Injective Envelopes of Real C*- and AW*-Algebras

Abdugafur Rakhimov; Laylo Ramazonova

doi:doi:10.11648/j.ijamtp.20251101.12

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Injective Envelopes of Real C- and AW-Algebras

Abdugafur Rakhimov^*

, Laylo Ramazonova

Published in International Journal of Applied Mathematics and Theoretical Physics (Volume 11, Issue 1)

Received: 26 March 2025 Accepted: 8 April 2025 Published: 29 April 2025

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Abstract

Injective (complex and real) W*- and C*- algebras, in particular, factors have been studied quite well. On the other hand, in an arbitrary case, i.e., in the non-injective case, it is quite difficult to study (up to isomorphism) the W*-algebras, in particular, it is known that there is a continuum of pairwise non-isomorphic non-injective factors of type II₁. Therefore, it seems interesting to study the so called maximal injective W* and C*-subalgebras or what is equivalent, the smallest injective C*-algebra containing a given algebra, which is called an injective envelope of C*- algebra. It is shown that every outer *-automorphism of a real C*-algebra can be uniquely extended to an injective envelope of real C*-algebra. It is proven that if a real C*-algebra is a simple, then its injective envelope is also simple, and it is a real AW*-factor. The example of a real C*-algebra that is not real AW*-algebra and the injective envelope is a real AW*-factor of type III, which is not a real W*-algebra is constructed. This leads to the interesting result that up to isomorphism, the class of injective real (resp. complex) AW*-factors of type III is at least one larger than the class injective real (resp. complex) W*-factors of type III.

Published in	International Journal of Applied Mathematics and Theoretical Physics (Volume 11, Issue 1)
DOI	10.11648/j.ijamtp.20251101.12
Page(s)	19-23
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

C*- Algebras, AW*-algebras, Injective Envelopes of Real C*-Algebras

1. Introduction

As is known, injective (complex and real) W*- and C*- algebras, in particular, factors have been studied quite well. On the other hand, in an arbitrary case, i.e., in the non-injective case, it is quite difficult to study (up to isomorphism) the W*-algebras, in particular, it is known that there is a continuum of pairwise non-isomorphic non-injective factors of type

{II}_{1}

. Therefore, it seems interesting to study the so called maximal injective W* and C*-subalgebras or what is equivalent, the smallest injective C*-algebra containing a given algebra, which is called an injective envelope of C*- algebra. In the complex case, such algebras were considered in the works M.Hamana, K.Saito, M.Wright, R.Kadison, J.Fang. In this paper, we consider the existence and uniqueness of injective envelope real C*-algebras for given real C*- algebras. By analogy with Saito and Joita we will show that every outer *-automorphism of a real C*-algebra can be uniquely extended to an injective envelope of real C*-algebra

[1, 2]

. Similar to Hamana, we will prove that if a real C*-, algebra is a simple, then its injective envelope is also simple and it is a real AW*-factor

[3]

. We will build an example of a real C*-algebra that is not real AW*- algebra and the injective envelope is a real AW*-factor of type III, which is not a real W*-algebra. This leads to the interesting result that up to isomorphism, the class of injective real (resp. complex) AW*-factors of type III is at least one larger than the class injective real (resp. complex) W*-factors of type III.

2. Injective Envelopes C*-Algebras

2.1. Preliminaries

Let

B (H)

be the algebra of all bounded linear operators acting on a complex Hilbert space

H

. Recall that a real *- subalgebra

R \subset B (H)

with the identity

l

is called a real W*- algebra, if it is weakly closed and

R \cap iR = {0}

Let

A

be a Banach *-algebra over the field

C

. The algebra

A

is called a C*-algebra, if

‖a a^{*}‖ = {‖a‖}^{2}

, for any

a \in A

. A real Banach *-algebra

R

is called a real C*-algebra, if

‖a a^{*}‖ = {‖a‖}^{2}

and an element

l + a a^{*}

is invertible for any

a \in R

. It is easy to see that

R

is a real C*-algebra if and only if a norm on

R

can be extended onto the complexification

A = R + iR

of the algebra

R

so that algebra

A

is a C*-algebra (see

[4, 5]

Denote by

M_{n} (A)

algebra of all

n \times n

matrices over

A

which is also a C*-algebra, relative to ordinary matrix operations. An element

a \in A

is called positive and denote by

a \geq 0

, if there exists a self-adjoint element

b \in A

, such that

a = b^{2}

. The set of all positive elements of

A

denoted by

A^{+}

A continuous linear map

φ

between two C*-algebras

A

and

B

is called completely positive, if for any

n \geq 1

, the natural map

φ_{n}

from the C*-algebra

A \otimes M_{n}

to the C*-algebra

B \otimes M_{n}

, defining by

φ_{n} ({(a_{ij})}_{i, j = 1}^{n}) = {(φ (a_{ij}))}_{i, j = 1}^{n}

is positive, where

M_{n}

is the C*-algebra of

n \times n

matrices over

C

A C*-algebra

A

with the unit

1

is said to be injective if whenever

B

is a unital C*-algebra and

S

is a self-adjoint subspace of

B

containing the unit, then each completely positive map

φ : S \to A

can be extended to a completely positive map

\overset{̅}{φ} : B \to A

. It was proved that for each Hilbert space

H

, the algebra of bounded operators on

H

is injective, i.e.,

B (H)

is injective.

It is known that any C*-algebra can be isomorphically embedded into some algebra

B (H)

in which it is uniformly closed. In 1967 in the work of Hakeda-Tomiyama, C*- algebras with property E are considered: A C*-algebra

A

has the property E (an extensional property) if there is a projection

P : B (H) \to A

such that

‖P‖ = 1

and

P (1) = 1

. In this case the map

P

is completely positive. It is known that these two definitions are equivalent to each other.

Thus, every C*-algebra lies in the injective C*-algebra

B (H)

and the smallest injective C*-algebra containing this algebra is called an injective enveloping C*-algebra. In

[3]

, the author proved that every C*-algebra

A

with identity has a unique injective enveloping C*-algebra, which is denoted as

I (A)

, i.e., the smallest injective C*-algebra containing

A

as a C*-subalgebra

[6-9]

2.2. Outer *-Automorphisms of Real C*-Algebras

Let

A

be a real or complex algebra. A subspace

I

of an algebra

A

is called an left ideal (resp. right ideal) if

xy \in I

(resp.

yx \in I

), for all

x \in A

and

y \in I

. A left and right ideal is called a two-sided ideal or ideal. An algebra

A

is said to be simple if it contains no non-trivial two-sided ideals and the multiplication operation is not zero (that is, there are a and b with

ab \neq 0

). If

α

is an automorphism of

A

, then

A

said to be

α

-simple if the only an

α

-invariant, closed, proper, two-sided ideal of

A

0

Let

A

be (complex) an *-algebra. A real subspace

I

A

is called a real ideal of

A

I ∙ A, A ∙ I \subset I_{c}

, where

I_{c} = I + iI

. Since each complex subspace of

A

is a real subspace, any complex ideal is automatically a real ideal of

A

. Let

I

be a real ideal of

A

. If there exists a real *-subalgebra

R

A

with

R + iR = A

, such that

I \subset R

, then

I

is called a pure real ideal of

A

. In this case, it is obvious that we have

I \cdot R \subset I

. Note that, the reverse is not true, i.e., from

I \cdot R \subset I

it does not follow

I \subset R

. But a complex subspace

J = I + iI

always is a complex ideal of

A

. On the other hand if

I \subset R

is a real subspace of

A

and

I + iI

is a complex ideal, then

I

is a pure real ideal, i.e., we obtain

I \cdot R \subset I

Let, now

I

and

Q

be pure real ideals of

A

. In general, the set

I + iQ

is not a (complex) subspace. More precisely the set

I + iQ

is a complex subspace if and only if

I = Q

. Therefore we consider the smallest complex subspace

J

A

, containing

I

and

Q

. Obviously

J

is equal to

(I + Q) + i (I + Q)

. Thus, if

I

and

Q

are real ideals, then

J = (I + Q) + i (I + Q)

is a complex ideal.

Theorem 3.1. Let

R

be a real algebra. Then

R

is simple if, and only if

R + iR

is simple.

Proof. Assume that real subalgebra

I

of a algebra

R

be a non-trivial two-sided ideal of a algebra

R

. Then, obviously that complex subspace

I + iI

is be a non-trivial two-sided ideal of a algebra

R + iR

. Inversely, let

J

is be non-trivial two-sided ideal in

R + iR

and let

a + ib \in J

, where

a, b \in R

. Since

J

is a two-sided ideal, then for

a - ib \in R + iR

we have

x = (a + ib) (a - ib) = a^{2} + b^{2} + iba - iab \in J

y = (a - ib) (a + ib) = a^{2} + b^{2} - iba + iab \in J

Hence,

x + y = 2 a^{2} + 2 b^{2} \in J

, therefore

J \cap R \neq \emptyset

. Assume that

I ≔ J \cap R

. It is easy to see, that real subspace

I

be a non-trivial two-sided ideal of algebra

R

. The proof is completed.

Let

α

be a *-automorphism of real *-algebra

R

. By

\tilde{α}

we denote the linear extension of

α

= R + iR

, which is defined as

\tilde{α} (x + y) = = α (x) + iα (y)

, where

x, y \in R

Proposition 3.1. R is an α-simple if and only if A is an α˜- simple.

Proof. Let

I

be a subspace of

R

. Then

I_{c} = I + iI

is (complex) subspace of

A

. It’s obvious that

I

is ideal of

R

if and only if

I_{c}

is ideal of

A

. Moreover,

I

is closed if and only if

I_{c}

is closed. It’s easy to see that if

α (I) \subset I

, then

\tilde{α} (I_{c}) = α (I) + iα (I) \subset I_{c}

, and obversely,

\tilde{α} (I_{c}) \subset I_{c}

implies

α (I) \subset I

. Finally, it is obvious that

I \neq R

(or

θ

)

\Leftrightarrow I_{c} \neq A

(or {

θ

}). The proof is completed.

Let us formulate one result from the work

[10]

Proposition 3.2. A real C*-algebra

R

is an injective if and only if the C*-algebra

R + iR

is an injective.

Now let’s prove the following result.

Proposition 3.3. Let

R

be a real C*-algebra. Then a real C*-algebra

B

is injective envelope of

R

if and only if

B + iB

is injective envelope C*-algebra of

A = R + iR

Proof. Let

B + iB

be an injective envelope of

R + iR

. By Proposition 3.2

B

is an injective real C*-algebra. If

S

is an injective real C*-algebra with

R \subset S

, then

A = R + iR \subset S + iS

and by Proposition 3.2

S + iS

also is an injective, hence

B + iB \subset S + iS .

Therefore

B \subset S

and

B

is injective envelope real C*-algebra of

R

Conversely, let

B

be an injective envelope real C*-algebra of

R

. Then

A = R + iR \subset B + iB

and by Proposition 3.2

B + iB

is an injective C*-algebra. It’s easy to see that

B + iB

is injective envelope of

A

. The proof is completed.

Hence, using the result of

[3]

we obtain the following corollary.

Corollary 1. Any real C*-algebra has a unique injective envelope real C*-algebra.

Proposition 3.4. Let R be a real C*-algebra,

α

be a *- automorphism of

R .

Then

α

is outer *-automorphism of

R

if and only if

\tilde{α}

is outer *-automorphism of

A = R + iR

Proof. If

α

is an inner *-automorphism of

R

, there is an unitary

u \in R

such that

α (x) = Adu (x) = ux u^{*}, \forall x \in R

. Hence, we obtain

\tilde{α} (x + iy) = α (x) + iα (y) = uxu * + iuy u^{*} = u (x + iy) u^{*} = Adu (x + iy),

i.e.,

α

is also an inner *-automorphism of

A

. Conversely, let

\tilde{α}

is an inner, i.e.

\tilde{α} (x + iy) = Adv (x + iy)

, where

v \in A

is an unitary. Since

\tilde{α} (R) \subset R

, then by Corollary 3.1 from

[11]

, there exists an unitary

u \in R

such that

\tilde{α} = Adu .

Therefore

α = Adu

, i.e.,

α

is also an inner. The proof is completed.

Now we will prove the main result of the section.

Theorem 3.2. Let

R

be real C*-algebra,

α

be a *- automorphism of

R

such that

R

α

-simple. Let

B

be the injective envelope real C*-algebra of

R

. If

α

is an outer *- automorphism of

R

, then

α

has a unique extension to an outer *-automorphism of

B

Proof. By Proposition 3.1 C*-algebra

A = R + iR

\tilde{α}

- simple. By Proposition 3.4 *-automorphism

\tilde{α}

also is an outer *-automorphism. By Proposition 3.3 C*-algebra

B_{c} = B + iB

is the injective envelope of

A

. Then by [1, Theorem 3.6] *-automorphism

\tilde{α}

has a unique extension

\overset{̅}{\tilde{α}}

to outer *- automorphism of

B_{c}

. It is obvious that the restriction of

\overset{̅}{\tilde{α}}

R

coincides with

α

, i.e.

{\overset{̅}{\tilde{α}}|}_{R} = α

. Then it directly shows that

\overset{̅}{α} = \overset{̅}{\tilde{α}} {​|}_{B}

is a unique extension of

α

to an outer *-automorphism of

B

. The proof is completed.

2.3. Injective Envelope of Real Simple C*-Algebras

To motivate the next definitions, suppose

A

is a *-ring with unity, and let

w

be a partial isometry in

A

. If

e = w^{*} w

, it results from

w = w w^{*} w

that

wy = 0

iff

ey = 0

iff

(1 - e) y = y

iff

y \in (1 - e) A

, thus the elements that right-annihilate

w

form a principal right ideal generated by a projection. If

S

is a nonempty subset of

A

, we

write

R (S) = {x \in A : sx = 0, \forall s \in S}

and call

R (S

) the right-annihilator of

S

. Similarly, the set

L (S) = {x \in A : xs = 0, \forall s \in S}

denotes the it left-annihilator of

S

A Baer *-ring is a *-ring

A

such that, for every nonempty subset

S

A,

R (S) = gA

for a suitable projection

g

. It follows that

L (S) = {(R (S^{*}))}^{*} = {(hA)}^{*} = Ah

for a suitable projection

h

. A real (resp. complex) AW*-algebra is a real (resp. complex) C*-algebra that is a Baer *-ring (for more details see

[12-14]

). An AW*-algebra

A

is called a factor if the center of

A

is trivial. It is known that, every W*-algebra is an AW*-algebra. The converse of it was shown to be false by J.Dixmier, who showed that exist commutative AW*-algebras that cannot be represented (*-isomorphically) as W*-algebras on any Hilbert space.

The following interesting result is true.

Theorem 4.1. Let

R

be a real C*-algebra with the unit and let

B

its injective envelope. If

R

is a simple algebra, then

B

is also simple, in which case

B

is a real AW*-factor.

Proof. By Proposition 3.3 C*-algebra

B + iB

is an injective envelope of

A = R + iR

. By Theorem 3.1, since

R

is a simple algebra, then

A

is also simple. By Proposition 4.15 from

[3]

B + iB

is also simple and

B + iB

is a (complex) AW*-factor. Then

B

is also simple and by Proposition 4.3.1 from

[4]

, algebra

B

is a real AW*-factor. The proof is completed.

Now, by analogy with Hamana’s work, we give an example of an injective non W*-, AW*-factor of type III

[3]

Consider the Calkin algebra:

A = B (H) / K (H)

, where

H

is an infinite-dimensional separable Hilbert space and

K (H)

is an algebra of all compact operators on

H

. Example 5.1 from

[3]

shows that if

B

is an injective envelope of

A

, then

B

is an injective AW*-factor of type III, which is not a W*-algebra. Let us recall here that the Calkin algebra

A

is not an AW*- algebra

[15]

Now let’s look at the real analogue of this example. Let

H_{r}

is an infinite-dimensional separable real Hilbert space. Then since

B (H) = B (H_{r}) + iB (H_{r})

and

K (H) = K (H_{r}) + iK (H_{r})

, where

H = H_{r} + i H_{r} .

It’s easy to see that

A_{r} = B (H_{r}) / K (H_{r})

is a real C*-algebra and we have

A_{r} + i A_{r} = A

. Moreover, if

B_{r}

is an injective real envelope of

A_{r}

, then

B_{r} + i B_{r} = B

. By analogy

[15]

, since

B

is not a W*-algebra, then

B_{r}

is also not real W*-algebra.

Thus

A_{r}

is a real C*-algebra that is not real AW*-algebra and the injective envelope

B_{r}

A_{r}

is an injective real AW*- factor of type III, which is not a real W*-algebra.

It is known that for each number

λ \in [0, 1]

the class of injective (complex) W*-factors of type

II I_{λ}

is unique, i.e. any two injective

II I_{λ}

-factors are isomorphic. From the Hamana’s example we get the following interesting result [3, Example 5.1].

Corollary 2. Up to isomorphism the class of injective (complex) AW*-factors of type

II I_{λ} (0 \leq λ \leq 1)

is not unique, i.e. there are at least two isomorphy classes of injective (complex) AW*-factors of type

II I_{λ}

In the real case: up to isomorphism there exist exactly two injective real W*-factors of type

II I_{λ} (0 < λ \leq 1

) and up to isomorphism there exists a unique injective real W*-factor of type

II I_{1}

. For the case of real injective type

II I_{0}

factor we can construct a countable number of pairwise non isomorphic real injective factors of type

II I_{0}

, with isomorphic enveloping (complex) W*-factors (see

[4, 5]

Thus, from the above example, in the real case we can state the following

Corollary 3. Up to isomorphism, the class of of injective real AW*-factors of type III is at least one larger than the class injective real W*-factors of type III.

3. Materials and Methods

To obtain the results, methods of operator algebras were used, as well as the method of transition to the enveloping von Neumann algebra.

4. Results

Outer *-automorphism of a real C*-algebra can be uniquely extended to an injective envelope of real C*-algebra is proved. It is proven that if a real C*-algebra is a simple, then its injective envelope is also simple, and it is a real AW*-factor. The example of a real C*-algebra that is not real AW*-algebra and the injective envelope is a real AW*-factor of type III, which is not a real W*-algebra is constructed. This leads to the interesting result that up to isomorphism, the class of injective real (resp. complex) AW*-factors of type III is at least one larger than the class injective real (resp. complex) W*-factors of type III.

5. Discussion

It is known that the class of injective complex von Neumann factors of type III is unique, i.e. any two injective

II I_{λ}

-factors are isomorphic. But, as shown in this paper the class of injective real (resp. complex) AW*-factors of type III is at least one larger than the class injective real (resp. complex) W*-factors of type III.

6. Conclusions

Recently, interest in the study of AW*-algebras has increased. The results obtained here will undoubtedly be of interest to specialists in the field of operator algebra theory. It is proven that if a real C*-algebra is a simple, then its injective envelope is also simple, and it is a real AW*-factor. The example of a real C*-algebra that is not real AW*-algebra and the injective envelope is a real AW*-factor of type III, which is not a real W*-algebra is constructed. It is known that the class of injective complex von Neumann factors of type III is unique, i.e. any two injective

II I_{λ}

Abbreviations

B(H)	Algebra of all Bounded Linear Operators Acting on a Complex Hilbert Space H
$M_{n} (A)$	Algebra of all $n \times n$ Matrices over $A$
property E	An Extensional Property
$H$	An Infinite-dimensional Separable Hilbert Space
$K (H)$	An Algebra of all Compact Operators on $H$

Author Contributions

Abdugafur Rakhimov: Conceptualization, Supervision, Writing - original draft, Data curation, Investigation, Validation, Formal Analysis, Methodology, Writing - review & editing

Laylo Ramazonova: Investigation, Methodology, Writing - original draft, Validation, Writing - review & editing, Formal Analysis

Conflicts of Interest

The authors declare no conflicts of interest.

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Rakhimov, A., Ramazonova, L. (2025). Injective Envelopes of Real C*- and AW*-Algebras. International Journal of Applied Mathematics and Theoretical Physics, 11(1), 19-23. https://doi.org/10.11648/j.ijamtp.20251101.12

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Rakhimov, A.; Ramazonova, L. Injective Envelopes of Real C*- and AW*-Algebras. Int. J. Appl. Math. Theor. Phys. 2025, 11(1), 19-23. doi: 10.11648/j.ijamtp.20251101.12

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AMA Style

Rakhimov A, Ramazonova L. Injective Envelopes of Real C*- and AW*-Algebras. Int J Appl Math Theor Phys. 2025;11(1):19-23. doi: 10.11648/j.ijamtp.20251101.12

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  title = {Injective Envelopes of Real C*- and AW*-Algebras
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  journal = {International Journal of Applied Mathematics and Theoretical Physics},
  volume = {11},
  number = {1},
  pages = {19-23},
  doi = {10.11648/j.ijamtp.20251101.12},
  url = {https://doi.org/10.11648/j.ijamtp.20251101.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20251101.12},
  abstract = {Injective (complex and real) W*- and C*- algebras, in particular, factors have been studied quite well. On the other hand, in an arbitrary case, i.e., in the non-injective case, it is quite difficult to study (up to isomorphism) the W*-algebras, in particular, it is known that there is a continuum of pairwise non-isomorphic non-injective factors of type II1. Therefore, it seems interesting to study the so called maximal injective W* and C*-subalgebras or what is equivalent, the smallest injective C*-algebra containing a given algebra, which is called an injective envelope of C*- algebra. It is shown that every outer *-automorphism of a real C*-algebra can be uniquely extended to an injective envelope of real C*-algebra. It is proven that if a real C*-algebra is a simple, then its injective envelope is also simple, and it is a real AW*-factor. The example of a real C*-algebra that is not real AW*-algebra and the injective envelope is a real AW*-factor of type III, which is not a real W*-algebra is constructed. This leads to the interesting result that up to isomorphism, the class of injective real (resp. complex) AW*-factors of type III is at least one larger than the class injective real (resp. complex) W*-factors of type III.
},
 year = {2025}
}

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TY  - JOUR
T1  - Injective Envelopes of Real C*- and AW*-Algebras

AU  - Abdugafur Rakhimov
AU  - Laylo Ramazonova
Y1  - 2025/04/29
PY  - 2025
N1  - https://doi.org/10.11648/j.ijamtp.20251101.12
DO  - 10.11648/j.ijamtp.20251101.12
T2  - International Journal of Applied Mathematics and Theoretical Physics
JF  - International Journal of Applied Mathematics and Theoretical Physics
JO  - International Journal of Applied Mathematics and Theoretical Physics
SP  - 19
EP  - 23
PB  - Science Publishing Group
SN  - 2575-5927
UR  - https://doi.org/10.11648/j.ijamtp.20251101.12
AB  - Injective (complex and real) W*- and C*- algebras, in particular, factors have been studied quite well. On the other hand, in an arbitrary case, i.e., in the non-injective case, it is quite difficult to study (up to isomorphism) the W*-algebras, in particular, it is known that there is a continuum of pairwise non-isomorphic non-injective factors of type II1. Therefore, it seems interesting to study the so called maximal injective W* and C*-subalgebras or what is equivalent, the smallest injective C*-algebra containing a given algebra, which is called an injective envelope of C*- algebra. It is shown that every outer *-automorphism of a real C*-algebra can be uniquely extended to an injective envelope of real C*-algebra. It is proven that if a real C*-algebra is a simple, then its injective envelope is also simple, and it is a real AW*-factor. The example of a real C*-algebra that is not real AW*-algebra and the injective envelope is a real AW*-factor of type III, which is not a real W*-algebra is constructed. This leads to the interesting result that up to isomorphism, the class of injective real (resp. complex) AW*-factors of type III is at least one larger than the class injective real (resp. complex) W*-factors of type III.

VL  - 11
IS  - 1
ER  -

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Author Information

Abdugafur Rakhimov

Algebra and Functional Analysis, National University of Uzbekistan, Tashkent, Uzbekistan; Department of Mathematics, Karadeniz Technical University, Trabzon, Turkey

Research Fields: Functional analysis, Theory of operator algebras, Theory of unbounded operators, Topology, Fuzzy topology

Contact Email

http://orcid.org/0000-0002-2675-1510
Laylo Ramazonova

V. I. Romanovsky Institute of Mathematics of the Academy of Sciences of Uzbekistan, Tashkent, Uzbekistan

Research Fields: Functional analysis, Theory of operator algebras, Theory of unbounded operators, Topology, Fuzzy topology

Contact Email

http://orcid.org/0009-0009-8772-1129

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APA Style

Rakhimov, A., Ramazonova, L. (2025). Injective Envelopes of Real C*- and AW*-Algebras. International Journal of Applied Mathematics and Theoretical Physics, 11(1), 19-23. https://doi.org/10.11648/j.ijamtp.20251101.12

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ACS Style

Rakhimov, A.; Ramazonova, L. Injective Envelopes of Real C*- and AW*-Algebras. Int. J. Appl. Math. Theor. Phys. 2025, 11(1), 19-23. doi: 10.11648/j.ijamtp.20251101.12

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AMA Style

Rakhimov A, Ramazonova L. Injective Envelopes of Real C*- and AW*-Algebras. Int J Appl Math Theor Phys. 2025;11(1):19-23. doi: 10.11648/j.ijamtp.20251101.12

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@article{10.11648/j.ijamtp.20251101.12,
  author = {Abdugafur Rakhimov and Laylo Ramazonova},
  title = {Injective Envelopes of Real C*- and AW*-Algebras
},
  journal = {International Journal of Applied Mathematics and Theoretical Physics},
  volume = {11},
  number = {1},
  pages = {19-23},
  doi = {10.11648/j.ijamtp.20251101.12},
  url = {https://doi.org/10.11648/j.ijamtp.20251101.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20251101.12},
  abstract = {Injective (complex and real) W*- and C*- algebras, in particular, factors have been studied quite well. On the other hand, in an arbitrary case, i.e., in the non-injective case, it is quite difficult to study (up to isomorphism) the W*-algebras, in particular, it is known that there is a continuum of pairwise non-isomorphic non-injective factors of type II1. Therefore, it seems interesting to study the so called maximal injective W* and C*-subalgebras or what is equivalent, the smallest injective C*-algebra containing a given algebra, which is called an injective envelope of C*- algebra. It is shown that every outer *-automorphism of a real C*-algebra can be uniquely extended to an injective envelope of real C*-algebra. It is proven that if a real C*-algebra is a simple, then its injective envelope is also simple, and it is a real AW*-factor. The example of a real C*-algebra that is not real AW*-algebra and the injective envelope is a real AW*-factor of type III, which is not a real W*-algebra is constructed. This leads to the interesting result that up to isomorphism, the class of injective real (resp. complex) AW*-factors of type III is at least one larger than the class injective real (resp. complex) W*-factors of type III.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Injective Envelopes of Real C*- and AW*-Algebras

AU  - Abdugafur Rakhimov
AU  - Laylo Ramazonova
Y1  - 2025/04/29
PY  - 2025
N1  - https://doi.org/10.11648/j.ijamtp.20251101.12
DO  - 10.11648/j.ijamtp.20251101.12
T2  - International Journal of Applied Mathematics and Theoretical Physics
JF  - International Journal of Applied Mathematics and Theoretical Physics
JO  - International Journal of Applied Mathematics and Theoretical Physics
SP  - 19
EP  - 23
PB  - Science Publishing Group
SN  - 2575-5927
UR  - https://doi.org/10.11648/j.ijamtp.20251101.12
AB  - Injective (complex and real) W*- and C*- algebras, in particular, factors have been studied quite well. On the other hand, in an arbitrary case, i.e., in the non-injective case, it is quite difficult to study (up to isomorphism) the W*-algebras, in particular, it is known that there is a continuum of pairwise non-isomorphic non-injective factors of type II1. Therefore, it seems interesting to study the so called maximal injective W* and C*-subalgebras or what is equivalent, the smallest injective C*-algebra containing a given algebra, which is called an injective envelope of C*- algebra. It is shown that every outer *-automorphism of a real C*-algebra can be uniquely extended to an injective envelope of real C*-algebra. It is proven that if a real C*-algebra is a simple, then its injective envelope is also simple, and it is a real AW*-factor. The example of a real C*-algebra that is not real AW*-algebra and the injective envelope is a real AW*-factor of type III, which is not a real W*-algebra is constructed. This leads to the interesting result that up to isomorphism, the class of injective real (resp. complex) AW*-factors of type III is at least one larger than the class injective real (resp. complex) W*-factors of type III.

VL  - 11
IS  - 1
ER  -

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