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Can anyone give me a relatively simple proof or Some reference for the following fact.(I know that there is a proof of this theorem in Gerard J. Murphy'book: "$C^*$-Algebras and Operator Theory", but I'm sure that there should be a simple proof of this.

Every hereditary $C^$-subalgebra C*-subalgebra of a simple $C^$-algebra C^*$-algebra is also simple! Maybe this is easy for someone, but it makes me confused for a long time. I am a novice! 3 added 9 characters in body; edited title # Question about hereditary C*-algebra.$C^*$-algebra. Can anyone give me a relatively simple proof or Some reference for the following fact.(I know that there is a proof of this theorem in Gerard J. Murphy'book: "C*-Algebras$C^*$-Algebras and Operator Theory", but I'm sure that there should be a simple proof of this. Every hereditary C*-subalgebra$C^$-subalgebra of a simple C*-algebra$C^\$-algebra is also simple!

Maybe this is easy for someone, but it makes me confused for a long time. I am a novice!

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