Displaying similar documents to “Elements of finite closed descent in Banach and Fréchet algebras”

Banach's school and topological algebras

Wiesław Żelazko (2009)

Banach Center Publications

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We present here some evidence of the activity of Banach Lwów School of functional analysis in the field of topological algebras. We shall list several results connected with such names as Stanisław Mazur (1905-1981), Maks (Meier) Eidelheit (1910-1943), Stefan Banach (1892-1945) and Andrzej Turowicz (1904-1989) showing that if the war had not interrupted this activity we could expect more interesting results in this direction.

Small deformations of topological algebras

Mati Abel, Krzysztof Jarosz (2003)

Studia Mathematica

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We investigate stability of various classes of topological algebras and individual algebras under small deformations of multiplication.

Ultragraph C*-algebras via topological quivers

Takeshi Katsura, Paul S. Muhly, Aidan Sims, Mark Tomforde (2008)

Studia Mathematica

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Given an ultragraph in the sense of Tomforde, we construct a topological quiver in the sense of Muhly and Tomforde in such a way that the universal C*-algebras associated to the two objects coincide. We apply results of Muhly and Tomforde for topological quiver algebras and of Katsura for topological graph C*-algebras to study the K-theory and gauge-invariant ideal structure of ultragraph C*-algebras.

Solved and unsolved problems in generalized notions of amenability for Banach algebras

Yong Zhang (2010)

Banach Center Publications

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We survey the recent investigations on approximate amenability/contractibility and pseudo-amenability/contractibility for Banach algebras. We will discuss the core problems concerning these notions and address the significance of any solutions to them to the development of the field. A few new results are also included.

Dual Banach algebras: representations and injectivity

Matthew Daws (2007)

Studia Mathematica

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We study representations of Banach algebras on reflexive Banach spaces. Algebras which admit such representations which are bounded below seem to be a good generalisation of Arens regular Banach algebras; this class includes dual Banach algebras as defined by Runde, but also all group algebras, and all discrete (weakly cancellative) semigroup algebras. Such algebras also behave in a similar way to C*- and W*-algebras; we show that interpolation space techniques can be used in place of...

On φ-inner amenable Banach algebras

A. Jabbari, T. Mehdi Abad, M. Zaman Abadi (2011)

Colloquium Mathematicae

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Generalizing the concept of inner amenability for Lau algebras, we define and study the notion of φ-inner amenability of any Banach algebra A, where φ is a homomorphism from A onto ℂ. Several characterizations of φ-inner amenable Banach algebras are given.

Amenability for dual Banach algebras

V. Runde (2001)

Studia Mathematica

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We define a Banach algebra 𝔄 to be dual if 𝔄 = (𝔄⁎)* for a closed submodule 𝔄⁎ of 𝔄*. The class of dual Banach algebras includes all W*-algebras, but also all algebras M(G) for locally compact groups G, all algebras ℒ(E) for reflexive Banach spaces E, as well as all biduals of Arens regular Banach algebras. The general impression is that amenable, dual Banach algebras are rather the exception than the rule. We confirm this impression. We first show that under certain conditions...

Fréchet algebras of power series

H. Garth Dales, Shital R. Patel, Charles J. Read (2010)

Banach Center Publications

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We consider Fréchet algebras which are subalgebras of the algebra 𝔉 = ℂ [[X]] of formal power series in one variable and of 𝔉ₙ = ℂ [[X₁,..., Xₙ]] of formal power series in n variables, where n ∈ ℕ. In each case, these algebras are taken with the topology of coordinatewise convergence. We begin with some basic definitions about Fréchet algebras, (F)-algebras, and other topological algebras, and recall some of their properties; we discuss Michael's problem from 1952 on the continuity...

Non-normal elements in Banach *-algebras

B. Yood (2004)

Studia Mathematica

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Let A be a Banach *-algebra with an identity, continuous involution, center Z and set of self-adjoint elements Σ. Let h ∈ Σ. The set of v ∈ Σ such that (h + iv)ⁿ is normal for no positive integer n is dense in Σ if and only if h ∉ Z. The case where A has no identity is also treated.

Commutators in Banach *-algebras

Bertram Yood (2008)

Studia Mathematica

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The set of commutators in a Banach *-algebra A, with continuous involution, is examined. Applications are made to the case where A = B(ℓ₂), the algebra of all bounded linear operators on ℓ₂.

When is L(X) topologizable as a topological algebra?

W. Żelazko (2002)

Studia Mathematica

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Let X be a locally convex space and L(X) be the algebra of all continuous endomorphisms of X. It is known (Esterle [2], [3]) that if L(X) is topologizable as a topological algebra, then the space X is subnormed. We show that in the case when X is sequentially complete this condition is also sufficient. In this case we also obtain some other conditions equivalent to the topologizability of L(X). We also exhibit a class of subnormed spaces X, called sub-Banach spaces, which are not necessarily...

Some homological properties of Banach algebras associated with locally compact groups

Mehdi Nemati (2015)

Colloquium Mathematicae

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We investigate some homological notions of Banach algebras. In particular, for a locally compact group G we characterize the most important properties of G in terms of some homological properties of certain Banach algebras related to this group. Finally, we use these results to study generalized biflatness and biprojectivity of certain products of Segal algebras on G.

A properly infinite Banach *-algebra with a non-zero, bounded trace

H. G. Dales, Niels Jakob Laustsen, Charles J. Read (2003)

Studia Mathematica

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A properly infinite C*-algebra has no non-zero traces. We construct properly infinite Banach *-algebras with non-zero, bounded traces, and show that there are even such algebras which are fairly "close" to the class of C*-algebras, in the sense that they can be hermitian or *-semisimple.

Approximate identities in Banach function algebras

H. G. Dales, A. Ülger (2015)

Studia Mathematica

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In this paper, we shall study contractive and pointwise contractive Banach function algebras, in which each maximal modular ideal has a contractive or pointwise contractive approximate identity, respectively, and we shall seek to characterize these algebras. We shall give many examples, including uniform algebras, that distinguish between contractive and pointwise contractive Banach function algebras. We shall describe a contractive Banach function algebra which is not equivalent to...

Normed "upper interval" algebras without nontrivial closed subalgebras

C. J. Read (2005)

Studia Mathematica

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It is a long standing open problem whether there is any infinite-dimensional commutative Banach algebra without nontrivial closed ideals. This is in some sense the Banach algebraists' counterpart to the invariant subspace problem for Banach spaces. We do not here solve this famous problem, but solve a related problem, that of finding (necessarily commutative) infinite-dimensional normed algebras which do not even have nontrivial closed subalgebras. Our examples are incomplete normed...

Perturbed linear rough differential equations

Laure Coutin, Antoine Lejay (2014)

Annales mathématiques Blaise Pascal

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We study linear rough differential equations and we solve perturbed linear rough differential equations using the Duhamel principle. These results provide us with a key technical point to study the regularity of the differential of the Itô map in a subsequent article. Also, the notion of linear rough differential equations leads to consider multiplicative functionals with values in Banach algebras more general than tensor algebras and to consider extensions of classical results such...

Fréchet algebras, formal power series, and automatic continuity

S. R. Patel (2008)

Studia Mathematica

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We describe all those commutative Fréchet algebras which may be continuously embedded in the algebra ℂ[[X]] in such a way that they contain the polynomials. It is shown that these algebras (except ℂ[[X]] itself) always satisfy a certain equicontinuity condition due to Loy. Using this result, some applications to the theory of automatic continuity are given; in particular, the uniqueness of the Fréchet algebra topology for such algebras is established.

Spatiality of derivations of Fréchet GB*-algebras

Martin Weigt, Ioannis Zarakas (2015)

Studia Mathematica

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We show that every continuous derivation of a countably dominated Fréchet GB*-algebra A is spatial whenever A is additionally an AO*-algebra.