Arens multiplication

Arens products

A pair of intrinsically defined products on the double dual space $ A ^ {* * } $ of any normed algebra $ A $, each making $ A ^ {* * } $ into a Banach algebra. In 1951, [a1] and [a2], R. Arens defined these products using an essentially categorical framework.

Let $ A $ be a normed algebra with dual Banach space (i.e., set of continuous linear functionals on $ A $) $ A ^ {*} $ and double dual space $ A ^ {* * } = ( A ^ {*} ) ^ {*} $. Let $ \kappa : A \rightarrow {A ^ {* * } } $ be the canonical isometric linear injection given by evaluation: $ \kappa ( a ) ( \omega ) = \omega ( a ) $ for all $ a \in A $, $ \omega \in A ^ {*} $.

The definition of the Arens products is in three steps. For any $ a \in A $ and $ \omega \in A ^ {*} $, define elements $ \omega _ {a} $ and $ _ {a} \omega $ of $ A ^ {*} $ by

$$ \tag{a1 } \omega _ {a} ( b ) = \omega ( ab ) , _ {a} \omega ( b ) = \omega ( ba ) , \quad \forall b \in A. $$

For any $ \omega \in A ^ {*} $ and $ f \in A ^ {* * } $, define elements $ _ {f} \omega $ and $ \omega _ {f} $ of $ A ^ {*} $ by

$$ \tag{a2 } _ {f} \omega ( a ) = f ( \omega _ {a} ) , \omega _ {f} ( a ) = f ( _ {a} \omega ) , \quad \forall a \in A. $$

Finally, for any $ f, g \in A ^ {* * } $, define elements $ fg $ and $ f \cdot g $ of $ A ^ {* * } $ by

$$ \tag{a3 } fg ( \omega ) = f ( _ {g} \omega ) , f \cdot g ( \omega ) = g ( \omega _ {f} ) , \quad \forall \omega \in A ^ {* * } . $$

The two products $ fg $ and $ f \cdot g $ in $ A ^ {* * } $ are often called the first and second Arens product, respectively. However, there is perfect symmetry between them. Each product makes $ A ^ {* * } $ into a Banach algebra, and $ \kappa $ is an injective homomorphism from $ A $ into $ A ^ {* * } $ with respect to either Arens product.

The subalgebra $ \kappa ( A ) $ is always a spectral subalgebra of $ A ^ {* * } $ with respect to either Arens product (cf. also Spectral decomposition of a linear operator). This means that the spectrum of an element $ \kappa ( a ) $ is the same whether calculated in $ \kappa ( A ) $ or in $ A ^ {* * } $. The two products agree whenever one of the factors is in $ \kappa ( A ) $. When the two products coincide on all of $ A ^ {* * } $, the algebra $ A $ is said to be Arens regular (cf. also Arens regularity). Subalgebras and quotient algebras of Arens-regular algebras are Arens regular.

If $ \varphi : A \rightarrow B $ is a continuous (anti-) homomorphism, then $ {\varphi ^ {* * } } : {A ^ {* * } } \rightarrow {B ^ {* * } } $ is a continuous (anti-) homomorphism with respect to either Arens product (respectively, the opposite Arens products) on $ A ^ {* * } $ and $ B ^ {* * } $. This shows that the involution on a Banach $ * $- algebra $ A $ extends naturally to an involution on $ A ^ {* * } $ if and only if $ A $ is Arens regular.

Examples of Arens products.

1) The classical Banach algebra $ c _ {0} $ of complex sequences converging to zero with pointwise multiplication has dual and double dual naturally isomorphic to $ {\mathcal l} ^ {1} $ and $ {\mathcal l} ^ \infty $, respectively. The Arens product on $ ( {\mathcal l} ^ {1} ) ^ {* * } $ corresponds to the usual pointwise multiplication in $ {\mathcal l} ^ \infty $.

2–3) The Banach space $ {\mathcal l} ^ {1} = {\mathcal l} ^ {1} ( \mathbf N ) $ 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 $ {\mathcal l} ^ {1} $ as the Banach space $ { \mathop{\rm ba} } ( \mathbf N ) \simeq \kappa ( {\mathcal l} ^ {1} ) \oplus { \mathop{\rm sba} } ( \mathbf N ) $ of all complex, bounded, finitely-additive set functions, where $ { \mathop{\rm sba} } ( \mathbf N ) $ is the subset of singular measures (i.e. those which vanish on all finite subsets of $ \mathbf N $). The duality is implemented by simply integrating the sequence in $ {\mathcal l} ^ \infty $ by the set function in $ { \mathop{\rm ba} } ( \mathbf N ) $. With pointwise multiplication, $ {\mathcal l} ^ {1} $ is Arens regular and $ \kappa $ is an isomorphism onto its range. Any product in $ ( {\mathcal l} ^ {1} ) ^ {* * } $ with one factor from $ { \mathop{\rm sba} } ( \mathbf N ) $ is zero, so $ { \mathop{\rm sba} } ( \mathbf N ) $ is the Gel'fand (i.e., Jacobson) radical (cf. Jacobson radical) of the commutative algebra $ ( {\mathcal l} ^ {1} ) ^ {* * } $.

If one regards $ {\mathcal l} ^ {1} $ as the dual of the space $ c $ of all bounded sequences, then $ {\mathcal l} ^ \infty $ can be viewed as the commutative $ C ^ {*} $- algebra $ C ( \beta \mathbf N ) $ of all continuous complex-valued functions on the Stone–Čech compactification $ \beta \mathbf N $ of $ \mathbf N $. Hence, $ ( {\mathcal l} ^ {1} ) ^ {* * } $ can be identified with the Banach space $ M ( \beta \mathbf N ) $ of regular Borel measures on $ \beta \mathbf N $. 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 $ A $ is a $ C ^ {*} $- algebra, then it is Arens regular and $ A ^ {* * } $ with its Arens product is the usual von Neumann enveloping algebra of $ A $( cf. von Neumann algebra). Surprisingly, the Arens regularity of $ C ^ {*} $- 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 $ C ^ {*} $- algebras was recognized very early and plays a significant role in von Neumann algebra theory. Let $ H $ be a Hilbert space. The trace on the ideal $ B _ {T} ( H ) \subseteq B ( H ) $ of trace-class operators in the algebra $ B ( H ) $ of all bounded linear operators establishes a natural isometric linear isomorphism of the Banach space $ B _ {T} ( H ) $ onto the dual Banach space $ B _ {K} ( H ) ^ {*} $ of the ideal of compact operators $ B _ {K} ( H ) $. It also defines a natural isometric linear isomorphism of $ B ( H ) $ onto $ B _ {T} ( H ) ^ {*} $. The resulting isometric linear isomorphism $ \Theta: B ( H ) \rightarrow B _ {K} ( H ) ^ {* * } $ is an algebra isomorphism with respect to both Arens products, which agree on $ B _ {K} ( H ) ^ {* * } $. Any $ K \in B _ {K} ( H ) $ satisfies $ \Theta ( K ) = \kappa ( K ) $.

Arens-regular algebras are rare. For a locally compact group (cf. also Compact group; Locally compact skew-field) $ G $, $ L _ {1} ( G ) $ is Arens regular only when $ G $ is finite. Even for Arens-regular algebras, $ A ^ {* * } $ is often intractable. Various natural quotients are often more useful. There is an intimate connection between the Arens products on $ A ^ {* * } $ and the double centralizer algebra of $ A $. An important technical property of the Arens product is the close connection between approximate identities in $ A $ and one-sided or actual identity elements in $ A ^ {* * } $. The case in which $ \kappa ( A ) $ is an ideal in $ A ^ {* * } $ 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 [a3], which contains numerous further references.

References

[a1]	R. Arens, "Operations induced in function classes" Monatsh. Math. , 55 (1951) pp. 1–19
[a2]	R. Arens, "The adjoint of a bilinear operation" Proc. Amer. Math. Soc. , 2 (1951) pp. 839–848
[a3]	T.W. Palmer, "Banach algebras and the general theory of $\star$-algebras I" , Encycl. Math. Appl. , 49 , Cambridge Univ. Press (1994)