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1.
In this paper a Cohen factorization theorem x = at · xt (t > 0) is proved for a Banach algebra A with a bounded approximate identity, where t ? at is a continuous one-parameter semigroup in A. This theorem is used to show that a separable Banach algebra B has a bounded approximate identity bounded by 1 if and only if there is a homomorphism θ from L1(R+) into B such that ∥ θ ∥ = 1 and θ(L1(R+)). B = B = B · θ(L1(R+)). Another corollary is that a separable Banach algebra with bounded approximate identity has a commutative bounded approximate identity, which is bounded by 1 in an equivalent algebra norm.  相似文献   

2.
Let A be a von Neumann algebra, let σ be a strongly continuous representation of the locally compact abelian group G as 1-automorphisms of A. Let M(σ) be the Banach algebra of bounded linear operators on A generated by ∝ σt(t) (μ?M(G)). Then it is shown that M(σ) is semisimple whenever either (i) A has a σ-invariant faithful, normal, semifinite, weight (ii) σ is an inner representation or (iii) G is discrete and each σt is inner. It is shown that the Banach algebra L(σ) generated by ∝ ?(t)σt dt (? ? L1(G)) is semisimple if a is an integrable representation. Furthermore, if σ is an inner representation with compact spectrum, it is shown that L(σ) is embedded in a commutative, semisimple, regular Banach algebra with isometric involution that is generated by projections. This algebra is contained in the ultraweakly continuous linear operators on A. Also the spectral subspaces of σ are given in terms of projections.  相似文献   

3.
M-ideals in a commutative Banach algebra A are shown to correspond to certain hermitian central projections in A7, and thus possess bounded approximate identities. This leads to a new characterization of M-ideals in function algebras.  相似文献   

4.
Our main result is an extension of a theorem due to Novodvorskii and Taylor; we give some special cases. Let A be a commutative Banach algebra with identity, and let Δ be its maximal ideal space. Let B be a Banach algebra with identity; let B?1 denote the invertible group in B and id B denote the set of idempotents in B. Let [(A \?bo B)?1] denote the set of path components of (A \?bo B)?1, and [Δ, B?1] denote the set of homotopy classes of continuous maps of Δ into B?1. We prove that the Gelfand transform on A induces a bijection of [(A \?bo B)?1] onto [Δ, B?1], and extend this result to prove a theorem of Davie. We show that the Gelfand transform induces a bijection of [id(A \?bo B)] onto [Δ, id B], and investigate consequences of this result for specific examples of the Banach algebra B.  相似文献   

5.
Let A be a Jordan algebra over the reals which is a Banach space with respect to a norm satisfying the requirements: (i) ∥ a ° b ∥ ≤ ∥ a ∥ ∥ b ∥, (ii) ∥ a2 ∥ = ∥ a2, (iii) ∥ a2 ∥ ≤ ∥ a2 + b2 ∥ for a, b?A. It is shown that A possesses a unique norm closed Jordan ideal J such that AJ has a faithful representation as a Jordan algebra of self-adjoint operators on a complex Hilbert space, while every “irreducible” representation of A not annihilating J is onto the exceptional Jordan algebra M38.  相似文献   

6.
The Schur product of two n×n complex matrices A=(aij), B=(bij) is defined by A°B=(aijbij. By a result of Schur [2], the algebra of n×n matrices with Schur product and the usual addition is a commutative Banach algebra under the operator norm (the norm of the operator defined on Cn by the matrix). For a fixed matrix A, the norm of the operator B?A°B on this Banach algebra is called the Schur multiplier norm of A, and is denoted by ∥Am. It is proved here that ∥A∥=∥U1AU∥m for all unitary U (where ∥·∥ denotes the operator norm) iff A is a scalar multiple of a unitary matrix; and that ∥Am=∥A∥ iff there exist two permutations P, Q, a p×p (1?p?n) unitary U, an (n?p)×(n?p)1 contraction C, and a nonnegative number λ such that
A=λPU00CQ;
and this is so iff ∥A°A?∥=∥A∥2, where ā is the matrix obtained by taking entrywise conjugates of A.  相似文献   

7.
Let A be a subalgebra of the full matrix algebra Mn(F), and suppose JA, where J is the Jordan block corresponding to xn. Let S be a set of generators of A. It is shown that the graph of S determines whether A is the full matrix algebra Mn(F).  相似文献   

8.
A Lie coalgebra is a coalgebra whose comultiplication Δ : MM ? M satisfies the Lie conditions. Just as any algebra A whose multiplication ? : A ? AA is associative gives rise to an associated Lie algebra L(A), so any coalgebra C whose comultiplication Δ : CC ? C is associative gives rise to an associated Lie coalgebra Lc(C). The assignment C ? Lc(C) is functorial. A universal coenveloping coalgebra Uc(M) is defined for any Lie Lie coalgebra M by asking for a right adjoint Uc to Lc. This is analogous to defining a universal enveloping algebra U(L) for any Lie algebra L by asking for a left adjoint U to the functor L. In the case of Lie algebras, the unit (i.e., front adjunction) 1 → L o U of the adjoint functor pair U ? L is always injective. This follows from the Poincaré-Birkhoff-Witt theorem, and is equivalent to it in characteristic zero (x = 0). It is, therefore, natural to inquire about the counit (i.e., back adjunction) Lc o Uc → 1 of the adjoint functor pair Lc ? Uc.Theorem. For any Lie coalgebra M, the natural mapLc(UcM) → M is surjective if and only if M is locally finite, (i.e., each element of M lies in a finite dimensional sub Lie coalgebra of M).An example is given of a non locally finite Lie coalgebra. The existence of such an example is surprising since any coalgebra C whose diagonal Δ is associative is necessarily locally finite by a result of that theory. The present paper concludes with a development of an analog of the Poincaré-Birkhoff-Witt theorem for Lie algebras which we choose to call the Dual Poincaré-Birkhoff-Witt Theorem and abbreviate by “The Dual PBWθ.” The constraints of the present paper, however, allow only a sketch of this theorem. A complete proof will appear in a subsequent paper. The reader may, however, consult [12], in the meantime, for details. The Dual PBWθ shows for any locally finite Lie coalgebra M the existence (in χ = 0) of a natural isomorphism of the graded Hopf algebras 0E(UcM) and 0E(ScM) associated to UcM and to ScM = Uc(TrivM) when Uc(M) and Sc(M) are given the Lie filtrations. [Just as Uc(M) is the analog of the enveloping algebra U(L) of a Lie algebra L, so Sc(V) is the analog of the symmetric algebra S(V) on a vector space V. Triv(M) denotes the trivial Lie coalgebra structure on the underlying vector space of M obtained by taking the comultiplication to be the zero map.]  相似文献   

9.
Let H(Δ) denote the Banach algebra of bounded analytic functions on the open unit disc, let M denote its maximal ideal space, and let ? denote its Shilov boundary. D. J. Newman has shown that a homomorphism ? in M will be in ? if and only if ? is unimodular on all Blaschke products. We answer a question of K. Hoffman by showing that ? will be in ? if and only if ? is unimodular on every Blaschke product whose zero set is an interpolating sequence. Our method is based on a construction due to L. Carleson, originally developed for the proof of the Corona theorem.  相似文献   

10.
The Ostrowski-Reich theorem states that for a system Ax =b of linear equations with A nonsingular, if A is hermitian and if the diagonal of A is positive, then the SOR method converges for each relaxation parameter in (0,2) if and only if A is positive definite. This is actually a special case of the Householder-John theorem, which states that for A=M?N with A,M nonsingular, if A is hermitian and M1+N is positive definite, then M?1N is a convergent matrix if and only if A is positive definite. Our purposes here are to generalize the Householder-John theorem and to provide an insight into how and why the SOR method can converge. As a result the Ostrowski-Reich theorem is extended in two directions; one is when A is hermitian but the diagonal of A is not necessarily positive, so that A is not necessarily positive definite, and the other is when A+A1 is positive definite but A is not necessarily hermitian. In the process, several other convergence results are obtained for general splittings of A. However, no claims are made concerning the case in which the convergence results obtained here can be applied to practical situations.  相似文献   

11.
Let A be a C1-algebra and X a Banach A-module. The module action of A on X gives rise to module actions of A7 on X1 and X7, and derivations of A into X (resp. X1) extend to derivations of A7 into X7 (resp. X1). If A is nuclear, and X is a dual Banach A-module with X1 weakly sequentially complete, then every derivation of A into X is inner. Under the same hypothesis on A, the extension to the finite part of A7 of any derivation of A into any dual Banach A-module is inner, as are all derivations of A into A1. Every derivation of a semifinite von Neumann algebra into its predual is inner.  相似文献   

12.
Let X be a Banach space, C a bounded closed subset of X, A a convex closed subset of X, E a complete metric space formed by all α-nonexpansive mappings fCA and M a complete metric space formed by α-nonexpansive differentiable mappings fCX. The following assertions are proved in this paper: (1) Properness of I ? f is a generic property in E (2)the subset of E formed by all α-contractive mappings is of Baire first category in E; and (3) for every y?X, the functional equation x ? f(x) = y has generically a finite number of solutions for f in M. Some applications to the fixed point theory and calculation of the topological degree are given.  相似文献   

13.
If A is a differential module, then the computation of its homology may frequently be simplified by finding a large acyclic submodule N, for then H(A)?H(AN) as modules, and hopefully AN is more tractable than A. The same idea works if A is a differential algebra, but in that case it is critical to factor out by an acyclic idealI?A, so that H(A)?H(AI) as algebras. This reduction technique in the classical (ungraded) case is used by Rees [7] and Tate [11], for example. I used a graded version in my thesis [8,9] to study the cohomology of two-stage Postnikov systems. Recently this Acyclic Ideal theorem has been used by Mann, May, Milgram and Sigaard [6] and there has also developed a body of work on the Koszul complex by Józefiak [2,3] and others in which this theorem fits naturally.  相似文献   

14.
Let A be a uniform algebra on a compact space X, let M be the maximal ideal space of A, and consider an element ? of A. Choose a component W of C??(X). In 1963 Bishop showed that {y in M ¦ ?(y) ? W} can be made into a one-dimensional complex analytic space provided there is a subset G of W having positive area such that for each λ in G {y in M ¦ ?(y) = λ} is finite. We show that the hypothesis of “positive area” may be replaced by “positive exterior capacity” and that no weaker condition will suffice.  相似文献   

15.
A formula for the resolvent R(λ, T) of a Baxter operator T, on a complex Banach algebra A with identity e, is obtained. With the parameter θ ≠ 0 and e, but under some restriction, this formula is analogous to that for the resolvent of an averaging operator. A counterexample is given, which shows that such a Baxter operator is not averaging in general. When θ is regular in A, a simple representation of T in terms of summation and averaging operators is obtained.  相似文献   

16.
Let L = d2dx2 ? q(x) be a Sturm-Liouville operator acting on functions defined on R. The authors have recently shown how to construct commutative associative algebras of distributions of compact support for which L is a centralizer (in the sense that L(f 1 g) = (Lf) 1 g for distributions f, g of compact support) when q is locally bounded. Here, it is assumed either that q is bounded and x → (1 + ¦ x ¦) q(x) is integrable, or that q is of bounded variation. A function ψ is then found such that Mψ={μ : μ is a measure on R and | μ |(ψ) < & infin;} becomes a Banach algebra containing the algebra of measures of compact support. The representation theory of Mψ is discussed and conditions for its semisimplicity are obtained.  相似文献   

17.
One says a commutative radical Banach algebra A has a lower bound if there is a lower growth condition on ∥xn1n for all nonzero elements x in A. If A is a separable algebra we give necessary and sufficient conditions for A to possess a lower bound.  相似文献   

18.
Let N be an observable Cox process on a locally compact space E directed by an unobservable random measure M. Techniques are presented for estimation of M, using the observations of N to calculate conditional expectations of the form E [M]|FA], where FA is the σ–algebra generated by the restriction of N to A. We introduce a random measure whose distribution depends on NA, from which we obtain both exact estimates and a recursive method for updating them as further observations become available. Application is made to the specific cases of estimation of an unknown, random scalar multiplier of a known measure, of a symmetrically distributed directing measure M and of a Markov–directed Cox process on R. By means of a Poisson cluster representation, the results are extended to treat the situation where N is conditionally additive and infinitely divisible given M.  相似文献   

19.
It has been conjectured that if A is a doubly stochastic nn matrix such that per A(i, j)≥perA for all i, j, then either A = Jn, then n × n matrix with each entry equal to 1n, or, up to permutations of rows and columns, A = 12(In + Pn), where Pn is a full cycle permutation matrix. This conjecture is proved.  相似文献   

20.
Let A be the Clifford algebra constructed over a quadratic n-dimensional real vector space with orthogonal basis {e1,…, en}, and e0 be the identity of A. Furthermore, let Mk(Ω;A) be the set of A-valued functions defined in an open subset Ω of Rm+1 (1 ? m ? n) which satisfy Dkf = 0 in Ω, where D is the generalized Cauchy-Riemann operator D = ∑i = 0m ei(??xi) and k? N. The aim of this paper is to characterize the dual and bidual of Mk(Ω;A). It is proved that, if Mk(Ω;A) is provided with the topology of uniform compact convergence, then its strong dual is topologically isomorphic to an inductive limit space of Fréchet modules, which in its turn admits Mk(Ω;A) as its dual. In this way, classical results about the spaces of holomorphic functions and analytic functionals are generalized.  相似文献   

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