Arens product

The Arens product , named after Richard Arens , is a construction based on the mathematical theory of Banach algebras . Strictly speaking, there are two products on the binary space of a Banach algebra , which continue the given product if one understands by virtue of the natural embedding of as a subspace of . Both products make a Banach algebra. If the two products match, the output algebra is called Arens-regular . ${\displaystyle A''}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle A\rightarrow A''}$ ${\displaystyle A''}$ ${\displaystyle A''}$ ${\displaystyle A}$

construction

First Arens product

Let it be a Banach algebra, its dual space and its dual space. As usual, by virtue of the isometric embedding ${\displaystyle A}$ ${\displaystyle A'}$ ${\displaystyle A''}$ ${\displaystyle A}$

${\displaystyle \Phi :A\rightarrow A'',\,\Phi (a):=\Phi _{a}:A'\rightarrow \mathbb {K} ,\,\Phi _{a}(f):=f(a)}$

understood as a subspace of . The construction of a product takes place in three steps: ${\displaystyle A''}$ ${\displaystyle A''}$

1. For and is defined by . ${\displaystyle x\in A}$ ${\displaystyle f\in A'}$ ${\displaystyle fx\in A'}$ ${\displaystyle (fx)(y):=f(xy),\,y\in A}$
2. For and is defined by . ${\displaystyle F\in A''}$ ${\displaystyle f\in A'}$ ${\displaystyle Ff\in A'}$ ${\displaystyle (Ff)(x):=F(fx),\,x\in A}$
3. For is defined by . ${\displaystyle F,G\in A''}$ ${\displaystyle FG\in A''}$ ${\displaystyle (FG)(f):=F(Gf),\,f\in A'}$

The marked link on to say the first Arens product. One can show that it is actually an associative multiplication, which turns into a Banach algebra. This multiplication is always used in the following . The formula , which is easy to recalculate, shows that this continues the product given on the initial algebra if, as mentioned above, it is understood as a subset of . ${\displaystyle (F,G)\mapsto FG}$ ${\displaystyle A''}$ ${\displaystyle A''}$ ${\displaystyle A''}$ ${\displaystyle \Phi (ab)=\Phi (a)\Phi (b)}$ ${\displaystyle A}$ ${\displaystyle A''}$

Second Arens product

The second Arens product results from the first by applying the above construction to the counter-algebra and then going back to the counter-algebra, i.e. H. one forms . This can also be described as a three-stage construction: ${\displaystyle A^{op}}$ ${\displaystyle ((A^{op})'')^{op}}$

1. For and is defined by . ${\displaystyle x\in A}$ ${\displaystyle f\in A'}$ ${\displaystyle xf\in A'}$ ${\displaystyle (xf)(y):=f(yx),\,y\in A}$
2. For and is defined by . ${\displaystyle F\in A''}$ ${\displaystyle f\in A'}$ ${\displaystyle fF\in A'}$ ${\displaystyle (fF)(x):=F(xf),\,x\in A}$
3. For is defined by . ${\displaystyle F,G\in A''}$ ${\displaystyle F\cdot G\in A''}$ ${\displaystyle (F\cdot G)(f):=F(fG),\,f\in A'}$

Again, this defines a multiplication that continues that of and turns it into a Banach algebra. ${\displaystyle A}$ ${\displaystyle A''}$

Arens regularity

While the first Arens product was written without linking signs, we have chosen a point for the second Arens product to distinguish it. Arens has already shown in his fundamental work that if one of the factors is off , that is , off. In general, the two Arens products do not match. This leads to the following definition: ${\displaystyle FG=F\cdot G}$ ${\displaystyle A}$ ${\displaystyle \Phi (A)\subset A''}$

A Banach algebra is called Arens-regular if the first and second Arens-Prooducts agree, that is, if for all . ${\displaystyle A''}$ ${\displaystyle FG=F\cdot G}$ ${\displaystyle F,G\in A''}$

A Banach algebra is Arens-regular if and only if the linear operator defined by is weakly compact for each . ${\displaystyle A}$ ${\displaystyle f\in A'}$ ${\displaystyle T_{f}(a):=fa}$ ${\displaystyle T_{f}:A\rightarrow A'}$

Examples

Group algebras

If is a locally compact group , then the group algebra is Arens-regular if and only if is finite. In particular, the convolutionalgebra is an example of a non-Arens-regular Banachalgebra. ${\displaystyle G}$ ${\displaystyle L^{1}(G)}$ ${\displaystyle G}$ ${\displaystyle \ell ^{1}(\mathbb {Z} )}$

C * algebras

S. Sherman and Z. Takeda have shown that C * -algebras are always Arens-regular, that the involution of the C * -algebra continues on the bidual and that this also becomes a C * -algebra, even a von -Neumann algebra . It can also be shown that this corresponds to the enveloping Von Neumann algebra .

properties

Approximation of the one

A Banach algebra has a bounded right approximation of one if and only if has a right one element. It follows: ${\displaystyle A}$ ${\displaystyle A''}$

An Arens-regular Banach algebra has a bounded approximation of one if and only if has a one element. ${\displaystyle A}$ ${\displaystyle A''}$

Commutativity

Commutativity is only passed on to the bidual in the case of Arens regularity. Is a commutative Banach algebra, then is commutative under one of the Arens products if and only if Arens is regular. ${\displaystyle A}$ ${\displaystyle A''}$ ${\displaystyle A}$

Inheritance properties

Let it be an Arens-regular Banach algebra, a closed subalgebra and a closed two-sided ideal . Then also , and Arens-regular. ${\displaystyle A}$ ${\displaystyle B\subset A}$ ${\displaystyle I\subset A}$ ${\displaystyle B}$ ${\displaystyle A/I}$

If a compact Hausdorff space and a Banach algebra, then the Banach algebra of the continuous functions with the pointwise explained connections is Arens-regular if and only if Arens-regular. The Arens regularity of the injective tensor product follows from the Arens regularity , because the latter agrees with . The projective tensor product of Arens-regular Banach algebras is in general not Arens-regular again. ${\displaystyle K}$ ${\displaystyle A}$ ${\displaystyle C(K,A)}$ ${\displaystyle K\rightarrow A}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle C(K)\otimes _{\varepsilon }A}$ ${\displaystyle C(K,A)}$

Individual evidence

1. ^ FF Bonsall , J. Duncan: Complete Normed Algebras . Springer-Verlag 1973, ISBN 3-540-06386-2 , § 10, Example 13 (v)
2. ^ R. Arens: The adjoint of a bilinear operation , Proceedings Amer. Math. Soc. Volume 2 (1951), pages 839-848
3. ↑ SLGulick: Commutativity and ideals in the biduals of topological algebras , Pacific J. Math. Volume 18 (1966), pages 121-137 (commutative case)
4. ↑ J. Hennefeld: A note on the Arens Products , Pacific J. Math. Volume 26 (1968), pages 115-119 (general case)
5. ^ NJ Young: The Irregularity of Multiplication in Group Algebras , Quart. J. Math. Oxford, Vol. 24 (1973), pp. 59-62
6. ^ FF Bonsall, J. Duncan: Complete Normed Algebras . Springer-Verlag 1973, ISBN 3-540-06386-2 , § 38, Theorem 19
7. ^ FF Bonsall, J. Duncan: Complete Normed Algebras . Springer-Verlag 1973, ISBN 3-540-06386-2 , § 29, sentence 7
8. ^ FF Bonsall, J. Duncan: Complete Normed Algebras . Springer-Verlag 1973, ISBN 3-540-06386-2 , § 29, Corollary 8
9. ^ J. Duncan, SAR Hosseiniun: The second dual of a Banach algebra , Proceedings Royal Soc. Edinburgh, Volume 84 (1979), pages 309-325, § 2, sentence 1
10. ^ J. Duncan, SAR Hosseiniun: The second dual of a Banach algebra , Proceedings Royal Soc. Edinburgh, Volume 84 (1979), pages 309-325, § 2, Corollary to Theorem 1
11. ↑ A. Ülger: Arens regularity of the algebra C (K, A) , Journal London Mathematical Society, Vol S2-42, Issue 2 (1989), pages 354-364
12. ↑ A. Ülger: Arens regularity of the algebra A⊗B , Trans. Amer. Math. Soc., 305, 623-639 (1988)