Norm of General Derivation on Tensor Product of 
C*-Algebras

Benjamin Kimeu Daniel; Kinyanjui Jeremiah Ndungu; Zachary Kaunda Kayiita

doi:doi:10.11648/j.pamj.20261501.12

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Norm of General Derivation on Tensor Product of C*-Algebras

Benjamin Kimeu Daniel^*

, Kinyanjui Jeremiah Ndungu

, Zachary Kaunda Kayiita

Published in Pure and Applied Mathematics Journal (Volume 15, Issue 1)

Received: 22 January 2026 Accepted: 4 February 2026 Published: 26 February 2026

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Abstract

The study of operator theory, specifically the determination of the norm of general derivations in C* algebras, has attracted significant attention in recent years. This problem has been approached through various methods across different spaces, yielding several results. The norm of a general derivation is crucial for understanding the structure and behavior of derivations in the context of C* algebras, with implications for functional analysis and operator theory. Despite previous efforts, a complete understanding remains elusive. The purpose of this paper is to investigate this problem using the finite rank operator in the tensor product of C*-algebras. By leveraging the tensor product structure, we explore how finite rank operators can provide insights into the norm of general derivations. The research utilizes a mathematical framework that examines the behavior of bounded linear operators within the tensor product of Hilbert spaces and C*-algebras. Through this approach, we aim to advance existing knowledge and offer new results that could potentially contribute to the field. The final conclusion of the study confirms that the use of finite rank operators in the tensor product of C*-algebras offers a valuable method for approximating and understanding the norm of general derivations, providing a more refined approach than previous techniques.

Published in	Pure and Applied Mathematics Journal (Volume 15, Issue 1)
DOI	10.11648/j.pamj.20261501.12
Page(s)	6-10
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), 2026. Published by Science Publishing Group

Keywords

General Derivation Operator, Finite Rank Operator, Tensor Product of C*-Algebra

1. Introduction

1.1. C* Algebras and Their Tensor Product Structure in Hilbert Spaces

Definition 1.1.1. Tensor product. Daniel et al.,

[6]

and Muiruri et al.,

[10]

Consider two complex Hilbert spaces, 𝑍 = {

u_{1}

u_{2}

…} and 𝐿 = {

v_{1}

v_{1}

… } with inner products defined as

〈u_{1}, u_{2}〉

and

〈v_{1}, v_{2}〉

respectively. A tensor product of 𝑍 and 𝐿 is a Hilbert space 𝑍 ⊗ 𝐿 where ⊗: 𝑍 × 𝐿 → 𝑍 ⊗ 𝐿,⊗ (𝑢, 𝑣) → 𝑢 ⊗ 𝑣 is a bilinear mapping: The vectors 𝑢 ⊗ 𝑣 form a total subset of 𝑍 ⊗ 𝐿 < 𝑢1 ⊗ 𝑣1, 𝑢2 ⊗ 𝑣2 >=<

u_{1} {, v}_{1}

> <

u_{2}

v_{2}

>, ∀

u_{1}

u_{2}

∈ 𝑍, 𝑣1, 𝑣2 ∈ 𝐿. This implies that

‖u \otimes v‖

‖u‖ ‖v‖

∀ 𝑢 ∈ 𝑍, 𝑣 ∈ 𝐿.If 𝐸 ∈ 𝐵(𝑍), 𝐹 ∈ 𝐵(𝐿), then 𝐵(𝑍 ⊗ 𝐿) is a Hilbert space and for 𝐸 ⊗ 𝐹 ∈ 𝐵(𝑍 ⊗ 𝐿) we have 𝐸 ⊗ 𝐹(𝑢 ⊗ 𝑣) = 𝐸𝑢 ⊗ 𝐹𝑣

𝑢 ∈ 𝑍, 𝑣 ∈ 𝐿.

Fundamental Properties of Operators in 𝐵(𝑍 ⊗ 𝐿) hold:

1) i).(𝐸 ⊗ 𝐹)(𝐺 ⊗ 𝐻) = 𝐸𝐺 ⊗ 𝐹𝐻, ∀ 𝐸 ∈ 𝐵(𝑍), 𝐺 ∈ 𝐵(𝑍) and 𝐹 ∈ 𝐵(𝐿), 𝐻 ∈ 𝐵(𝐿).(Associativity and commutativity).

2) ii).

‖E \otimes F‖

‖E‖ ‖F‖

∀ 𝐸 ∈ 𝐵(𝑍) and 𝐹 ∈ 𝐵(𝐿) (Distributivity under tensor product).

3) iii). Linearity of the Tensor Product Map ⊗ (𝑢, 𝑣) → 𝑢 ⊗ 𝑣 and its Properties:

a) (

u_{1}

u_{2}

) ⊗ 𝑣 = (

u_{1}

⊗ 𝑣) + (

u_{2}

⊗ 𝑣)

b) (𝜓𝑢) ⊗ 𝑣 = 𝜓(𝑢 ⊗ 𝑣).

c) 𝑢 ⊗ (

v_{1}

v_{2}

) = 𝑢 ⊗

v_{1}

+ 𝑢 ⊗

v_{2}

d) 𝑢 ⊗ (𝜓𝑣) = 𝜓(𝑢 ⊗ 𝑣).

The set of all vectors ⊗ (𝑢, 𝑣), 𝑢𝜖𝑍 and 𝑣𝜖𝐿 form a total subset of 𝑍 ⊗ 𝐿.

Definition 1.1.2. Elementary operator in a tensor product. Daniel et al.,

[5]

and King’ang’I et al.,

[7]

Let

Z

and

L

be complex Hilbert spaces, and

B (Z \otimes L)

denote the set of all bounded linear operators on the tensor product space

Z \otimes L

. For fixed elements

E \otimes F

and

G \otimes H

B (Z \otimes L),

where

E, G \in B (Z)

and

F, H \in B (L),

we define the elementary operator as follows:

E_{n}

(𝑍 ⊗ 𝐿) =

\sum_{n}

(𝐸𝑖 ⊗ 𝐹𝑖)(𝑈 ⊗ 𝑉)(𝐺𝑖 ⊗ 𝐻𝑖)

for every 𝑈 ⊗ 𝑉𝜖𝐵(𝑍 ⊗ 𝐿), 𝐸𝑖 ⊗ 𝐹𝑖, 𝐺𝑖 ⊗ 𝐻𝑖

\in

𝐵(𝑍 ⊗ 𝐿).

Substituting

n = 1,

we obtain the basic elementary operator:

𝐸(𝑍⊗𝐿) = (𝐸⊗𝐹)(𝑈⊗𝑉)(𝐺⊗𝐻)(1)

From equation (1), the basic elementary operator can be expressed as,

E (Z \otimes L) = (E \otimes F) (U \otimes V) (G \otimes H) = (EUG) \otimes (FVH)

Definition 1.1.3: General Derivation. Nyamwala and Agure,

[11]

and Okelo and Ambongo,

[12]

The general derivation operator is defined by

δ_{A, B} = AX - XB

where

A

and

B

are coefficient operators in

B (X) .

1.2. Norm of General Derivation

Stampfli

[14]

and Timoney

[15]

determined the

{‖δ‖}_{D}

in a Banach algebra

B (L)

by maximal numerical range and proved theorem 1.2.1:

Theorem 1.2.1. Stampfli

[14]

and Timoney

[15]

Let A, B ∈ B(H), the algebra of all bounded linear operators on a Hilbert space H. Then

‖δ_{A, B}‖

= sup {

‖AX - XB‖

: X ∈ B(H),

‖X‖ = 1

}

= i {nf}_{λ ϵ C} ‖A - λI‖ + ‖B - λI‖

Let A be an irreducible C^∗-algebra on H. Let A ∈ B(A). Then

‖δ_{A}‖ = 2 \inf_{λϵ C} ‖A - λI‖ = 2 d (A)

Nyamwala and Agure

[11]

and Society

[13]

determined the converse of Stampfli

[14]

, results, and established that the inner derivation derived from hyponormal operators has its range closed whenever the spectrum of the operator is defined.

Kittaneh

[8]

extended the work of Stampfli

[14]

in norm ideals and determined the range and kernel of a normal derivation restricted to unitary invariant norms.

Barraa and Pedersen

[2]

determined the conditions necessary for the product

δ_{(C, D)}, δ_{(A, B)}

of 2 general derivations to satisfy the conditions of a derivation and proved theorem 1.2.2:

Theorem 1.2.2. Barraa and Pedersen

[2]

Let A, B, C, and D be bounded operators on a Banach space

T

1) If

A \notin T

and

B \notin T

, then

δ_{C, D} δ_{A, B}

is a generalized derivation if and only iff

C = aA + cI and D = - aB + dI

for some scalars

a, c

, and

d .

2) If A ∈

T

and B

\notin T

, then

δ_{C, D} δ_{A, B}

is a generalized derivation if and only if C ∈

T

3) If A

\notin

T

and B ∈

T

, then

δ_{C, D} δ_{A, B}

is a generalized derivation if and only if D ∈

T

4) If

A \in T

and

B \in T

, then

δ_{C, D} δ_{A, B}

is a generalized derivation.

Barraa

[1]

and Barraa and Pedersen

[2]

determined the norm of the sum of two bounded operators on a Hilbert space and provided sufficient conditions under which the norm equals the sum of their norms

Muhol et al.

[9]

determined the norm of a general derivation acting on a norm ideal and proved Theorem 1.2.3.

Theorem 1.2.3. Muhol et al.,

[9]

Let

A, B \in B (H)

be S-universal operators and let

J

be a norm ideal in

B (H)

. Then

‖δ_{A, B}‖ = ‖δ_{A, B, J}‖

Bonyo and Agure

[4]

determined the norms of derivations implemented by S-universal operators. Bonyo and Agure

[4]

focused on understanding the behavior of derivations within norm ideals, particularly the conditions under which these derivations can be analyzed using S-universal operators. Their work contributes to the broader field of operator theory by providing insights into the relationship between derivations and norm ideals, helping to establish criteria for when these derivations attain their norms.

Beatrice et al.

[3]

determined the norm of a general derivation on an infinite-dimensional complex Hilbert space H using finite rank one operators and proved Theorem 1.2.4.

Theorem 1.2.4. Beatrice et al.

[3]

Let

A, B \in B (H)

and δ_{A, B}:

B (H) \to B (H) .

Then

δ_{A, B} B (H) = A + B .

Further, Beatrice et al.

[3]

showed that this equality holds using Stampfli’s maximal numerical range and proved Theorem 1.2.5.

Theorem 1.2.5. Beatrice et al.,

[3]

Let

d (A) = Inf {‖A - λI‖ : λ \in C},

and

d (B) = \inf {‖B - λI‖ : λ \in C} .

Then

‖δ_{A, B} B (H)‖ = ‖A‖ + ‖B‖

2. Main Results

Norm of General Derivation

δ

on a Tensor Product of C^∗-Algebras

Let

H \otimes K

be the tensor product of Hilbert spaces

H

and

K,

and

B (H \otimes K)

be the set of bounded linear operators on

H \otimes K

. Then, for

X \otimes Y \in B (H \otimes K)

with

X \otimes Y = 1

, we have:

δ_{A \otimes C, B \otimes D} | (B (H \otimes K) = A B + C D

Proof

By definition,

‖δ_{A \otimes C, B \otimes D}‖ = Sup {‖δ (A \otimes C, B \otimes D) (X \otimes Y)‖ : X \otimes Y \in B (H \otimes K), ‖X \otimes Y‖ = 1

= sup {

‖(A \otimes B) (X \otimes Y) - (X \otimes Y) (C \otimes D‖

: X ⊗ Y ∈ B(H ⊗ K),

‖X \otimes Y‖ = 1

Therefore,

δ_{A \otimes C, B \otimes D}

|| = sup {

δ_{A \otimes B, C \otimes D}

(X ⊗ Y)

: X ⊗ Y ∈ B(H ⊗ K), ||X ⊗ Y || = 1}

Taking an arbitrary ϵ > 0, we have

δ_{A \otimes B, C \otimes D}

|| − ϵ

<

δ_{A \otimes B, C \otimes D}

(X ⊗ Y)

for

all X \in B (H), Y \in B (K),

and

for

all X \in B (H), Y \in B (K),

and

X \otimes Y = 1 .

δ_{A \otimes B, C \otimes D}

−ϵ

<

(A⊗B)(X⊗Y)− (X⊗Y)(C⊗D)

Since

(A \otimes B) (X \otimes Y) - (X \otimes Y) (C \otimes D) \leq A B + C D

and letting

ϵ \to 0

, then we have

δ_{A \otimes B, C \otimes D}

≤

(2)

Conversely, let there exist a unit vector

e \otimes f

where

e \in H

and

f \in K

, then we have

δ_{A \otimes B, C \otimes D}

(X ⊗ Y)(e ⊗ f)

≤

δ_{A \otimes B, C \otimes D}

(X ⊗ Y)

\leq ‖δ_{A \otimes B, C \otimes D} (X \otimes Y)‖ ‖e‖ ‖f‖

given that

X, Y, e,

and

f

are unit operators and vectors, respectively, and

X \otimes Y = X Y

then

δ_{_{A \otimes B, C \otimes D}}

||||X ⊗ Y

e ⊗ f

δ_{A \otimes B, C \otimes D}

Thus,

δ_{A \otimes B, C \otimes D}

≥

δ_{A \otimes B, C \otimes D}

(X ⊗ Y)(e ⊗ f)

≥

{ (A ⊗ B)(X ⊗ Y) − (X ⊗ Y)(C ⊗ D)}(e ⊗ f)

By tensor product property, (A ⊗ B)(X ⊗ Y) = AX ⊗ BY we have

AX⊗BY

(3)

From equation (3), then

(AX ⊗ BY)(e ⊗ f) − (X ⊗ Y)(C ⊗ D)(e ⊗ f)

AXe ⊗ BY f − CXe ⊗ DY f

since

e ⊗ f

= 1

therefore,

δ_{A \otimes B, C \otimes D} \geq AXe \otimes BY f - CXe \otimes DY f

Squaring both sides, we get

{‖δ_{(A \otimes B, C \otimes D)}‖}^{2} \geq {‖AXe \otimes BY f - CXe \otimes DY f‖}^{2}

= ⟨ AXe \otimes BY f - CXe \otimes DY f, AXe \otimes BY f - CXe \otimes DY f ⟩

= ⟨ AXe \otimes BY f, AXe \otimes BY f ⟩ - ⟨ AXe \otimes BY f, CXe \otimes DY f ⟩ - ⟨ CXe \otimes DY f, AXe \otimes BY f ⟩ + ⟨ CXe \otimes DY f, CXe \otimes DY f ⟩

= {‖AXe \otimes BY f‖}^{2} - ⟨ AXe \otimes BY f, CXe \otimes DY f ⟩ - ⟨ CXe \otimes DY f, AXe \otimes BY f ⟩ + {‖CXe \otimes DY f‖}^{2} .

By properties of the tensor product, ⟨

x_{1}

⊗

y_{1}

x_{2}

⊗

y_{2}

⟩ = ⟨

x_{1}

x_{2}

⟩⟨

y_{1}

y_{2}

⟩ ∀

x_{1}, x_{2}

∈ H

and ∀

y_{1}, y_{2}

∈ K we have;

{‖δ_{(A \otimes B, C \otimes D)}‖}^{2} \geq {‖AXe‖}^{2} {‖BY f‖}^{2} - ⟨ AXe \otimes BY f, CXe \otimes DY f ⟩ - ⟨ CXe \otimes DY f, AXe \otimes BY f ⟩ + {‖CXe‖}^{2} {‖DY f‖}^{2}

.(4)

µ : H \to R

and

ν : K \to R

are functionals. For unit vectors y ∈ H and z ∈ K, define rank one operators:

A = µ \otimes y, y \in H, y = 1, by Ae = (µ \otimes y) e = µ (e) y

B = ν \otimes z, z \in K, z = 1, by Bf = (ν \otimes z) f = ν (f) z

Observe that:

A = \sup {(µ \otimes y) e : e \in H}

= \sup {µ (e) y : e \in H}

= \sup {| µ (e) | e : e \in H

= \sup \{|µ (e)| : e \in H\}

= | µ (e) |

Thus,

A = | µ (e) | : e \in H .

Similarly, using the same concept above, the norm of

B

‖B‖ = | ν (f) | : f \in K .

Moreover,

{‖ AXe ‖}^{2} = {‖(µy) xe‖}^{2}

= {‖µ (e) yxe‖}^{2}

{= |µ (e)|}^{2} {‖xe‖}^{2}

{= |µ (e)|}^{2}

since

x, e

are unit vectors

{‖xe‖}^{2} = 1

thus

{|µ (e)|}^{2} = {‖A‖}^{2}

Thus

{‖ AXe ‖}^{2} = {‖A‖}^{2}

(5)

Using the same concept from equation (5), we have Now,

{‖BY f‖}^{2} = {‖B‖}^{2}

(6)

{‖CXe‖}^{2} = {‖C‖}^{2}

(7)

{‖Dyf‖}^{2} = {‖D‖}^{2}

(8)

Now, from equation (4)

= - ⟨ Axe \otimes Byf, Cxe \otimes Dyf ⟩

= - ⟨ Axe \otimes Byf, Cxe \otimes Dyf ⟩

= - ⟨ Axe, Cxe ⟩ ⟨ Byf, Dyf ⟩

hence

- ⟨ Axe, Cxe ⟩ = ⟨ - (µ \otimes y) eX, (ν \otimes y) eX ⟩

= ⟨ - µ (e) yx, µ (e) yx ⟩

= - µ (e) x \cdot µ (e) x . ⟨ y, y ⟩

since

⟨y, y⟩ = 1

= - µ (e) x \cdot µ (e) x

but

µ (e)

are positive real numbers, therefore

- µ (e) x = | µ (e) | = µ (e) x

Therefore,

- µ (e) = | µ (e) | = A

, and

µ (e) = | µ (e) | = C .

Hence,

- ⟨ Axe, Cxe ⟩ = A C

, Similarly,

⟨ Byf, Dyf ⟩ = B D .

Thus,

- ⟨ Axe, Cxe ⟩ ⟨ Byf, Dyf ⟩ = A B C D .

(9)

- ⟨ Cxe \otimes Dyf, Axe \otimes Byf ⟩ = - ⟨ Cxe, Axe ⟩ ⟨ Dyf, Byf ⟩

= C A D B .

= A B C D .

(10)

Substituting equations (5), (6), (7), (8), (9), and (10) in (4), we have

{‖δ_{(A \otimes B, C \otimes D)}‖}^{2} \geq {‖A‖}^{2} {‖B‖}^{2} + 2 ‖A‖ ‖B‖ ‖C‖ ‖D‖ + {‖C‖}^{2} {‖D‖}^{2}

Taking the square roots of both sides, we get:

‖δ_{(A \otimes B, C \otimes D)} | (B (H \otimes K)‖ \geq ‖A‖ ‖B‖ + ‖C‖ ‖D‖

(11)

Therefore, from (2) and (11)

δ_{A \otimes C, B \otimes D} | (B (H \otimes K) = A B + C D

3. Conclusion

From the study, it can be concluded that using a finite rank operator, the Norm of general derivation in the tensor product of C*Algebras is

δ_{A \otimes C, B \otimes D} | (B (H \otimes K) = A B + C D .

Abbreviations

C* algebras	C-star Algebras
$H$	Hilbert Space
$B (H)$	Bounded Linear Operators on a Hilbert Space H
$B (Z \otimes L)$	Bounded Linear Operators on the Tensor Product of Hilbert Spaces Z and L
$δ_{A, B}$	General Derivation Between Two Operators A and B
$S - universal operators$	A Class of Operators Used in Norm Ideals
$‖.‖$	Norm Notation
$I$	Identity Operator
$J$	Norm Ideal in B(H)
$x ⨂ y$	Tensor Product of Operators X and Y
$\in$	Arbitrary Small Positive Number (Epsilon)
$λ$	Scalar Value
$R$	Real Numbers

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Barraa, M. (2002). Convexoid and generalized derivations. Linear algebra and its applications, 350(1-3): 289–292. https://doi.org/10.1016/s0024-3795(02)00311-7
[2]	Barraa, M. and Pedersen, S. (1999). On the product of two generalized derivations. Proceedings of the American Mathematical Society, pages 2679–2683. https://www.jstor.org/stable/119569
[3]	Beatrice, O. A., Agure, J., and Nyamwala, F. (2019). On the norm of a generalized derivation. https://doi.org/10.12988/pms.2019.9810
[4]	Bonyo, J. and Agure, J. (2011). Norms of derivations implemented by s-universal operators.
[5]	Daniel, B. K., Wabomba, M. S., and Ndungu, K. J. (2023). An application of maximal numerical range on norm of an elementary operator of length two in tensor product. International Journal Of Mathematics And Computer Research, 11(11): 3837–3842.
[6]	Daniel, B., Musundi, S., and Ndungu, K. (2022). An application of maximal numerical range on norm of basic elementary operator in tensor product. Journal of Progressive Research in Mathematics, 19(1): 73–81.
[7]	King’ang’i, D. N., Agure, J. O., and Nyamwala, F. O. (2014). On the norm of elementary operator. http://dx.doi.org/10.4236/apm.2014.47041
[8]	Kittaneh, F. (1995). Normal derivations in norm ideals. Proceedings of the American Mathematical Society, 123(6): 1779–1785. https://doi.org/10.2307/2160991
[9]	Muholo, J., Bonyo, J., and Agure, J. (2019). Norm properties of generalized derivations on norm ideals.
[10]	Muiruri, P., King'ang'i, D., & Musundi, S. (2019). On the Norm of Basic Elementary Operator in a Tensor Product.
[11]	Nyamwala, F. and Agure, J. (2008). Norms of elementary operators in banach algebras. Int. Journal of Math. Analysis, 2(9): 411–424.
[12]	Okelo, N., A. J. and Ambongo, D. (2010). norm of elementary operator and characterization of norm-attained operators. int. Journal of Math. Analysis, 4(11971204).
[13]	Society, L. M. (1926). Journal of the London Mathematical Society, volume 1. London Mathematical Society.
[14]	Stampfli, J. (1970). The norm of a derivation. Pacific journal of mathematics, 33(3), 737-747.
[15]	Timoney, R. M. (2007). Some formulae for norms of elementary operators. Journal of Operator Theory, pages 121–145. https://www.jstor.org/stable/24715761

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APA Style

Daniel, B. K., Ndungu, K. J., Kayiita, Z. K. (2026). Norm of General Derivation on Tensor Product of C*-Algebras. Pure and Applied Mathematics Journal, 15(1), 6-10. https://doi.org/10.11648/j.pamj.20261501.12

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ACS Style

Daniel, B. K.; Ndungu, K. J.; Kayiita, Z. K. Norm of General Derivation on Tensor Product of C*-Algebras. Pure Appl. Math. J. 2026, 15(1), 6-10. doi: 10.11648/j.pamj.20261501.12

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AMA Style

Daniel BK, Ndungu KJ, Kayiita ZK. Norm of General Derivation on Tensor Product of C*-Algebras. Pure Appl Math J. 2026;15(1):6-10. doi: 10.11648/j.pamj.20261501.12

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@article{10.11648/j.pamj.20261501.12,
  author = {Benjamin Kimeu Daniel and Kinyanjui Jeremiah Ndungu and Zachary Kaunda Kayiita},
  title = {Norm of General Derivation on Tensor Product of 
C*-Algebras},
  journal = {Pure and Applied Mathematics Journal},
  volume = {15},
  number = {1},
  pages = {6-10},
  doi = {10.11648/j.pamj.20261501.12},
  url = {https://doi.org/10.11648/j.pamj.20261501.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20261501.12},
  abstract = {The study of operator theory, specifically the determination of the norm of general derivations in C* algebras, has attracted significant attention in recent years. This problem has been approached through various methods across different spaces, yielding several results. The norm of a general derivation is crucial for understanding the structure and behavior of derivations in the context of C* algebras, with implications for functional analysis and operator theory. Despite previous efforts, a complete understanding remains elusive. The purpose of this paper is to investigate this problem using the finite rank operator in the tensor product of C*-algebras. By leveraging the tensor product structure, we explore how finite rank operators can provide insights into the norm of general derivations. The research utilizes a mathematical framework that examines the behavior of bounded linear operators within the tensor product of Hilbert spaces and C*-algebras. Through this approach, we aim to advance existing knowledge and offer new results that could potentially contribute to the field. The final conclusion of the study confirms that the use of finite rank operators in the tensor product of C*-algebras offers a valuable method for approximating and understanding the norm of general derivations, providing a more refined approach than previous techniques.},
 year = {2026}
}

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TY  - JOUR
T1  - Norm of General Derivation on Tensor Product of 
C*-Algebras
AU  - Benjamin Kimeu Daniel
AU  - Kinyanjui Jeremiah Ndungu
AU  - Zachary Kaunda Kayiita
Y1  - 2026/02/26
PY  - 2026
N1  - https://doi.org/10.11648/j.pamj.20261501.12
DO  - 10.11648/j.pamj.20261501.12
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 6
EP  - 10
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20261501.12
AB  - The study of operator theory, specifically the determination of the norm of general derivations in C* algebras, has attracted significant attention in recent years. This problem has been approached through various methods across different spaces, yielding several results. The norm of a general derivation is crucial for understanding the structure and behavior of derivations in the context of C* algebras, with implications for functional analysis and operator theory. Despite previous efforts, a complete understanding remains elusive. The purpose of this paper is to investigate this problem using the finite rank operator in the tensor product of C*-algebras. By leveraging the tensor product structure, we explore how finite rank operators can provide insights into the norm of general derivations. The research utilizes a mathematical framework that examines the behavior of bounded linear operators within the tensor product of Hilbert spaces and C*-algebras. Through this approach, we aim to advance existing knowledge and offer new results that could potentially contribute to the field. The final conclusion of the study confirms that the use of finite rank operators in the tensor product of C*-algebras offers a valuable method for approximating and understanding the norm of general derivations, providing a more refined approach than previous techniques.
VL  - 15
IS  - 1
ER  -

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Author Information

Benjamin Kimeu Daniel

Department of Pure & Applied Sciences, Kirinyaga University, Kerugoya, Kenya

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http://orcid.org/0000-0003-0569-7364
Kinyanjui Jeremiah Ndungu

Department of Pure & Applied Sciences, Kirinyaga University, Kerugoya, Kenya

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http://orcid.org/0009-0002-6768-1990
Zachary Kaunda Kayiita

Department of Pure & Applied Sciences, Kirinyaga University, Kerugoya, Kenya

Contact Email

http://orcid.org/0009-0005-4376-2528

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APA Style

Daniel, B. K., Ndungu, K. J., Kayiita, Z. K. (2026). Norm of General Derivation on Tensor Product of C*-Algebras. Pure and Applied Mathematics Journal, 15(1), 6-10. https://doi.org/10.11648/j.pamj.20261501.12

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ACS Style

Daniel, B. K.; Ndungu, K. J.; Kayiita, Z. K. Norm of General Derivation on Tensor Product of C*-Algebras. Pure Appl. Math. J. 2026, 15(1), 6-10. doi: 10.11648/j.pamj.20261501.12

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AMA Style

Daniel BK, Ndungu KJ, Kayiita ZK. Norm of General Derivation on Tensor Product of C*-Algebras. Pure Appl Math J. 2026;15(1):6-10. doi: 10.11648/j.pamj.20261501.12

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@article{10.11648/j.pamj.20261501.12,
  author = {Benjamin Kimeu Daniel and Kinyanjui Jeremiah Ndungu and Zachary Kaunda Kayiita},
  title = {Norm of General Derivation on Tensor Product of 
C*-Algebras},
  journal = {Pure and Applied Mathematics Journal},
  volume = {15},
  number = {1},
  pages = {6-10},
  doi = {10.11648/j.pamj.20261501.12},
  url = {https://doi.org/10.11648/j.pamj.20261501.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20261501.12},
  abstract = {The study of operator theory, specifically the determination of the norm of general derivations in C* algebras, has attracted significant attention in recent years. This problem has been approached through various methods across different spaces, yielding several results. The norm of a general derivation is crucial for understanding the structure and behavior of derivations in the context of C* algebras, with implications for functional analysis and operator theory. Despite previous efforts, a complete understanding remains elusive. The purpose of this paper is to investigate this problem using the finite rank operator in the tensor product of C*-algebras. By leveraging the tensor product structure, we explore how finite rank operators can provide insights into the norm of general derivations. The research utilizes a mathematical framework that examines the behavior of bounded linear operators within the tensor product of Hilbert spaces and C*-algebras. Through this approach, we aim to advance existing knowledge and offer new results that could potentially contribute to the field. The final conclusion of the study confirms that the use of finite rank operators in the tensor product of C*-algebras offers a valuable method for approximating and understanding the norm of general derivations, providing a more refined approach than previous techniques.},
 year = {2026}
}

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TY  - JOUR
T1  - Norm of General Derivation on Tensor Product of 
C*-Algebras
AU  - Benjamin Kimeu Daniel
AU  - Kinyanjui Jeremiah Ndungu
AU  - Zachary Kaunda Kayiita
Y1  - 2026/02/26
PY  - 2026
N1  - https://doi.org/10.11648/j.pamj.20261501.12
DO  - 10.11648/j.pamj.20261501.12
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 6
EP  - 10
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20261501.12
AB  - The study of operator theory, specifically the determination of the norm of general derivations in C* algebras, has attracted significant attention in recent years. This problem has been approached through various methods across different spaces, yielding several results. The norm of a general derivation is crucial for understanding the structure and behavior of derivations in the context of C* algebras, with implications for functional analysis and operator theory. Despite previous efforts, a complete understanding remains elusive. The purpose of this paper is to investigate this problem using the finite rank operator in the tensor product of C*-algebras. By leveraging the tensor product structure, we explore how finite rank operators can provide insights into the norm of general derivations. The research utilizes a mathematical framework that examines the behavior of bounded linear operators within the tensor product of Hilbert spaces and C*-algebras. Through this approach, we aim to advance existing knowledge and offer new results that could potentially contribute to the field. The final conclusion of the study confirms that the use of finite rank operators in the tensor product of C*-algebras offers a valuable method for approximating and understanding the norm of general derivations, providing a more refined approach than previous techniques.
VL  - 15
IS  - 1
ER  -

Copy | Download