\) are constructed that are generalizations of complex Banach and Hilbert spaces.

1. We shall agree to regard \(Z\subset \widetilde Z\), where \(Z\) is a \(K\)-space with unit \(1\), \(\widetilde Z=C_\infty(Q)\) is its maximal extension \((^{2,3})\), which is a field \((^4)\). Let us associate with the function \(\lambda^{-1}\) of a real argument the function \(z^{-1}\) in \(\widetilde Z\). Then it makes sense to consider in \(Z\) abstract functions of arguments from \(Z\), generated \((^2)\) in \(\widetilde Z\) by superposition of continuous functions of real arguments and functions of the form \(\lambda^{-1}\), if the values of the abstract functions belong to \(Z\), although certain intermediate operations are not always realizable without leaving \(Z\) (cf. Definition 1 in \((^1)\)). For example, \(\sqrt{z^2+u^2}\) exists in \(Z\) for any \(z,u\in Z\); for the existence in \(Z\) of the quotient \((^1)\) \(zz'/z^2+u^2\), it is sufficient that \(z'/z\) exist. In establishing such facts here, the properties of the quotient \((^{1,4})\), preservation of relations for generated \((o)\)-continuous abstract functions \((^2)\), and the normality of the embedding of \(Z\) in \(\widetilde Z\) \((^{2,3})\) are used.

2. Suppose that in \(Z\), for each direction \((^3)\) \(A\), there are defined in some way nonempty classes of positive generalized sequences of elements \(\alpha\), \(z_\alpha\ge 0\), converging to zero in the direction \(A\). Put \(z_\alpha\to z\) if \(|z_\alpha-z|\to 0\). The sequence \(z_\alpha\) is called convergent in itself if \(u_{\alpha\beta}=z_\alpha-z_\beta\to 0\) in the direction \(A^2\) of all pairs \((\alpha,\beta)\), where we put \((\alpha,\beta)<(\gamma,\delta)\) in \(A^2\) if \(\alpha<\gamma\) and \(\beta<\delta\) in \(A\). We shall require (cf. \((^1)\)) the fulfillment of the conditions for positive sequences:

a) if \(z_\alpha\to 0\) and all \(z_\alpha=z\), then \(z=0\);

b) if \(z_\alpha\le u_\alpha\to 0\), then \(z_\alpha\to 0\) (normality of convergence);

c) if \(z_\alpha\to 0\), \(u_\alpha\to 0\), then \(z_\alpha+u_\alpha\to 0\).

1. It follows from this that a stationary sequence \(z_\alpha=z\) converges to \(z\); the operations of addition and subtraction are continuous in the aggregate of both arguments; and the limit is unique.

2. Moreover, conclusions 1–3 of Theorem 2 from \((^5)\) are also valid for generalized sequences converging in the sense of § 2 of the present paper.

3. Conditions a)—c) are satisfied by all types of convergence (also for generalized sequences) indicated in § 4 of \((^1)\), as well as by \((t)\)- and \((r)\)-convergence \((^3)\); \((r)\)-convergence with a constant regulator \((^3)\) \(1\); \((b)\)-convergence, if \(Z\) is simultaneously a \(KN\)-linear \((^3)\); \((bk)\)-convergence, if \(Z\) is structurally normed by means of an Archimedean \(K\)-linear \((^3)\) \(W\) in such a way that from the comparability \(|z|\le |u|\) in \(Z\) there follows the comparability \(\|z\|\le \|u\|\) of the generalized norms in \(W\).

4. These types of convergence also satisfy the following conditions. If \(z_\alpha\to 0\) in the direction \(A\), then the sequence \(z_\beta\to 0\) in any

to a cofinal subdirection in \(A\) of elements \(\beta\). If \(z_\delta \to 0\) along any subdirection consisting of all \(\delta > \gamma\) in \(A\), where \(\gamma\) is any fixed element, then \(z_\alpha \to 0\) along the direction \(A\). If \(z_\alpha \to 0\), \(u_\alpha \to 0\), then \(z'_{\alpha\beta}=z_\alpha \pm u_\beta \to 0\) along the direction \(A^2\); hence convergence follows from convergence to itself. If \(|z_\alpha-z_\beta|\leq u_\gamma \to 0\) for all \(\alpha>\gamma,\ \beta>\gamma\) in \(A\), then \(z_\alpha\) converges to itself. The operation of multiplication by a number is continuous with respect to the totality of both factors.

1. We shall further write \(u_\alpha \downarrow u\) if \(u_\alpha \to u\) in one of the senses of item 5 and \(u_\alpha \geq u_\beta\) for all \(\alpha<\beta\) in \(A\). If \(u_\alpha \downarrow u\), then all \(u_\alpha \geq u\). If \(|z_\alpha-z|\leq u_\alpha \downarrow 0\), then \(z_\alpha \xrightarrow{(0)} z\) \((^3)\). In particular, this also applies to monotone sequences \(\pm u_\alpha \downarrow u\). If \(Z\) is complete (with respect to convergence), then the principle of contracting segments holds in it (segments in the sense of \((^3)\), the sequence of segments generalized). Under certain additional conditions imposed on the sequence of products \(z_\alpha u_\alpha\), it converges to \(\lim z_\alpha \lim u_\alpha\).

2. Form the set \(V=Z^2\) of all pairs (in ordered notation) of elements \(z,u\in Z\), denoting the pair by \(z+ui\). Everywhere below \(v=z+ui\), \(w=z'+u'i\); \(\lambda,\mu\) are real numbers.

Definition 1. If in \(Z^2\) the operations are defined by

\[ v+\lambda w=(z+\lambda z')+(u+\lambda u')i,\qquad vw=(zz'-uu')+(zu'+z'u)i \]

(under the condition of existence of all products indicated in the right-hand side of the last equality), then \(V=Z^2\) will be called a complex partially ordered space or a \(K^2\)-space.

By the product \((\lambda+\mu i)v\) we understand \((\lambda 1+\mu 1 i)v\).

The meaning of the embedding of the \(K^2\)-space \(V=Z^2\) into the \(K^2\)-space \(\widetilde V=\widetilde Z^2\) is clear; \(\widetilde V\) is an absolute ring (the product always exists). In the \(K^2\)-space \(V\) there are preserved, evidently, the relations (cf. \((^2)\)) between the generating continuous functions of complex arguments.

If \(z\geq 0\), then there exists a unique \(\sqrt{z+0i}\) (with positive component of the pair), and it is equal to \(\sqrt z+0i\).

Identify the set \(V_1\) of all pairs \(z+0i\) with \(Z\), regarding \(z+0i=z\). We introduce the notions \(\operatorname{Re}v=z\), \(\operatorname{Im}v=u\), \(|v|=\sqrt{z^2+u^2}\), \(\bar v=z-ui\). They make sense in \(Z\) or \(V\) for any \(v\in V\). The following relations hold:

\[ \bigl||v|-|w|\bigr|\leq |v\pm w|\leq |v|+|w|, \]

\[ |z|\vee |u|\leq |v|\leq |z|+|u|,\qquad |v|=\sqrt{\bar v v}. \]

The embedding of \(V\) into \(\widetilde V\) is normal: if \(v\in\widetilde V\), \(w\in V\), and \(|v|\leq |w|\) in \(\widetilde Z\), then \(v\in V\).

Definition 2. The trace \(e_v\) of an element \(v\in\widetilde Z^2\) is the trace \((^2,^3)\) \(e_{|v|}\) of the element \(|v|\in Z\). If \(|v|\,d\,|w|\) \((^2,^3)\), then we shall call \(v\) and \(w\) disjoint, \(v\,d\,w\).

It follows from the definition that: \(e_v=e_z\vee e_u=e_{|z|+|u|}\), \(e_{v+w}\leq e_v\vee e_w\), \(e_{vw}=e_v\wedge e_w\), since \(|vw|=|v|\,|w|\) in \(\widetilde Z\). Therefore the following four relations are equivalent: 1) \(v\,d\,w\); 2) \(z\,d\,z'\) and \(u\,d\,u'\); 3) \(z\,d\,u'\) and \(z'\,d\,u\); 4) \(vw=0\). If \(v\,d\,w\), then \(|v+w|=|v|+|w|\) and \(e_{v+w}=e_v\vee e_w=e_v+e_w\).

Definition 3. If for \(v\in V\) there exists an element \(w\in V\) such that \(e_w=vw=e_v\), then \(w\) will be called the inverse with respect to \(v\), \(w=v^{-1}\).

From the existence of \(v^{-1}\) its uniqueness follows. In \(\widetilde V\) every element has an inverse.

Definition 4. If the result \(wv^{-1}\) of operations in \(\widetilde V\) on elements from \(V\) belongs to \(V\), then we shall call it a quotient or fraction \(w/v\).

According to the definition we obtain the well-known formulas:

\[ |w/v|=|w|/|v|, \]

\[ w/v=w\bar v/|v|^2=(zz'+uu'/z^2+u^2)+(zu'-z'u/z^2+u^2)i \]

and so on; \(e_{w/v}=e_w\wedge e_v\); the relations \(v\,d\,w\) and \(w/v=0\) are equivalent.

The \(K^2\)-space \(\widetilde V\) is an absolute field (the quotient always exists, but \(v/v=e_v\)). For the existence in \(V\) of the fraction \(z_1+z_2 i/u_1+u_2 i\), it is sufficient that all fractions \(z_m/u_n\) \((m,n=1,2)\) exist in \(Z\). From the relations \(|v|\ge |v'|\), \(|w|\le |w'|\), \(w'/v'\in V\), it follows that \(w/v\in V\). All operations defining the rings \(Z,V\) and the fields \(\widetilde Z,\widetilde V\) are related to one another in the usual way and to the structural relations in \(Z\) and \(\widetilde Z\) (cf. \((1\text{--}4)\)).

Let us define convergence in the direction \(A\) of a generalized sequence \(v_\alpha\to v\) in \(V\), if \(|v_\alpha-v|\to0\) in \(Z\) in the sense of item 2; the sequence \(w_\alpha\) will be called convergent in itself if \(z_{\alpha\beta}=|w_\alpha-w_\beta|\to0\) in the direction \(A^2\). In order that \(v_\alpha\to v\), it is necessary and sufficient that \(\operatorname{Re}v_\alpha\to z\), \(\operatorname{Im}v_\alpha\to u\) simultaneously. Using this, it is not difficult to establish the properties of convergence in \(V\).

1. By analogy with \((^1)\), we introduce the notion of a generalized module \(X\) over \(V\) (the products \(vx\) exist, in general, not for all pairs of elements \(v\in V\), \(x\in X\)).

Definition 5 (see Definition 2 in \((^1)\)). A generalized two-sided module \(X\) over the \(K^2\)-space \(V\), where \(vx=xv\), \(1x=x\), and from the existence of \(wx\) in \(X\) and the comparability \(|v|\le |w|\) there follows the existence of \(vx\) in \(X\), will be called an \(M_V\)-module.

Here equalities of the form \(v(x+y)=vx+vy\), etc. (which define the module \(X\)) are postulated under the condition that all products occurring in at least one part of each such equality exist. By the product \((\lambda+\mu i)x\) we mean \((\lambda 1+\mu 1 i)x\).

1. By analogy with Definition 5 \((^1)\), we introduce the notion of a \(B_V\)-module, where, in particular, the equality \(|vx|=|v|\,|x|\) is postulated for generalized norms \(|x|\in Z_+\) (provided \(vx\) or \(v|x|\) exists). As before, we define the trace \(e_x=e_{|x|}\), disjointness \(x\,d\,y\), if \(|x|\,d\,|y|\); \(v\,d\,x\), if \(v\,d\,|x|\).

Let us define convergence \(x_\alpha\to x\) in \(X\) and in itself, if \(|x_\alpha-x|\to0\) in \(Z\) in the direction \(A\), or, respectively, \(z_{\alpha\beta}=|x_\alpha-x_\beta|\to0\) in the direction \(A^2\) in the sense of item 2. For convergence in \(X\), all conditions of the type listed in item 3 are fulfilled. If \(x_\alpha\to x\), then \(|x_\alpha|\to |x|\). If, in \(Z\), all conditions of item 6 are also fulfilled, then similar conditions are fulfilled in \(X\) as well. We give the notions of modules of type \(B_V\), \(B_{K^\ast}\), \(B_{B_k^2}\) (cf. Definition 7 in \((^1)\)), where convergence in itself (in the direction \(A^2\)) implies convergence (in the direction \(A\)). In modules of type \(B_{K^\ast}\), \(B_{B_k^2}\), convergence in \(Z\) means \((o)\)- or \((bk)\)-convergence (see item 4 in \((^1)\), generalized sequences).

From the identification \(V_1=Z\) it follows that a \(B_V\)-module also has all properties of a \(B_Z\)-module \((^1)\). In particular, a module of type \(B_{K^\ast}\) is a space of type \(B_K\).

1. The definitions given in item 7 of \((^1)\) are transferred in the same way. In the definition of an \(H_V\)-module (cf. Definition 10 in \((^1)\)), instead of symmetry of the scalar product, conjugacy is postulated: \((x,y)=\overline{(y,x)}\in V\). In the axiomatics of an \(H_V\)-module there is the requirement \((vx,y)=v(x,y)\), if \(vx\) or \(v(x,y)\) exists. The well-known elementary properties hold: \((x,vy)=\bar v(x,y)\), the parallelogram formula is valid, etc. We retain the previous definition \((^1)\) of a complete orthonormal system \(\{\varepsilon_n\}\) of an \(H_V\)-module (in particular, \(|\varepsilon_n|\) are unit elements in \(Z\); in general, the inequality \(|\varepsilon_n|\ne1\) is possible).

1. Quite analogously to (1), absolute \(M_V\)-, \(B_V\)-, \(H_V\)-modules are defined (where \(Z=\widetilde Z\) and all products \(vx\) exist); modules of type \(H_V\), \(H_{K^2}\), \(H_{B_K^2}\) (complete) are defined as special cases of modules of type \(B_V\), etc.; we define \(M\)-, \(B\)-, \(H\)-extensions.

After replacing the indices \(Z, K, B_K\) in the notation for modules (for example, the index \(K\) for a module of type \(B_K\) (1)), respectively, by the indices \(V, K^2, B_K^2\), all the assertions in §§ 5–7 of (1) remain in force (but in Theorem 7

\[ |x|^2=\sum_{n=1}^{\infty} |(x,e_n)|^2). \]

In proving the existence of an \(M\)-extension \(\widetilde X\) for any \(M_V\)-module (cf. Theorem 1 of (1)), the product

\[ \widetilde v x=x\widetilde v=\{v_nx_n\} \]

is defined in \(\widetilde X\), where \(v_n=z_n+u_n i\), and the cononunique \(z_n, u_n\) \((o)\)-converge in \(\widetilde Z\), respectively, to \(\operatorname{Re} v\), \(\operatorname{Im} v\).

If in \(Z\) convergence is meant in one of the senses of § 5 of the present paper and \(v_\alpha\to v\) in \(V\), \(x_\alpha\to x\) in a \(B_V\)-module \(X\), then the products \(v_\alpha x_\alpha\to vx\) in \(X\) under certain additional conditions imposed on the sequence \(v_\alpha x_\alpha\). The same also applies to scalar products in an \(H_V\)-module.

1. In (1) and here, instead of the \(K\)-spaces \(Z\) and \(\widetilde Z\), one may take a \(K_\sigma\)-space with unit and its maximal extension (3).

Gorky State University
named after N. I. Lobachevsky

Received
19 V 1962

References

1. S. N. Slugin, DAN, 139, No. 5, 1059 (1961).
2. L. V. Kantorovich, B. Z. Vulikh, A. G. Pinsker, Functional Analysis in Semiordered Spaces, 1950.
3. B. Z. Vulikh, Introduction to the Theory of Semiordered Spaces, 1961.
4. B. Z. Vulikh, Matem. sbornik, 22 (64), 1, 27 (1948).
5. S. N. Slugin, DAN, 131, No. 6, 1261 (1960).