Definition 3

A full subalgebra of A is called a unital subalgebra B such that every element of B which is invertible in A is invertible in B.

In other words, B is a full subalgebra of A if and only if Sp$_B(x) =$ Sp$_A(x)$ for every $x\in B$.

The intersection of a family of full subalgebras of A is a full subalgebra of A.

Let M be a subset of A. The intersection of the full subalgebras of A containing M is the smallest full subalgebra of A containing M; it is called the full subalgebra of A generated by M. The commutant $M'$ of M in A is a full subalgebra of A (for, if $x$ is invertible in A and commutes with M, then $x^{-1}$ commutes with M). Hence the bicommutant $M''$ of M is a full subalgebra of A which contains the full subalgebra of A generated by M.

If the elements of M are pairwise commuting, one has $M\subset M'$ and $M''\subset M'''$; the algebra $M''$ is therefore commutative and the same is then true of the full algebra generated by M.

A maximal commutative subalgebra of A is a full subalgebra, since it is equal to its commutant.