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Nik Weaver
Nik Weaver
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This is always true. The key point is that the projection map $a \mapsto a_m$ from the direct sum onto $A_m$ is a $*$-homomorphism, so its restriction to $C^*(G)$ is a $*$-homomorphism, and the image of any C-algebra under a $*$-homomorphism is a C${}^*$-algebra under a $*$-homomorphism is a C*-algebra.

This is always true. The key point is that the projection map $a \mapsto a_m$ from the direct sum onto $A_m$ is a $*$-homomorphism, so its restriction to $C^*(G)$ is a $*$-homomorphism, and the image of any C-algebra under a $*$-homomorphism is a C-algebra.

This is always true. The key point is that the projection map $a \mapsto a_m$ from the direct sum onto $A_m$ is a $*$-homomorphism, so its restriction to $C^*(G)$ is a $*$-homomorphism, and the image of any C${}^*$-algebra under a $*$-homomorphism is a C*-algebra.

Source Link
Nik Weaver
Nik Weaver
  • 42.8k
  • 3
  • 112
  • 213

This is always true. The key point is that the projection map $a \mapsto a_m$ from the direct sum onto $A_m$ is a $*$-homomorphism, so its restriction to $C^*(G)$ is a $*$-homomorphism, and the image of any C-algebra under a $*$-homomorphism is a C-algebra.