Equivalent definitions of the multiplier algebra For a $C^*$-algebra $A$, G.J. Murphy (in his book page 38) Defines the multiplier algebra $M(A)$ of $A$ to be the set of its double centralizers.
In this book, they define $M(A)$ as the collection of all mapping $T:A \rightarrow A$ such that for any $x ,y \in A$, $x(Ty)=(Tx)y$.(Definition 1.4.10)
Another definition is This by the universal property.
Are there other definitions that are equivalent to the above or somehow related?
 A: There are many equivalent definitions of the multiplier algebra $M(A)$. I'll list some, but I'll only describe the underlying sets.
$(1)$ The set of double centralizers $(L,R)$ which consist of maps $A\to A$ satisfying $R(a)b = aL(b)$ for all $a,b \in A$.
$(2)$ (Essentially the same as $(1)$) The set of maps $L: A \to A$ for which there exists a map $R: A \to A$ such that $R(a)b = aL(b)$ for all $a,b \in A$.
$(3)$ View the $C^*$-algebra $A$ as a right Hilbert module over itself in the obvious way. Then we can define $M(A) = \mathcal{L}_A(A)$, the $A$-adointable operators.
$(4)$ The multiplier $C^*$-algebra as the solution to a universal problem: $M(A)$ is the largest $C^*$-algebra that contains the $C^*$-algebra $A$ as an essential two-sided ideal, in the sense that if $j: A \to C$ is a $*$-morphism with $j(A)$ an essential ideal in $C$, there exists a unique $*$-morphism $M(A) \to C$ extending the map $j:A \to C$.
$(5)$ If $A \subseteq B(H)$ is a faithful non-degenerate $*$-representation (we can also represent on a Hilbert module instead of a Hilbert space), we can define
$$M(A) = \{x \in B(H): x A \subseteq A \text{ and } Ax \subseteq A\}\subseteq B(H).$$
All these interpretations are canonically $*$-isomorphic. The way to see this is to show that each one of them satisfies the universal property satisfied in $(4)$ (though it is immediately clear that the interpretations $(1), (2), (3)$ give the same $C^*$-algebra). A good reference to see the equivalence $(3) \iff (4) \iff (5)$ (which is basically the interesting part) is Lance's book "Hilbert $C^*$-modules" chapter 2.
