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''Arens products''

A pair of intrinsically defined products on the double dual space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a1106601.png" /> of any [[Normed algebra|normed algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a1106602.png" />, each making <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a1106603.png" /> into a [[Banach algebra|Banach algebra]]. In 1951, [[#References|[a1]]] and [[#References|[a2]]], R. Arens defined these products using an essentially categorical framework.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a1106604.png" /> be a normed algebra with dual [[Banach space|Banach space]] (i.e., set of continuous linear functionals on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a1106605.png" />) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a1106606.png" /> and double dual space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a1106607.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a1106608.png" /> be the canonical isometric linear injection given by evaluation: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a1106609.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066011.png" />.

The definition of the Arens products is in three steps. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066013.png" />, define elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066015.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066016.png" /> by

<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/a/a110/a110660/a11066017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066019.png" />, define elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066021.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066022.png" /> by

<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/a/a110/a110660/a11066023.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

Finally, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066024.png" />, define elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066026.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066027.png" /> by

<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/a/a110/a110660/a11066028.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>

The two products <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066030.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066031.png" /> are often called the first and second Arens product, respectively. However, there is perfect symmetry between them. Each product makes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066032.png" /> into a Banach algebra, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066033.png" /> is an injective [[Homomorphism|homomorphism]] from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066034.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066035.png" /> with respect to either Arens product.

The subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066036.png" /> is always a spectral subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066037.png" /> with respect to either Arens product (cf. also [[Spectral decomposition of a linear operator|Spectral decomposition of a linear operator]]). This means that the spectrum of an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066038.png" /> is the same whether calculated in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066039.png" /> or in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066040.png" />. The two products agree whenever one of the factors is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066041.png" />. When the two products coincide on all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066042.png" />, the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066043.png" /> is said to be Arens regular (cf. also [[Arens regularity|Arens regularity]]). Subalgebras and quotient algebras of Arens-regular algebras are Arens regular.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066044.png" /> is a continuous (anti-) homomorphism, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066045.png" /> is a continuous (anti-) homomorphism with respect to either Arens product (respectively, the opposite Arens products) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066047.png" />. This shows that the involution on a Banach <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066048.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066049.png" /> extends naturally to an involution on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066050.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066051.png" /> is Arens regular.

===Examples of Arens products.===


1) The classical Banach algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066052.png" /> of complex sequences converging to zero with pointwise multiplication has dual and double dual naturally isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066054.png" />, respectively. The Arens product on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066055.png" /> corresponds to the usual pointwise multiplication in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066056.png" />.

2–3) The Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066057.png" /> is a commutative Banach algebra under either pointwise or convolution multiplication. It is Arens regular under the first but not the second. For pointwise multiplication one wishes to represent the double dual of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066058.png" /> as the Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066059.png" /> of all complex, bounded, finitely-additive set functions, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066060.png" /> is the subset of singular measures (i.e. those which vanish on all finite subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066061.png" />). The duality is implemented by simply integrating the sequence in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066062.png" /> by the set function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066063.png" />. With pointwise multiplication, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066064.png" /> is Arens regular and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066065.png" /> is an isomorphism onto its range. Any product in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066066.png" /> with one factor from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066067.png" /> is zero, so <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066068.png" /> is the Gel'fand (i.e., Jacobson) radical (cf. [[Jacobson radical|Jacobson radical]]) of the commutative algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066069.png" />.

If one regards <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066070.png" /> as the dual of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066071.png" /> of all bounded sequences, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066072.png" /> can be viewed as the commutative [[C*-algebra|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066073.png" />-algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066074.png" /> of all continuous complex-valued functions on the [[Stone–Čech compactification|Stone–Čech compactification]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066075.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066076.png" />. Hence, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066077.png" /> can be identified with the Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066078.png" /> of regular Borel measures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066079.png" />. With this interpretation, it is clear that the two Arens products do not agree. In essence, this construction extends to all discrete semi-group algebras.

4) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066080.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066081.png" />-algebra, then it is Arens regular and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066082.png" /> with its Arens product is the usual von Neumann enveloping algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066083.png" /> (cf. [[Von Neumann algebra|von Neumann algebra]]). Surprisingly, the Arens regularity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066084.png" />-algebras depends only on their Banach space structure and not at all on the nature of their product. A special case of Arens products on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066085.png" />-algebras was recognized very early and plays a significant role in von Neumann algebra theory. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066086.png" /> be a [[Hilbert space|Hilbert space]]. The trace on the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066087.png" /> of trace-class operators in the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066088.png" /> of all bounded linear operators establishes a natural isometric linear isomorphism of the Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066089.png" /> onto the dual Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066090.png" /> of the ideal of compact operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066091.png" />. It also defines a natural isometric linear isomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066092.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066093.png" />. The resulting isometric linear isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066094.png" /> is an algebra isomorphism with respect to both Arens products, which agree on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066095.png" />. Any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066096.png" /> satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066097.png" />.

Arens-regular algebras are rare. For a locally compact group (cf. also [[Compact group|Compact group]]; [[Locally compact skew-field|Locally compact skew-field]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066098.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a11066099.png" /> is Arens regular only when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a110660100.png" /> is finite. Even for Arens-regular algebras, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a110660101.png" /> is often intractable. Various natural quotients are often more useful. There is an intimate connection between the Arens products on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a110660102.png" /> and the double centralizer algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a110660103.png" />. An important technical property of the Arens product is the close connection between approximate identities in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a110660104.png" /> and one-sided or actual identity elements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a110660105.png" />. The case in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a110660106.png" /> is an ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a110660107.png" /> has been studied and characterized. The following theorem is an important special case: A semi-simple annihilator Banach algebra is an ideal in its double dual with respect to either Arens product.

The most comprehensive recent exposition is [[#References|[a3]]], which contains numerous further references.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Arens,   "Operations induced in function classes"  ''Monatsh. Math.'' , '''55'''  (1951)  pp. 1–19</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Arens,   "The adjoint of a bilinear operation"  ''Proc. Amer. Math. Soc.'' , '''2'''  (1951)  pp. 839–848</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  T.W. Palmer,   "Banach algebras and the general theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110660/a110660108.png" />-algebras I" , ''Encycl. Math. Appl.'' , '''49''' , Cambridge Univ. Press  (1994)</TD></TR></table>