Concrete C*-algebra - Imported Knowlpedia Content

Definition

Let $\mathcal H$ be a complex Hilbert space. A concrete $C^*$ -algebra on $\mathcal H$ is a norm-closed subalgebra $A\subseteq B(\mathcal H)$ of the bounded operators that is closed under operator adjoints. With the inherited operator norm and involution, $A$ is a $C^*$ -algebra. The definition does not require $A$ to contain the identity operator unless “unital concrete $C^*$ -algebra” is explicitly stated.

Relation to abstract

C^*

-algebras

The Gelfand--Naimark representation theorem says that every abstract $C^*$ -algebra admits an isometric $*$ -isomorphism onto a concrete $C^*$ -algebra Murphy, Theorem 3.4.1. Thus the abstract axioms capture exactly the norm-closed adjoint-stable operator algebras, although a given abstract algebra can have many inequivalent realizations on Hilbert spaces. “Concrete” records a chosen faithful operator realization, not an additional algebraic axiom.

Degeneracy and units

The action of $A$ on $\mathcal H$ is nondegenerate when $\overline{A\mathcal H}=\mathcal H$ . A unital abstract algebra can be represented degenerately with its unit acting as a proper projection rather than as $I_{\mathcal H}$ ; a unital representation convention rules this out. Likewise, a concrete algebra may possess an identity that is a projection onto $\overline{A\mathcal H}$ even when it does not contain the ambient identity operator.

Examples and non-examples

The algebras $B(\mathcal H)$ and $K(\mathcal H)$ are concrete $C^*$ -algebras. Multiplication operators by functions in $C_0(X)$ give commutative concrete models on suitable $L^2$ -spaces. The upper-triangular matrices are norm closed but not adjoint closed, so they are an operator algebra but not a concrete $C^*$ -algebra.

References

Gerard J. Murphy, $C^*$ -Algebras and Operator Theory, Academic Press, 1990. DOI record. Relevant: §2.1 and Theorem 3.4.1 on concrete algebras and faithful representations.
Gert K. Pedersen, $C^*$ -Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. DOI record. Relevant: Chapters 1 and 3 on abstract $C^*$ -algebras and representations.