Über Banachverbandsalgebren mit multiplikativen Zerlegungseigenschaft. (Banach lattice algebras with multiplicative decomposition property)
Let A be a real Banach lattice algebra and denote its canonical complexification by \(A_{{\mathbb{C}}}\). Let \(L^ r(A,A')\) denote the order- complete Banach lattice of regular linear mappings (i.e. expressible as difference of positive linear mappings) of A into its topological dual and let M: A\({}'\to L^ r(A,A')\) be the linear operator defined by \[ (M\mu)(x)(z)=\mu (xz)\quad for\quad all\quad x,z\in A,\quad \mu \in A'. \] The same symbol M is used for its canonical complexification. A multiplicative decomposition property for \(A_{{\mathbb{C}}}\) was defined by the author in J. Funct. Anal. 37, 382-400 (1988; Zbl 0478.46017), and is proved to be equivalent to the property used by \textit{J. L. Taylor} [Trans. Am. Math. Soc. 119, 150-166 (1965; Zbl 0128.355)] in his definition of abstract convolution measure algebras (whose Banach structures are L-spaces) and also equivalent to M being a lattice homomorphism. The latter result is evident from the tensor-product formulation of his property by \textit{J. L. Taylor} [Trans. Am. Math. Soc. 135, 105-113 (1969; Zbl 0189.140)]. A Banach lattice algebra is called an L-algebra of its underlying set is an abstract L-space (which the author usually, but not always, calls an AL-space). The author does not bother to define AL-algebras, a species of L-algebras, which he used in the earlier article (loc. cit.). The absolute kernel N(\(\mu)\) of a positive linear form \(\mu\) on A is defined to be \(\{\) \(x\in A: \mu (| x|)=0\}\). For a positive non-trivial submultiplicative form A/N(\(\mu)\) is an L-algebra, denoted \(A^{\mu}\), and if \(\mu\) is multiplicative it is an AL-algebra. If A has the multiplicative decomposition property then also \(A^{\mu}\) does. The author is thus able to use the identification of the dual of an L-space of continuous functions on an extremely disconnected set K. The results of this paper generalize those of the author's previous article (loc. cit.) where he used AL-algebras, and are an attempt to construct a Gelfand spectral theory. The objects concerned are (i) the positive elements of M, the set of multiplicative linear forms (which the author sometimes calls complex homomorphisms) on \(A_{{\mathbb{C}}},\) (ii) what he calls central homomorphisms, i.e. those algebraic endomorphisms T of \(A_{{\mathbb{C}}}\) for which \(| Tz| <c| z|\) for \(z\in A_{{\mathbb{C}}}\), \(c>0\), the modulus as defined in \textit{H. H. Schaefer}'s book ``Banach lattices and positive operators'' (1974; Zbl 0296.47023), and (iii) maximal structural ideals, i.e. two-sided ring ideals of \(A_{{\mathbb{C}}}\) which are also vector lattice ideals. (Note that in his earlier article (loc. cit.) the author had stipulated that the elements of M be non-trivial.) The author shows that if \(\mu_ 1\) and \(\mu_ 2\) are positive multiplicative linear forms on A such that \(N(\mu_ 1)\neq N(\mu_ 2)\) then \(\mu_ 1\) and \(\mu_ 2\) are orthogonal; if A has unit element and the multiplicative decomposition property then the maximal structural ideals are absolute kernels of positive multiplicative linear forms. When \(A_{{\mathbb{C}}}\) has the multiplicative decomposition property and unit element, the author constructs moduli and conjugations of central homomorphisms T such that \(| T|\) and T are central homomorphisms and \(| T| xy=| T| x.| T| y\), \(Txy=Tx.Ty\).
order- complete Banach lattice of regular linear mappings
scientific article; zbMATH DE number 4131036