Let $A$ be a $C^*$ algebra. A $C^*$ subalgebra $C\subset A$ is said to be $C^*$ algebraic complemented of $A$ if there exist a $C^*$ subalgebra $D\subset A$ with $A=C\oplus D$ and the obvios mapping $A\mapsto C\oplus D$ preserves all algebraic structure.

In this question we consider the particular case $A=\ell^\infty$. Inspired by the result of Joram Lindenstrauss On complemented subspace of m, Israel Journal of Mathematics, volume 5, pages153{156, 1967. which says that the only complemented subspace of $\ell^\infty$ are those Banach subspace which are isomorphism to $\ell^\infty$ we ask the following $C^*$ algebraic question:

Is it true to say that an infinite dimensional $C^*$ subalgebra of $\ell^\infty$ is $C^*$ algebraic complemented if and only if it is $C^*$ isomorphic to $\ell^\infty$?

Note: This would suggest characterization of all $C^*$ algebras whose only infinite dimensional $C^*$ algebraic complemented subalgebras are isomorphic to itself.

Let $A$ be a $C^*$ algebra. A $C^*$ subalgebra $C\subset A$ is said to be $C^*$ algebraic complemented of $A$ if there exist a $C^*$ subalgebra $D\subset A$ with $A=C\oplus D$ and the obvios mapping $A\mapsto C\oplus D$ preserves all algebraic structure.

In this question we consider the particular case $A=\ell^\infty$. Inspired by the result of Joram Lindenstrauss On complemented subspace of m, Israel Journal of Mathematics, volume 5, pages153{156, 1967. which says that the only infinit dimensional complemented subspace of $\ell^\infty$ are those Banach subspace which are isomorphism to $\ell^\infty$ we ask the following $C^*$ algebraic question:

Is it true to say that an infinite dimensional $C^*$ subalgebra of $\ell^\infty$ is $C^*$ algebraic complemented if and only if it is $C^*$ isomorphic to $\ell^\infty$?

Note: This would suggest characterization of all $C^*$ algebras whose only infinite dimensional $C^*$ algebraic complemented subalgebras are isomorphic to itself.

Let $A$ be a $C^*$ algebra. A $C^*$ subalgebra $C\subset A$ is said to be $C^*$ algebraic complemented of $A$ if there exist a $C^*$ subalgebra $D\subset A$ with $A=C\oplus D$ and the obvios mapping $A\mapsto C\oplus D$ preserves all algebraic structure.

In this question we consider the particular case $A=\ell^\infty$. Inspired by the result of Joram Lindenstrauss On complemented subspace of m, Israel Journal of Mathematics, volume 5, pages153{156, 1967. which says that the only complemented subspace of $\ell^\infty$ are those Banach subspace which are isomorphism to $\ell^\infty$ we ask the following $C^*$ algebraic question:

Is it true to say that a $C^*$ subalgebra of $\ell^\infty$ is $C^*$ algebraic complemented if and only if it is $C^*$ isomorphic to $\ell^\infty$?

Let $A$ be a $C^*$ algebra. A $C^*$ subalgebra $C\subset A$ is said to be $C^*$ algebraic complemented of $A$ if there exist a $C^*$ subalgebra $D\subset A$ with $A=C\oplus D$ and the obvios mapping $A\mapsto C\oplus D$ preserves all algebraic structure.

In this question we consider the particular case $A=\ell^\infty$. Inspired by the result of Joram Lindenstrauss On complemented subspace of m, Israel Journal of Mathematics, volume 5, pages153{156, 1967. which says that the only complemented subspace of $\ell^\infty$ are those Banach subspace which are isomorphism to $\ell^\infty$ we ask the following $C^*$ algebraic question:

Is it true to say that an infinite dimensional $C^*$ subalgebra of $\ell^\infty$ is $C^*$ algebraic complemented if and only if it is $C^*$ isomorphic to $\ell^\infty$?

Note: This would suggest characterization of all $C^*$ algebras whose only infinite dimensional $C^*$ algebraic complemented subalgebras are isomorphic to itself.