Matrix C*-algebra - Imported Knowlpedia Content

Definition

Let $A$ be a $C^*$ -algebra and let $n\geq1$ . The matrix $C^*$ -algebra $M_n(A)$ consists of $n\times n$ matrices with entries in $A$ , with matrix addition and multiplication and involution

(a_{ij})^*=(a_{ji}^*).

Its canonical norm can be obtained by representing $A$ faithfully on a Hilbert space $H$ and taking the operator norm of the induced action on $H^n$ ; this norm is independent of the faithful representation. With this norm, $M_n(A)$ is a $C^*$ -algebra. For $A=\mathbb C$ , it is the full matrix algebra $M_n(\mathbb C)$ .

Finite-dimensional structure

The algebra $M_n(\mathbb C)$ is unital, simple, and finite-dimensional. Its involution is conjugate transpose, its positive elements are the positive semidefinite matrices, and its norm is the largest singular value. Every finite-dimensional $C^*$ -algebra is isomorphic to a finite direct sum of such full matrix algebras, although that classification is a theorem rather than part of the definition Murphy, §2.1.

Matrices over a general algebra

The standard Hilbert $A$ -module $A^n$ identifies $M_n(A)$ with its algebra of compact module operators. When $A$ is unital, these are all adjointable endomorphisms of $A^n$ ; for nonunital $A$ , the adjointable algebra is generally larger. If $A$ is unital, the unit of $M_n(A)$ is $\operatorname{diag}(1_A,\ldots,1_A)$ ; if $A$ is nonunital, then $M_n(A)$ is nonunital. Matrix formation also preserves ideals and quotients: for a closed two-sided ideal $I$ , the kernel of $M_n(A)\to M_n(A/I)$ is $M_n(I)$ .

Amplification and convention

A linear map $\phi:A\to B$ has an entrywise amplification $\phi_n:M_n(A)\to M_n(B)$ . Positivity of all these amplifications defines complete positivity, so the canonical matrix norms carry information not visible at level $n=1$ . The phrase “matrix algebra” sometimes means only $M_n(\mathbb C)$ ; writing $M_n(A)$ is essential when the coefficient algebra is not the scalars.

References

Gerard J. Murphy, $C^*$ -Algebras and Operator Theory, Academic Press, 1990. DOI record. Relevant: §§2.1–2.2 on matrix examples and finite-dimensional $C^*$ -algebras.