| Title:
|
On homomorphisms between $C^*$-algebras and linear derivations on $C^*$-algebras (English) |
| Author:
|
Park, Chun-Gil |
| Author:
|
Chu, Hahng-Yun |
| Author:
|
Park, Won-Gil |
| Author:
|
Wee, Hee-Jeong |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
55 |
| Issue:
|
4 |
| Year:
|
2005 |
| Pages:
|
1055-1065 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
It is shown that every almost linear Pexider mappings $f$, $g$, $h$ from a unital $C^*$-algebra $\mathcal A$ into a unital $C^*$-algebra $\mathcal B$ are homomorphisms when $f(2^n uy)=f(2^n u)f(y)$, $g(2^n uy)=g(2^nu)g(y)$ and $h(2^n uy)=h(2^n u)h(y)$ hold for all unitaries $u \in \mathcal A$, all $y \in \mathcal A$, and all $n\in \mathbb{Z}$, and that every almost linear continuous Pexider mappings $f$, $g$, $h$ from a unital $C^*$-algebra $\mathcal A$ of real rank zero into a unital $C^*$-algebra $\mathcal B$ are homomorphisms when $f(2^n uy)=f(2^n u)f(y)$, $g(2^n uy)=g(2^n u)g(y)$ and $h(2^n uy)=h(2^n u)h(y)$ hold for all $u \in \lbrace v\in \mathcal A\mid v=v^*\hspace{5.0pt}\text{and}\hspace{5.0pt}v\hspace{5.0pt}\text{is} \text{invertible}\rbrace $, all $y\in \mathcal A$ and all $n\in \mathbb{Z}$. Furthermore, we prove the Cauchy-Rassias stability of $*$-homomorphisms between unital $C^*$-algebras, and $\mathbb{C}$-linear $*$-derivations on unital $C^*$-algebras. (English) |
| Keyword:
|
$C^*$-algebra homomorphism |
| Keyword:
|
$C^*$-algebra |
| Keyword:
|
real rank zero |
| Keyword:
|
$\mathbb{C}$-linear $*$-derivation |
| Keyword:
|
stability |
| MSC:
|
39B52 |
| MSC:
|
39B82 |
| MSC:
|
46L05 |
| MSC:
|
47B48 |
| idZBL:
|
Zbl 1081.39025 |
| idMR:
|
MR2184383 |
| . |
| Date available:
|
2009-09-24T11:30:14Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128044 |
| . |
| Reference:
|
[1] L. Brown and G. Pedersen: $C^*$-algebras of real rank zero.J. Funct. Anal. 99 (1991), 131–149. MR 1120918, 10.1016/0022-1236(91)90056-B |
| Reference:
|
[2] P. Găvruta: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings.J. Math. Anal. Appl. 184 (1994), 431–436. MR 1281518, 10.1006/jmaa.1994.1211 |
| Reference:
|
[3] B. E. Johnson: Approximately multiplicative maps between Banach algebras.J. London Math. Soc. 37 (1988), 294–316. Zbl 0652.46031, MR 0928525, 10.1112/jlms/s2-37.2.294 |
| Reference:
|
[4] K. Jun, B. Kim and D. Shin: On Hyers-Ulam-Rassias stability of the Pexider equation.J. Math. Anal. Appl. 239 (1999), 20–29. MR 1719096 |
| Reference:
|
[5] R. V. Kadison and G. Pedersen: Means and convex combinations of unitary operators.Math. Scand. 57 (1985), 249–266. MR 0832356, 10.7146/math.scand.a-12116 |
| Reference:
|
[6] R. V. Kadison and J. R. Ringrose: Fundamentals of the Theory of Operator Algebras. Elementary Theory.Academic Press, New York, 1994. MR 0719020 |
| Reference:
|
[7] C. Park and W. Park: On the Jensen’s equation in Banach modules.Taiwanese J. Math. 6 (2002), 523–531. MR 1937477, 10.11650/twjm/1500407476 |
| Reference:
|
[8] Th. M. Rassias: On the stability of the linear mapping in Banach spaces.Proc. Amer. Math. Soc. 72 (1978), 297–300. Zbl 0398.47040, MR 0507327, 10.1090/S0002-9939-1978-0507327-1 |
| . |
