Vector functions, algebra of

An arbitrary set $A$ of functions $x=\{x(t)\}$ on a topological space $T$ assuming at each point $t\in T$ values in some algebra $A(t)$ (usually dependent on $t$), and forming an algebra with respect to the usual operations. If all algebras $A(t)$ are Banach algebras, $A$ will be an algebra of vector functions if, for any function $x=\{x(t)\}\in A$, the function $t\to\Vert x(t)\Vert$ is continuous on $T$. The most important general problems in the theory of algebras of vector functions include the description of ideals in $A$ in terms of ideals in the algebras $A(t)$ and the establishment of a criterion for a function $x=\{x(t)\}$ to belong to the algebra $A$. A more frequently considered case is when $A$ is a Banach algebra with respect to the norm

$$\Vert x\Vert=\sup_{t\in T}\Vert x(t)\Vert_A(t),$$

while $T$ is a locally compact or paracompact space. Of special interest is the algebra of vector functions connected with a set of $C^*$-algebras (cf. $C^*$-algebra) $A(t)$; in this particular case non-commutative analogues of the Stone–Weierstrass theorem and certain theorems on the realization of $C^*$-algebras (in particular, $C^*$-algebras with continuous trace) as algebras of vector functions are known. These theorems in turn permit one to prove, in some cases, the commutativity of all the operators which commute with the operators of symmetric representations of algebras with an involution (the continuous analogue of the Schur lemma).

References

[a1]	M.A. Naimark, "Normed rings" , Reidel (1984) (Translated from Russian)