THE RELATION BETWEEN REAL AW*-FACTORS AND ANTI-AUTOMORPHISMS OF INVOLUTIVE (I.E. WITH PERIOD 2) *-(COMPLEX) AW*-FACTORS

ISSN: 2582-4686 SJIF 2021-3.261,SJIF 20222.889, 2024-6.875 ResearchBib IF: 9.948 / 2024 VOLUME-5, ISSUE-11 139 THE RELATION BETWEEN REAL AW*-FACTORS AND ANTI-AUTOMORPHISMS OF INVOLUTIVE (I.E. WITH PERIOD 2) *-(COMPLEX) AW*-FACTORS (1) Kh. Kh. Boltaev, (2) F. B. Rasulova (1,2) National Pedagogical University of Uzbekistan named after Nizami and (1) Tashkent International University, Tashkent, Uzbekistan [email protected] rasulovaferuza1[email protected] ABSTRACT. The paper of the is to initiate the study of real AW*-algebras in the framework of the theory of real C*-algebras and W*-algebras. It happens that in some aspects real AW*-algebras behave unlike complex AW*-algebras and sometimes their properties are completely different also from corresponding properties of real W*-algebras. We prove that if the complexification of a real C*-algebra A is a (complex) AW*-algebra then A itself is a real AW∗-algebra. By modifying the Takenouchi’s examples of complex non-W*, AW*-factors we show that there exist real non-W*, AW*-factors. The correspondence between real AW*-factors and involutive (i.e. with period 2) *- anti-automorphisms of (complex) AW*-factors is established. We give the decomposition of real AW*-algebras into types I, II and III similar to the case of complex AW*-algebras or W*-algebras. It is proved that if A is a real AW*-factor and its complexification is also an AW*-algebra (and therefore an AW*-factor) thenthetypesof A and M coincide. KEYWORDS: AW*-algebra, C*-algebra, factor, involutive *-antiautomorphism, complex Hilbert space, commutant, complexification, linear *-automorphism, conjugate, bicommutant, quaternions algebra, projection, isomorphic. 1. INTRODUCTION The theory of operator algebras was initiated in a series of papers by Murray and von Neumann in thirties. Later such algebras were called von Neumann algebras or W*-algebras. These algebras are self-adjoint unital subalgebras M of the algebra B(H) of bounded linear operators on a complex Hilbert space H, which is closed in the weak operator topology. Equivalently M is a von Neumann algebra in B(H) if it is equal to the commutant of its commutant (von Neumann’s bicommutant theorem). A factor (or W*-factor) is a von Neumann algebra with trivial center and investigation of general W*-algebras can be reduced to the case of W*-factors, which are classified into types I, II and III. Real operator algebra is a *-algebra consisting of bounded (real) linear operators on a real Hilbert space H. If it is closed in the weak operator topology we have real W*-algebra, and if it is uniformly closed (i.e. in the norm topology) then we come to the notion of the real C*-algebra. In his monograph [7] Li Bing-Ren has set up the fundamentals of real operator algebras and gave a systematic discussion of the real counterpart for the theory of W*- and C*-algebras. A slightly different (but almost the same up to *-isomorphism) definition of real W*-algebras was given by E.Størmer [13,14]: A real von Neumann algebra (or real W*-algebra) is a real *-algebra of bounded linear operators on a complex Hilbert space containing the identity operator 1, which is ISSN: 2582-4686 SJIF 2021-3.261,SJIF 20222.889, 2024-6.875 ResearchBib IF: 9.948 / 2024 VOLUME-5, ISSUE-11 140 closed in the weak operator topology and satisfies the condition The smalles (complex) von Neumann algebra containing coincides with its complexification , i.e. . Moreover generates a natural involutive (i.e. of order 2) *- antiautomorphism of , namely , where It is clear that . Conversely, given a (complex) von Neumann algebra U and any involutive *-antiautomorphism α on U, the set is a real von Neumann algebra in the above sense. It is not diffucul to see that two real von Neumann algebras generating the same (complex) von Neumann algebra are isomorphic if and only if the corresponding involutive *-antiautomorphisms are conjugate. Thus the study of the above real von Neumann algebra can be reduced to the study of pairs (U, α), where U is a (complex) von Neumann algebra and α - its involutive *-antiautomorphism. 2. PRELIMINARIES Let H be a complex Hilbert space, denote the algebra of all bounded linear operators on H. The weak (operator) topology on is the locally convex topology, generated by semi norms of the form: W*-algebra is a weakly closed complex *- algebra of operators on a Hilbert space H containing the identity operator 1. Recall that W∗-algebras are also called von Neumann algebras. Let further M be a W*-algebra. The set of all elements from commuting with each element from M is called the commutant of the algebra M. The center of a W*-algebra M is the set of elements of M, commuting with each element from M. It is easy to see that . Elements of are called central elements. A W*-algebra M is called factor, if consists of the complex multiples of 1, i.e if We say that a W*-algebra M is injective if thereexists a projection P in onto M such that and 1. This isequivalent to the condition that M is hyperfinite, i.e., that there exists an increasing sequence of matrix subalgebras of the algebra M containing 1 and such that the union is weakly dense in M. Let be projections from M. We say that is equivalent to , and write , if for some partial isometry from M. A projection is called: finite, if implies ; infinite - otherwise; purely infinite, if doesn’t have any nonzero finite subprojection; abelian, if the algebra is an abelian W∗-algebra. A W∗-algebra M is called finite, infinite, purely infinite, if 1 is a finite, infinite, purely infinite respectively; M is σ-finite, if any family of pairwise orthogonal projections from M is at most countable; semifinite, if each projection ISSN: 2582-4686 SJIF 2021-3.261,SJIF 20222.889, 2024-6.875 ResearchBib IF: 9.948 / 2024 VOLUME-5, ISSUE-11 141 in M contains a nonzero finite subprojection; properly infinite, if every nonzero projection from is infinite; discrete, or of type I, if it contains a faithful abelian projection (i.e. an abelian projection with the central support 1); continuous, if there is no abelian projection in M except zero; M is of type II, if M is semifinite and continuous; type (respectively ), if M is of type I and finite (respectively properly infinite); type (respectively type ), if M is of type II and finite (respectively properly infinite); type III, if M is purely infinite. It is known that any W*-algebra has a unique decomposition along its center into the direct sum of W*-algebras of the and III types. A linear mapping is called a *-automorphism (respectively a *-antiautomorphism) if and (respectively , for all . A mapping α is called involutive if . A *-automorphism α is called inner if there exists a unitary in M, such that , for all . A *-automorphism is called centrally trivial if *-strongly as for any central sequence . We shall denote by the group of all *-automorphisms, by the group of all *antiautomorphisms, by the group of all inner *-automorphisms, and by Ct(R) the subgroup of its centrally trivial *-automorphisms of M. Two *-automorphisms or *-antiautomorphisms α and β are said to be conjugate (or outer conjugate), if (respectively ) for some *- automorphism θ (and an inner *-automorphism Adu). A linear functional ω on M is called positive, if for all . A positive linear functional ω on M with is called a state. Let be the positive part of M. A weight on M is a homogeneous additive function (we suppose that ). A weight (or a state) ω is called: faithful, if for any implies ; normal, if for any net in M, increasing to an element x, we have ; finite, if for all ; semifinite, if for any there exists a net of elements , such that , and in σ - weak topology; ω is a trace, if for all and each unitary . The type of a W*-algebra is tightly connect with the existence of traces on it. Namely a W*-algebra M is a finite if and only if it possesses a separating family of finite normal traces; it is semifinite if and only if it possesses a faithful semifinite normal trace; M is purely infinite if and only if there is no nonzero semifinite normal trace on M (see [15]). Definition. [4]. By a real C*-algebra we mean a real Banach *-algebra R such that the relation holds and the element is invertible for any ISSN: 2582-4686 SJIF 2021-3.261,SJIF 20222.889, 2024-6.875 ResearchBib IF: 9.948 / 2024 VOLUME-5, ISSUE-11 142 Definition*. [8,9]. A real C*-algebra R such that R+iR is a complex W*-algebra is referred to as a real W*-algebra. We proceed with another definition of a real W*-algebra, which can be found in papers of Størmer. Definition∗∗. [2,13]. A unital weakly closed real *-algebra R in such that is called a real W*-algebra. A real W*-algebra R is called a (real) factor if its center Z(R) consists of elements λ1, We say that a real W*-algebra R is of type and III, if the enveloping W*-algebra (i.e., the least W*-algebra containing R) is of the corresponding type with respect to the usual classification of W*-algebras. 3. MAIR RESULTS Let A be a real C*-algebra, with the complexification . Then M is a complex C*-algebra and, as we have seen in the previous section, if A is a real AW*-algebra M may not be a (complex) AW*-algebra. Now let us consider the converse problem if is an AW*-algebra is A necessarily a real AW∗-algebra? The following result gives a positive answer to this problem. Proposition 1. Let A be a real C*-algebra and let be its complexification. Suppose that M is an AW*-algebra. Then A is a real AW*-algebra. Proof. As we have mentioned in the first section, A coincides with the fixed point set under the conjugate linear *-automorphism of M, where , i.e. If S is a nonempty subset in A then for its right-annihilator (with respect to M) we have and because This means that if and only if . Now suppose that M is an AW*-algebra, then for a suitable projection . Since from above it follows that . Therefore is a projection and , i.e. . Thus i.e. This means that . But then i.e. A is a real AW*-algebra. Proposition 2. There exist real AW*-factors which are not real W*-factors. Theorem 1. A real AW ∗ -algebra A is a real W*-algebra if and only if (i) A possesses a separating family of normal states; ISSN: 2582-4686 SJIF 2021-3.261,SJIF 20222.889, 2024-6.875 ResearchBib IF: 9.948 / 2024 VOLUME-5, ISSUE-11 143 (ii) its complexification is an AW*-algebra. Proof. Necessity is obvious, since if A is a real W*-algebra, then is a (complex) W*- algebra (see [7, Chap.5]). Therefore, M is an AW*-algebra and it possesses a separating family of normal states, the restrictions of which on A give a separating family of normal states on A. Sufficiency. Let be an AW*-algebra and let A possess a separating family of normal states, which we denote by , i.e. for any exists with For we put A straightforward calculation shows that α is an involutive (i.e. with period 2) *-anti-automorphism of M, and The extension of by linearity on M we denote by , and we shall show, that the family is a separating family of normal states on M. For we have , and since is hermicitian we obtain since . Thus, for we have and since If , then (because α is a *-anti-automorphism), and hence i.e. . Therefore , i.e. all functionals are positive on M. Moreover, we have , i.e. is a family of states on M. Now let us show, that each state is normal. If is an arbitrary net with then since α is an order isomorphism of M, we have Therefore and . Since is a normal we obtain i.e. all functionals are normal on M. Finally, let and for all γ. Then , and since is a separating family of states, . Hence we have i.e. ISSN: 2582-4686 SJIF 2021-3.261,SJIF 20222.889, 2024-6.875 ResearchBib IF: 9.948 / 2024 VOLUME-5, ISSUE-11 144 . Thus, the AW*-algebra M possesses a separating family of normal states . By the theorem of Pedersen [10] M is a W∗-algebra. Therefore, by [7] A is a real W∗-algebra. Now, let M be a (complex) AW*-factor, α its involutive *-anti-automorphism. Then as it was mentioned above the set is a real C*-algebra such that (actually in terms of operation ”-”) and from Proposition 4.3.1 it follows that A is a real AW*-factor. It is known that two real W*-algebras generating the same (complex) W*-algebra, are isomorphic if and only if the corresponding involutive *-anti-automorphisms are conjugate [2,13,14]. A similar result is also valid for real AW*-algebras: Proposition 3. Let α and β be involutive *-anti-automorphisms of a (complex) AW*-factor M. Then the real AW*-factors and are real *-isomorphic if and only if the involutive *-anti-automorphisms α and β are conjugate, i.e. for a suitable *-automorphism of the AW*-factor M. Proof. Let A and B be real *-isomorphic with a *-isomorphism . Then can be naturally extended to a (complex) *-isomorphism θ of their complexifications and both coincide with M. Therefore θ is a *-automorphism of M and , i.e. if and only if . Thus for we have , i.e. for all Since is a *-automorphism on M which is identical on A and any real *automorphism of A can be uniquely extended to a complex *-automorphism of M, it follows that on whole M, i.e. and , i.e. α and β are conjugate. Conversely, if α and β are conjugate, i.e. for a suitable complex *-automorphism θ of M, then and if and only if , i.e. . Therefore, θ restricted on A gives the needed *-isomorphism between real AW∗-factors A and B. Now we consider one of the main results of this section. Theorem 2. Let A be a real AW*-algebra and its complexification is a (complex) AW*- algebra. Then A is of type I if and only if M of type I. Corollary. Let A be a real AW*-algebra of type I, and its complexification is a (complex) AW*-algebra. Then A is a real W*-algebra if and only if its center is a real W*- algebra. Proof. If A is a real W*-algebra then, obviously, its center is a real W*-algebra. Conversely, let A be an AW*-algebra of type I and its center is a W*-algebra. Then by Theorem 4.5.2, ISSN: 2582-4686 SJIF 2021-3.261,SJIF 20222.889, 2024-6.875 ResearchBib IF: 9.948 / 2024 VOLUME-5, ISSUE-11 145 is an AW*-algebra of type I, and its center is a W*-algebra. From Kaplanskys theorem [6, Theorem 2] it follows that M is an W*-algebra. Therefore, A is a real W*-algebra. REFERENCES 1. Ayupov Sh.A., Azamov N.A. Commutators and Lie isomorphisms of skew elements in prime operator algebras. Comm. Algebra, 1996, 24, N 4, pp.1501-1520. 2. Ayupov Sh.A., Rakhimov A.A., Usmanov Sh.M. 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