View source for Vector functions, algebra of

An arbitrary set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096460/v0964601.png" /> of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096460/v0964602.png" /> on a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096460/v0964603.png" /> assuming at each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096460/v0964604.png" /> values in some algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096460/v0964605.png" /> (usually dependent on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096460/v0964606.png" />), and forming an algebra with respect to the usual operations. If all algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096460/v0964607.png" /> are Banach algebras, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096460/v0964608.png" /> will be an algebra of vector functions if, for any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096460/v0964609.png" />, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096460/v09646010.png" /> is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096460/v09646011.png" />. The most important general problems in the theory of algebras of vector functions include the description of ideals in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096460/v09646012.png" /> in terms of ideals in the algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096460/v09646013.png" /> and the establishment of a criterion for a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096460/v09646014.png" /> to belong to the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096460/v09646015.png" />. A more frequently considered case is when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096460/v09646016.png" /> is a [[Banach algebra|Banach algebra]] with respect to the norm

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096460/v09646017.png" /></td> </tr></table>

while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096460/v09646018.png" /> is a locally compact or paracompact space. Of special interest is the algebra of vector functions connected with a set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096460/v09646019.png" />-algebras (cf. [[C*-algebra|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096460/v09646020.png" />-algebra]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096460/v09646021.png" />; in this particular case non-commutative analogues of the [[Stone–Weierstrass theorem|Stone–Weierstrass theorem]] and certain theorems on the realization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096460/v09646022.png" />-algebras (in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096460/v09646023.png" />-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|Schur lemma]]).



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.A. Naimark,   "Normed rings" , Reidel  (1984)  (Translated from Russian)</TD></TR></table>