Theorem: C*-Algebra embeds into Multiplier Algebra as C*-Subalgebra

Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $\struct {\map M A, \ast, \norm {\, \cdot \,}_{\map M A} }$ be the multiplier algebra of $A$.
Define $\phi : A \to \map M A$ by:
:$\map \phi a = \tuple {L_a, R_a}$
for each $a \in A$, where $\tuple {L_a, R_a}$ is the double centralizer generated by $a$.
Then $\phi \sqbrk A$ is a $\text C^\ast$-subalgebra of $\map M A$.

Proof:
We show that $\phi$ is a linear isometry.
Let $a, b \in A$ and $\lambda \in \C$.
Then we have for $c \in A$:
{{begin-eqn}}
{{eqn	| l = \map {L_{a + \lambda b} } c
		| r = \paren {a + \lambda b} c
}}
{{eqn	| r = a c + \lambda b c
}}
{{eqn	| r = \map {L_a} c + \lambda \map {L_b} c
}}
{{end-eqn}}
and:
{{begin-eqn}}
{{eqn	| l = \map {R_{a + \lambda b} } c
		| r = c \paren {a + \lambda b}
}}
{{eqn	| r = c a + \lambda c b 
}}
{{eqn	| r = \map {R_a} c + \lambda \map {R_b} c
}}
{{end-eqn}}
Hence:
:$L_{a + \lambda b} = L_a + \lambda L_b$
and:
:$R_{a + \lambda b} = R_a + \lambda R_b$
We therefore have: 
{{begin-eqn}}
{{eqn	| l = \map \phi a
		| r = \tuple {L_{a + \lambda b}, R_{a + \lambda b} }
}}
{{eqn	| r = \tuple {L_a + \lambda L_b, R_a + \lambda R_b}
}}
{{eqn	| r = \tuple {L_a, R_a} + \lambda \tuple {L_b, R_b}
		| c = {{Defof|Multiplier Algebra}}
}}
{{eqn	| r = \map \phi a + \lambda \map \phi b
}}
{{end-eqn}}
so $\phi$ is a linear transformation.
We also have:
{{begin-eqn}}
{{eqn	| l = \norm {\map \phi a}_{\map M A}
		| r = \norm {\tuple {L_a, R_a} }_{\map M A}
}}
{{eqn	| r = \norm {L_a}_{\map M A}
		| c = {{Defof|Multiplier Algebra}}
}}
{{eqn	| r = \norm a
		| c = Norm of Double Centralizer Generated by Element of C*-Algebra
}}
{{end-eqn}}
hence $\phi$ is a linear isometry.
We now show that $\phi$ is an algebra homomorphism.
Let $a, b, c \in A$.
We have:
{{begin-eqn}}
{{eqn	| l = \map {L_{a b} } c
		| r = a b c 
}}
{{eqn	| r = \map {L_a} {b c}
}}
{{eqn	| r = \map {L_a} {\map {L_b} c}
}}
{{end-eqn}}
so:
:$L_{a b} = L_a L_b$
Similarly we have:
{{begin-eqn}}
{{eqn	| l = \map {R_{a b} } c
		| r = c a b 
}}
{{eqn	| r = \paren {\map {R_a} c} b
}}
{{eqn	| r = \map {R_b} {\map {R_a} c}
}}
{{end-eqn}}
so:
:$R_{a b} = R_b R_a$
Hence we have:
{{begin-eqn}}
{{eqn	| l = \map \phi {a b}
		| r = \tuple {L_{a b}, R_{a b} }
}}
{{eqn	| r = \tuple {L_a L_b, R_b R_a} 
}}
{{eqn	| r = \tuple {L_a, R_a} \tuple {L_b, R_b}
		| c = {{Defof|Multiplier Algebra}}
}}
{{end-eqn}}
Hence $\phi$ is an algebra homomorphism.
We now show that $\phi$ is a $\ast$-algebra homomorphism.
We have:
{{begin-eqn}}
{{eqn	| l = \map {L_a^\ast} b 
		| r = \paren {\map {L_a} {b^\ast} }^\ast 
}}
{{eqn	| r = \paren {a b^\ast}^\ast 
}}
{{eqn	| r = b a^\ast 
		| c = $(\text C^\ast 1)$, $(\text C^\ast 3)$ in {{Defof|Involution on Algebra}}
}}
{{eqn	| r = \map {R_{a^\ast} } b 
}}
{{end-eqn}}
and similarly:
{{begin-eqn}}
{{eqn	| l = \map {R_a^\ast} b
		| r = \paren {\map {R_a} {b^\ast} }^\ast 
}}
{{eqn	| r = \paren {b^\ast a}^\ast 
}}
{{eqn	| r = a^\ast b 
		| c = $(\text C^\ast 1)$, $(\text C^\ast 3)$ in {{Defof|Involution on Algebra}}
}}
{{eqn	| r = \map {L_{a^\ast} } b 
}}
{{end-eqn}}
So we have:
:$L_a^\ast = R_{a^\ast}$
and:
:$R_a^\ast = L_{a^\ast}$
We therefore have:
{{begin-eqn}}
{{eqn	| l = \paren {\map \phi a}^\ast
		| r = \tuple {L_a, R_a}^\ast 
}}
{{eqn	| r = \tuple {R_a^\ast, L_a^\ast}
		| c = {{Defof|Multiplier Algebra}}
}}
{{eqn	| r = \tuple {L_{a^\ast}, R_{a^\ast} }
}}
{{eqn	| r = \map \phi {a^\ast} 
}}
{{end-eqn}}
Now, from Image of *-Algebra under *-Algebra Homomorphism is *-Subalgebra, $\phi \sqbrk A$ is a $\ast$-subalgebra of $\map M A$. 
From Image of Closed Set under Linear Isometry from Banach Space to Normed Vector Space is Closed, $\phi \sqbrk A$ is closed in $\map M A$. 
Hence from Closed *-Subalgebra of C*-Algebra is C*-Algebra, $\phi \sqbrk A$ is a $\text C^\ast$-subalgebra of $\map M A$. 
{{qed}}
Category:Multiplier Algebras
Category:C*-Algebras
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