Projective and flat ideals in function algebras and their relation to analytic structures

Pugach, L. I.

doi:10.1007/bf01158131

Mathematical Notes of the Academy of Sciences of the USSR

1982

DOI: 10.1007/bf01158131

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Projective and flat ideals in function algebras and their relation to analytic structures

L. I. Pugach

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“…We have ||/m|| < C||/w||, because | f\ < C on . See, for example, the standard argument in [5]. Since ||gm|| < A:||w|| we can conclude that \g\ < k on each annulus E,.…”

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confidence: 99%

On the C-Projectivity of ideals in Banach algebras

Pugach

1998

Glasgow Math. J.

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The notion of projective Banach module was defined by Helemskii in [1]-the paper which properly founded the homological theory of Banach algebras. The same author introduced the definition of the (relatively) flat Banach module in [2]. Recently M. C. White [3] modified both of those definitions, introducing so called C-projective and C-flat Banach modules.For a given constant C > 0 the Banach module Zover a Banach algebra A, (abbreviated below as "module"), is called C-projective gives us the more useful equivalent definition of C-projectivity. Namely, a module X is C-projective if and only if the morphism of external multiplication n : A ® X ->• X, defined by the formula n(a x) = ax, has a right inverse morphism p such that ||p|| < C. Here the symbol® denotes the projective tensor product of Banach spaces [4]. If ||p|| = C and there is no right inverse with a norm smaller than C, then it is natural to say that X is exactly C-projective. In this paper we give answers to two questions that (directly or not) were put in [3]. First, for arbitrary C > 1, we give an example of an exactly C-projective Banach ,4-module. (Moreover, it is a maximal ideal in a uniform algebra A.) Note that C-projectivity is impossible for C < 1 and for C = 1 there exist trivial examples: consider for example any maximal ideal in the disc-algebra, corresponding to an inner point of the disc. Second, we shall show that C-projectivity does not possess the same "continuity property" as C-flatness [3]: that is, there exists a module (again a maximal ideal in the uniform algebra) that is (C + c)-projective for all e > 0 but not C-projective.As usual, we denote by A(E) the uniform algebra of functions that are continuous on the given compact subset £ c C and analytic in its interior. EXAMPLE 1. Consider the compact subset A: = D U £ o / C x R , where D -{(z, 0): \z\ < 1} is the closed disc and E = {(z, i) : |C|~' < \z\ < 1, 0 < t < 1} is the cylindric annulus. (We denote by E, its section, where / is constant.) Consider the uniform algebra Proof. (1) Consider two functions: h e M such that h(z, t) = z for (z, t) € K and f{z, t) = l/z for (z, t)eK\O.Note that \\h\\ = 1 and, for each meM,fm is defined on K\O and we extend the definition to K by continuity. We have ||/m|| < C||/w||, because | f\ < C on f This work was partially supported by ISF (Soros foundation) grant M95300 and Russian RFFI grant 93-01-00156.

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“…We have ||/m|| < C||/w||, because | f\ < C on . See, for example, the standard argument in [5]. Since ||gm|| < A:||w|| we can conclude that \g\ < k on each annulus E,.…”

mentioning

confidence: 99%

On the C-Projectivity of ideals in Banach algebras

Pugach

1998

Glasgow Math. J.

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“…Introduction. This work can be considered as one in the series of papers by the ®rst author which study homological properties of non-idempotent maximal ideals in commutative Banach algebras ( [8], [9]). From another point of view it continues the long series of paper by many authors which studied conditions on the vanishing of cohomology groups of Banach algebras.…”

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confidence: 99%

Homology and cohomology groups of commutative Banach algebras and analytic polydiscs

Pugach

White

2000

Glasgow Math. J.

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Abstract. In this paper we deduce the existence of analytic structure in a neighbourhood of a maximal ideal M in the spectrum of a commutative Banach algebra, A, from homological assumptions. We assume properties of certain of the cohomology groups H n AY AaM, rather than the stronger conditions on the homological dimension of the maximal ideal the ®rst author has considered in previous papers. The conclusion is correspondingly weaker: in the previous work one deduces the existence of a Gel'fand neighbourhood with analytic structure, here we deduce only the existence of a metric neighbourhood with analytic structure. The main method is to consider products of certain co-cycles to deduce facts about the symmetric second cohomology, which is known to be related to the deformation theory of algebras.1991 Mathematics Subject Classi®cation. 46J20, 46M201. Introduction. This work can be considered as one in the series of papers by the ®rst author which study homological properties of non-idempotent maximal ideals in commutative Banach algebras ([8], [9]). From another point of view it continues the long series of paper by many authors which studied conditions on the vanishing of cohomology groups of Banach algebras.Throughout this paper A denotes a commutative Banach algebra with unit 1, M & A is a ®xed maximal ideal. We denote by w A the space of maximal ideals of A and consider two topologies on w A ± the Gelfand topology and norm (metric) topology, induced by the norm from A Ã . The letter 3 stands for the point in w A , that corresponds to M and for the multiplicative functional annihilating M.The symbol denotes the projective tensor product of Banach spaces [4], and % X M M 3 M is the operator of multiplication. We denote by M 2 the algebraic square of M (that is the linear space generated by ab, (aY b P M)). We also consider the topological square M 2 , which is the closure of M 2 . The ideal M is called idempotent if M M 2 and non-idempotent otherwise. Note that the quotient space MaM 2 is the predual of the space of bounded point derivations at the point 3.

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Projective modules in classical and quantum functional analysis

Helemskii

2009

J Math Sci

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Along with the classical version, there are two "quantized" versions of the theory of operator algebras. In these lectures, the fundamental homological notion of a projective module is described in the framework of these three theories. Our initial definitions of projectivity do not go far from their prototypes in abstract algebra; however, the principal results concern essentially functional-analytic objects and, as a rule, have no purely algebraic analogs.

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Projective and flat ideals in function algebras and their relation to analytic structures

Cited by 4 publications

References 4 publications

On the C-Projectivity of ideals in Banach algebras

On the C-Projectivity of ideals in Banach algebras

Homology and cohomology groups of commutative Banach algebras and analytic polydiscs

Projective modules in classical and quantum functional analysis

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