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Journal articles on the topic 'Banach algebra'

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Published: 4 June 2021

Last updated: 3 March 2023

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1

Miller, John Boris. "Strictly real Banach algebras." Bulletin of the Australian Mathematical Society 47, no. 3 (June 1993): 505–19. http://dx.doi.org/10.1017/s000497270001532x.

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A complex Banach algebra is a complexification of a real Banach algebra if and only if it carries a conjugation operator. We prove a uniqueness theorem concerning strictly real selfconjugate subalgebras of a given complex algebra. An example is given of a complex Banach algebra carrying two distinct but commuting conjugations, whose selfconjugate subalgebras are both strictly real. The class of strictly real Banach algebras is shown to be a variety, and the manner of their generation by suitable elements is proved. A corollary describes some strictly real subalgebras in Hermitian Banach star algebras, including C* algebras.

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2

Gourdeau, Frédéric. "Amenability of Lipschitz algebras." Mathematical Proceedings of the Cambridge Philosophical Society 112, no. 3 (November 1992): 581–88. http://dx.doi.org/10.1017/s0305004100071267.

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In this article, we study the amenability of Banach algebras in general, and that of Lipschitz algebras in particular. After introducing an alternative definition of amenability, we extend a result of [5], thereby proving a new characterization of amenability for Banach algebras. This characterization relates the amenability of a Banach algebra A to the space of bounded homomorphisms from A into another Banach algebra B (Theorem 4). This result allows us to solve the problem of amenability for virtually all Lipschitz algebras (of complex or Banach algebra valued functions), a class of algebras which has been studied in [2], [4] and [5].

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3

Khodakarami, Wania, Hoger Ghahramani, and Esmaeil Feizi. "Relative amenability of Banach algebras." Filomat 36, no. 6 (2022): 2091–103. http://dx.doi.org/10.2298/fil2206091k.

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Let A be a Banach algebra and I be a closed ideal of A. We say that A is amenable relative to I, if A/I is an amenable Banach algebra. We study the relative amenability of Banach algebras and investigate the relative amenability of triangular Banach algebras and Banach algebras associated to locally compact groups. We generalize some of the previous known results by applying the concept of relative amenability of Banach algebras, especially, we present a generalization of Johnson?s theorem in the concept of relative amenability.

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4

ABTAHI, F., and A. GHAFARPANAH. "A NOTE ON CYCLIC AMENABILITY OF THE LAU PRODUCT OF BANACH ALGEBRAS DEFINED BY A BANACH ALGEBRA MORPHISM." Bulletin of the Australian Mathematical Society 92, no. 2 (June 16, 2015): 282–89. http://dx.doi.org/10.1017/s0004972715000544.

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Let $T$ be a Banach algebra homomorphism from a Banach algebra ${\mathcal{B}}$ to a Banach algebra ${\mathcal{A}}$ with $\Vert T\Vert \leq 1$. Recently, Bhatt and Dabhi [‘Arens regularity and amenability of Lau product of Banach algebras defined by a Banach algebra morphism’, Bull. Aust. Math. Soc.87 (2013), 195–206] showed that cyclic amenability of ${\mathcal{A}}\times _{T}{\mathcal{B}}$ is stable with respect to $T$, for the case where ${\mathcal{A}}$ is commutative. In this note, we address a gap in the proof of this stability result and extend it to an arbitrary Banach algebra ${\mathcal{A}}$.

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5

Lee, Yang-Hi. "Stability of -Jordan Homomorphisms from a Normed Algebra to a Banach Algebra." Abstract and Applied Analysis 2013 (2013): 1–5. http://dx.doi.org/10.1155/2013/691025.

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We establish the hyperstability of -Jordan homomorphisms from a normed algebra to a Banach algebra, and also we show that an -Jordan homomorphism between two commutative Banach algebras is an -ring homomorphism.

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6

MONFARED, MEHDI SANGANI. "Character amenability of Banach algebras." Mathematical Proceedings of the Cambridge Philosophical Society 144, no. 3 (May 2008): 697–706. http://dx.doi.org/10.1017/s0305004108001126.

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AbstractWe introduce the notion of character amenable Banach algebras. We prove that character amenability for either of the group algebra L1(G) or the Fourier algebra A(G) is equivalent to the amenability of the underlying group G. Character amenability of the measure algebra M(G) is shown to be equivalent to G being a discrete amenable group. We also study functorial properties of character amenability. For a commutative character amenable Banach algebra A, we prove all cohomological groups with coefficients in finite-dimensional Banach A-bimodules, vanish. As a corollary we conclude that all finite-dimensional extensions of commutative character amenable Banach algebras split strongly.

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7

Yost, David. "Strictly Convex Banach Algebras." Axioms 10, no. 3 (September 11, 2021): 221. http://dx.doi.org/10.3390/axioms10030221.

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We discuss two facets of the interaction between geometry and algebra in Banach algebras. In the class of unital Banach algebras, there is essentially one known example which is also strictly convex as a Banach space. We recall this example, which is finite-dimensional, and consider the open question of generalising it to infinite dimensions. In C∗-algebras, we exhibit one striking example of the tighter relationship that exists between algebra and geometry there.

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8

LIU, CHENG–KAI. "The structure of triple homomorphisms onto prime algebras." Mathematical Proceedings of the Cambridge Philosophical Society 168, no. 2 (October 23, 2018): 345–60. http://dx.doi.org/10.1017/s0305004118000737.

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AbstractTriple homomorphisms on C*-algebras and JB*-triples have been studied in the literature. From the viewpoint of associative algebras, we characterise the structure of triple homomorphisms from an arbitrary ⋆-algebra onto a prime *-algebra. As an application, we prove that every triple homomorphism from a Banach ⋆-algebra onto a prime semisimple idempotent Banach *-algebra is continuous. The analogous results for prime C*-algebras and standard operator *-algebras on Hilbert spaces are also described.

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9

Ludkovsky, S., and B. Diarra. "Spectral integration and spectral theory for non-Archimedean Banach spaces." International Journal of Mathematics and Mathematical Sciences 31, no. 7 (2002): 421–42. http://dx.doi.org/10.1155/s016117120201150x.

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Banach algebras over arbitrary complete non-Archimedean fields are considered such that operators may be nonanalytic. There are different types of Banach spaces over non-Archimedean fields. We have determined the spectrum of some closed commutative subalgebras of the Banach algebraℒ(E)of the continuous linear operators on a free Banach spaceEgenerated by projectors. We investigate the spectral integration of non-Archimedean Banach algebras. We define a spectral measure and prove several properties. We prove the non-Archimedean analog of Stone theorem. It also contains the case ofC-algebrasC∞(X,𝕂). We prove a particular case of a representation of aC-algebra with the help of aL(Aˆ,μ,𝕂)-projection-valued measure. We consider spectral theorems for operators and families of commuting linear continuous operators on the non-Archimedean Banach space.

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10

Garimella, Ramesh V. "On continuity of derivations and epimorphisms on some vector-valued group algebras." Bulletin of the Australian Mathematical Society 56, no. 2 (October 1997): 209–15. http://dx.doi.org/10.1017/s0004972700030938.

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For a locally compact Abelian group G and a commutative Banach algebra B, let L1(G, B) be the Banach algebra of all Bochner integrable functions. We show that if G is compact and B is a nonunital Banach algebra without nontrivial zero divisors, then (i) all derivations on L1(G, B) are continuous if and only if all derivations on B are continuous, and (ii) each epimorphism from a Banach algebra X onto L1(G, B) is continuous provided every epimorphism from X onto B is continuous. If G is noncompact then every derivation on L1(G, B) and every epimorphism from a commutative Banach algebra onto L1(G, B) are continuous. Our results extend the results of Neumann and Velasco for nonunital Banach algebras.

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11

ESHAGHI GORDJI, MADJID, ALI JABBARI, ALI EBADIAN, and SAEID OSTADBASHI. "AUTOMATIC CONTINUITY OF 3-HOMOMORPHISMS ON TERNARY BANACH ALGEBRAS." International Journal of Geometric Methods in Modern Physics 10, no. 10 (October 8, 2013): 1320013. http://dx.doi.org/10.1142/s0219887813200132.

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In this paper we consider automatic continuity of 3-homomorphism and anti-3-homomorphism between non-unital ternary Banach algebras. We show that every surjective 3-homomorphism (anti-3-homomorphism) from ternary Banach algebra A into semisimple ternary Banach algebra B is continuous.

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12

NIKOU, AZADEH, and ANTHONY G. O'FARRELL. "BANACH ALGEBRAS OF VECTOR-VALUED FUNCTIONS." Glasgow Mathematical Journal 56, no. 2 (August 13, 2013): 419–26. http://dx.doi.org/10.1017/s0017089513000359.

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AbstractWe introduce the concept of an E-valued function algebra, a type of Banach algebra that consists of continuous E-valued functions on some compact Hausdorff space, where E is a Banach algebra. We present some basic results about such algebras, having to do with the Shilov boundary and the set of peak points of some commutative E-valued function algebras. We give some specific examples.

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13

İnceboz, Hülya, and Berna Arslan. "The first module (σ,τ)-cohomology group of triangular Banach algebras of order three." Journal of Algebra and Its Applications 17, no. 12 (December 2018): 1850225. http://dx.doi.org/10.1142/s0219498818502250.

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The notion of module amenability for a class of Banach algebras, which could be considered as a generalization of Johnson’s amenability, was introduced by Amini in [Module amenability for semigroup algebras, Semigroup Forum 69 (2004) 243–254]. The weak module amenability of the triangular Banach algebra [Formula: see text], where [Formula: see text] and [Formula: see text] are Banach algebras (with [Formula: see text]-module structure) and [Formula: see text] is a Banach [Formula: see text]-module, is studied by Pourabbas and Nasrabadi in [Weak module amenability of triangular Banach algebras, Math. Slovaca 61(6) (2011) 949–958], and they showed that the weak module amenability of [Formula: see text] triangular Banach algebra [Formula: see text] (as an [Formula: see text]-bimodule) is equivalent with the weak module amenability of the corner algebras [Formula: see text] and [Formula: see text] (as Banach [Formula: see text]-bimodules). The main aim of this paper is to investigate the module [Formula: see text]-amenability and weak module [Formula: see text]-amenability of the triangular Banach algebra [Formula: see text] of order three, where [Formula: see text] and [Formula: see text] are [Formula: see text]-module morphisms on [Formula: see text]. Also, we give some results for semigroup algebras.

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14

Mouton, S. "A spectral problem in ordered Banach algebras." Bulletin of the Australian Mathematical Society 67, no. 1 (February 2003): 131–44. http://dx.doi.org/10.1017/s0004972700033591.

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We recall the definition and properties of an algebra cone C of a complex unital Banach algebra A. It can be shown that C induces on A an ordering which is compatible with the algebraic structure of A, and A is then called an ordered Banach algebra. The Banach algebra ℒ(E) of all bounded linear operators on a complex Banach lattice E is an example of an ordered Banach algebra, and an interesting aspect of research in ordered Banach algebras is that of investigating in an ordered Banach algebra-context certain problems that originated in ℒ(E). In this paper we investigate the problems of providing conditions under which (1) a positive element a with spectrum consisting of 1 only will necessarily be greater than or equal to 1, and (2) f (a) will be positive if a is positive, where f (a) is the element defined by the holomorphic functional calculus.

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15

Alhefthi, Reem K., Akhlaq A. Siddiqui, and Fatmah B. Jamjoom. "Quasi-Jordan Banach Algebras." Abstract and Applied Analysis 2014 (2014): 1–11. http://dx.doi.org/10.1155/2014/690806.

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We initiate a study of quasi-Jordan normed algebras. It is demonstrated that any quasi-Jordan Banach algebra with a norm1unit can be given an equivalent norm making the algebra isometrically isomorphic to a closed right ideal of a unital split quasi-Jordan Banach algebra; the set of invertible elements may not be open; the spectrum of any element is nonempty, but it may be neither bounded nor closed and hence not compact. Some characterizations of the unbounded spectrum of an element in a split quasi-Jordan Banach algebra with certain examples are given in the end.

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16

Magyar, Zoltán, and Zoltán Sebestyén. "On the Definition of C*-Algebras II." Canadian Journal of Mathematics 37, no. 4 (August 1, 1985): 664–81. http://dx.doi.org/10.4153/cjm-1985-035-7.

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The theory of noncommutative involutive Banach algebras (briefly Banach *-algebras) owes its origin to Gelfand and Naimark, who proved in 1943 the fundamental representation theorem that a Banach *-algebra with C*-condition(C*)is *-isomorphic and isometric to a norm-closed self-adjoint subalgebra of all bounded operators on a suitable Hilbert space.At the same time they conjectured that the C*-condition can be replaced by the B*-condition.(B*)In other words any B*-algebra is actually a C*-algebra. This was shown by Glimm and Kadison [5] in 1960.

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17

Dadakhodjaev, R. A., and A. A. Rakhimov. "2-Local derivations of real AW*-algebras are derivation." Positivity 25, no. 4 (March 1, 2021): 1351–56. http://dx.doi.org/10.1007/s11117-021-00815-8.

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Abstract2-Local derivations on real matrix algebras over unital semi-prime Banach algebras are considered. Using the real analogue of the result that any 2-local derivation on the algebra $$M_{2^n}(A)$$ M 2 n ( A ) ($$n\ge 2$$ n ≥ 2 ) is a derivation, it is shown that any 2-local derivation on real AW$$^*$$ ∗ -algebra for which the enveloping algebra is (complex) AW*-algebra, is a derivation, where A is a unital semi-prime Banach algebra with the inner derivation property.

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18

Ghorbai, M., and Davood Ebrahimi Bagha. "Amenability of A⊕_T X as an extension of Banach algebra." Mathematica Montisnigri 49 (2020): 39–48. http://dx.doi.org/10.20948/mathmontis-2020-49-3.

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Let 𝐴𝐴,𝑋𝑋,𝔘𝔘 be Banach algebras and 𝐴𝐴 be a Banach 𝔘𝔘-bimodule also 𝑋𝑋 be a Banach 𝐴𝐴−𝔘𝔘-module. In this paper we study the relation between module amenability, weak module amenability and module approximate amenability of Banach algebra 𝐴𝐴⊕𝑇𝑇𝑋𝑋 and that of Banach algebras 𝐴𝐴,𝑋𝑋. Where 𝑇𝑇: 𝐴𝐴×𝐴𝐴→𝑋𝑋 is a bounded bi-linear mapping with specificconditions.

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19

Pfaffenberger, W. E., and J. Phillips. "Commutative Gelfand Theory for Real Banach Algebras: Representations as Sections of Bundles." Canadian Journal of Mathematics 44, no. 2 (April 1, 1992): 342–56. http://dx.doi.org/10.4153/cjm-1992-023-4.

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AbstractWe are concerned here with the development of a more general real case of the classical theorem of Gelfand ([5], 3.1.20), which represents a complex commutative unital Banach algebra as an algebra of continuous functions defined on a compact Hausdorff space.In § 1 we point out that when looking at real algebras there is not always a one-to-one correspondence between the maximal ideals of the algebra B, denoted ℳ, and the set of unital (real) algebra homomorphisms from B into C, denoted by ΦB. This simple point and subsequent observations lead to a theory of representations of real commutative unital Banach algebras where elements are represented as sections of a bundle of real fields associated with the algebra (Theorem 3.5). After establishing this representation theorem, we look into the question of when a real commutative Banach algebra is already complex. There is a natural topological obstruction which we delineate. Theorem 4.8 gives equivalent conditions which determine whether such an algebra is already complex.Finally, in § 5 we abstractly characterize those section algebras which appear as the target algebras for our Gelfand transform. We dub these algebras “almost complex C*- algebras” and provide a natural classification scheme.

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20

Noreldeen, Alaa Hassan. "On the Homology Theory of Operator Algebras." International Journal of Mathematics and Mathematical Sciences 2012 (2012): 1–13. http://dx.doi.org/10.1155/2012/368527.

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We investigate the cyclic homology and free resolution effect of a commutative unital Banach algebra. Using the free resolution operator, we define the relative cyclic homology of commutative Banach algebras. Lemmas and theorems of this investigation are studied and proved. Finally, the relation between cyclic homology and relative cyclic homology of Banach algebra is deduced.

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21

Srivastava, Neeraj, S. Bhattacharya, and S. N. Lal. "2-normed algebras-II." Publications de l'Institut Math?matique (Belgrade) 90, no. 104 (2011): 135–43. http://dx.doi.org/10.2298/pim1104135s.

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In the first part of the paper [5], we gave a new definition of real or complex 2-normed algebras and 2-Banach algebras. Here we give two examples which establish that not all 2-normed algebras are normable and a 2-Banach algebra need not be a 2-Banach space. We conclude by deriving a new and interesting spectral radius formula for 1-Banach algebras from the basic properties of 2-Banach algebras and thus vindicating our definitions of 2-normed and 2-Banach algebras given in [5].

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22

Alahmari, Abdullah, Falih A. Aldosray, and Mohamed Mabrouk. "Banach algebras satisfying certain chain conditions on closed ideals." Studia Scientiarum Mathematicarum Hungarica 57, no. 3 (October 20, 2020): 290–97. http://dx.doi.org/10.1556/012.2020.57.3.1465.

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AbstractLet 𝔄 be a unital Banach algebra and ℜ its Jacobson radical. This paper investigates Banach algebras satisfying some chain conditions on closed ideals. In particular, it is shown that a Banach algebra 𝔄 satisfies the descending chain condition on closed left ideals then 𝔄/ℜ is finite dimensional. We also prove that a C*-algebra satisfies the ascending chain condition on left annihilators if and only if it is finite dimensional. Moreover, other auxiliary results are established.

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23

Martinez-Moreno, J., and A. Rodriguez-Palacios. "Imbedding elements whose numerical range has a vertex at zero in holomorphic semigroups." Proceedings of the Edinburgh Mathematical Society 28, no. 1 (February 1985): 91–95. http://dx.doi.org/10.1017/s0013091500003229.

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If a is an element of a complex unital Banach algebra whose numerical range is confined to a closed angular region with vertex at zero and angle strictly less than π, we imbed a in a holomorphic semigroup with parameter in the open right half plane.There has been recently a great development in the theory of semigroups in Banach algebras (see [6]), with attention focused on the relation between the structure of a given Banach algebra and the existence of continuous or holomorphic non-trivial semigroups with certain properties with range in this algebra. The interest of this paper arises from the fact that we relate in it, we think for the first time, this new point of view in the theory of Banach algebras with the already classic one of numerical ranges [2,3]. The proofs of our results use, in addition to some basic ideas from numerical ranges in Banach algebras, the concept of extremal algebra Ea(K) of a compact convex set K in ℂ due to Bollobas [1] and concretely the realization of Ea(K) achieved by Crabb, Duncan and McGregor [4].

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24

PIRKOVSKII, A. YU. "Approximate characterizations of projectivity and injectivity for Banach modules." Mathematical Proceedings of the Cambridge Philosophical Society 143, no. 2 (September 2007): 375–85. http://dx.doi.org/10.1017/s0305004107000163.

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AbstractWe characterize projective and injective Banach modules in approximate terms, generalizing thereby a characterization of contractible Banach algebras given by F. Ghahramani and R. J. Loy. As a corollary, we show that each uniformly approximately amenable Banach algebra is amenable. Some applications to homological dimensions of Banach modules and algebras are also given.

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25

Paravicini, Walther. "Induction for Banach Algebras, Groupoids and KKban." Journal of K-theory 4, no. 3 (October 23, 2009): 405–68. http://dx.doi.org/10.1017/is009010006jkt073.

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AbstractGiven two equivalent locally compact Hausdorff groupoids, We prove that the Bost conjecture with Banach algebra coefficients is true for one if and only if it is true for the other. This also holds for the Bost conjecture with C*- coefficients. To show these results, the functoriality of Lafforgue's KK-theory for Banach algebras and groupoids with respect to generalised morphisms of groupoids is established. It is also shown that equivalent groupoids have Morita equivalent L1-algebras (with Banach algebra coefficients).

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26

Burlando, Laura. "Comparisons between different spectra of an element in a Banach algebra." International Journal of Mathematics and Mathematical Sciences 16, no. 4 (1993): 819–22. http://dx.doi.org/10.1155/s0161171293001036.

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In this paper we study the relationships among the spectra of the cosets of an element of a Banach algebra in some quotient algebras. We also characterize the spectrum of anya∈M(whereMis an ideal of a Banach algebra with identity and moreover has an identity) in the whole algebra in terms of the spectrum ofainM.

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27

Gourdeau, Frédéric. "Amenability of Banach algebras." Mathematical Proceedings of the Cambridge Philosophical Society 105, no. 2 (March 1989): 351–55. http://dx.doi.org/10.1017/s0305004100067840.

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We consider the problem of amenability for a commutative Banach algebra. The question of amenability for a Banach algebra was first studied by B. E. Johnson in 1972, in [5]. The most recent contributions, to our knowledge, are papers by Bade, Curtis and Dales [1], and by Curtis and Loy [3]. In the first, amenability for Lipschitz algebras on a compact metric space K is studied. Using the fact, which they prove, that LipαK is isometrically isomorphic to the second dual of lipαK, for 0 < α < 1, they show that lipαK is not amenable when K is infinite and 0 < α < 1. In the second paper, the authors prove, without using any serious cohomology theory, some results proved earlier by Khelemskii and Scheinberg [8] using cohomology. They also discuss the amenability of Lipschitz algebras, using the result that a weakly complemented closed two-sided ideal in an amenable Banach algebra has a bounded approximate identity. Their result is stronger than that of [1].

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28

SAMEA, H. "ESSENTIAL AMENABILITY OF ABSTRACT SEGAL ALGEBRAS." Bulletin of the Australian Mathematical Society 79, no. 2 (March 13, 2009): 319–25. http://dx.doi.org/10.1017/s0004972708001329.

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AbstractA number of well-known results of Ghahramani and Loy on the essential amenability of Banach algebras are generalized. It is proved that a symmetric abstract Segal algebra with respect to an amenable Banach algebra is essentially amenable. Applications to locally compact groups are given.

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29

NASR-ISFAHANI, RASOUL, and MEHDI NEMATI. "ESSENTIAL CHARACTER AMENABILITY OF BANACH ALGEBRAS." Bulletin of the Australian Mathematical Society 84, no. 3 (November 1, 2011): 372–86. http://dx.doi.org/10.1017/s0004972711002620.

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AbstractFor a Banach algebra 𝒜 and a character ϕ on 𝒜, we introduce and study the notion of essential ϕ-amenability of 𝒜. We give some examples to show that the class of essentially ϕ-amenable Banach algebras is larger than that of ϕ-amenable Banach algebras introduced by Kaniuth et al. [‘On ϕ-amenability of Banach algebras’, Math. Proc. Cambridge Philos. Soc.144 (2008), 85–96]. Finally, we characterize the essential ϕ-amenability of various Banach algebras related to locally compact groups.

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30

JAIN, RANJANA, and AJAY KUMAR. "IDEALS IN OPERATOR SPACE PROJECTIVE TENSOR PRODUCT OF C*-ALGEBRAS." Journal of the Australian Mathematical Society 91, no. 2 (October 2011): 275–88. http://dx.doi.org/10.1017/s1446788711001479.

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AbstractLet A and B be C*-algebras. We prove the slice map conjecture for ideals in the operator space projective tensor product $A \mathbin {\widehat {\otimes }} B$. As an application, a characterization of the prime ideals in the Banach *-algebra $A\mathbin {\widehat {\otimes }} B$ is obtained. In addition, we study the primitive ideals, modular ideals and the maximal modular ideals of $A\mathbin {\widehat {\otimes }} B$. We also show that the Banach *-algebra $A\mathbin {\widehat {\otimes }} B$ possesses the Wiener property and that, for a subhomogeneous C*-algebra A, the Banach * -algebra $A \mathbin {\widehat {\otimes }} B$ is symmetric.

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31

Lin, Pei-Kee. "A counterexample of hermitian liftings." Proceedings of the Edinburgh Mathematical Society 32, no. 2 (June 1989): 255–59. http://dx.doi.org/10.1017/s0013091500028650.

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Let X be a complex Banach space, and let and denote respectively the algebras of bounded and compact operators on X. The quotient algebra is called the Calkin algebra associated with X. It is known that both and are complex Banach algebras with unit e. For such unital Banach algebras B, setand define the numerical range of x ∈ B asx is said to be hermitian if W(x)⊆R. It is known thatFact 1. ([4 vol. I, p. 46]) x is hermitian if and only if ‖eiαx‖ = (or ≦)1 for all α ∈ R, where ex is defined by

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32

Roh, Jaiok, and Ick-Soon Chang. "Approximate Derivations with the Radical Ranges of Noncommutative Banach Algebras." Abstract and Applied Analysis 2015 (2015): 1–7. http://dx.doi.org/10.1155/2015/594075.

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We consider the derivations on noncommutative Banach algebras, and we will first study the conditions for a derivation on noncommutative Banach algebra. Then, we examine the stability of functional inequalities with a derivation. Finally, we take the derivations with the radical ranges on noncommutative Banach algebras.

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33

Abtahi, F., and S. Rahnama. "Essential amenability of Fréchet algebras." Ukrains’kyi Matematychnyi Zhurnal 72, no. 7 (July 15, 2020): 867–76. http://dx.doi.org/10.37863/umzh.v72i7.830.

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UDC 517.98 Essential amenability of Banach algebras have been defined and investigated. Here, this concept will be introduced for Frechet algebras. Then a number of well-known results of essential amenability of Banach algebras are generalized for Fréchet algebras. Moreover, related results about Segal–Fréchet algebras are provided. As the main result, it is provedthat if ( 𝒜 , p ℓ ) is an amenable Fréchet algebra with a uniformly bounded approximate identity, then every symmetric Segal – Fréchet algebra in ( 𝒜 , p ℓ ) is essentially amenable.

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34

El Harti, R. "The structure of a subclass of amenable banach algebras." International Journal of Mathematics and Mathematical Sciences 2004, no. 55 (2004): 2963–69. http://dx.doi.org/10.1155/s0161171204401069.

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We give sufficient conditions that allow contractible (resp., reflexive amenable) Banach algebras to be finite-dimensional and semisimple algebras. Moreover, we show that any contractible (resp., reflexive amenable) Banach algebra in which every maximal left ideal has a Banach space complement is indeed a direct sum of finitely many full matrix algebras. Finally, we characterize Hermitian*-algebras that are contractible.

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35

Tomiuk, B. J. "Isomorphisms of multiplier algebras." Glasgow Mathematical Journal 28, no. 1 (January 1986): 73–77. http://dx.doi.org/10.1017/s0017089500006364.

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Let A and B be semisimple Banach algebras, and let M1(A) (resp. M1(B)) be the algebra of left multipliers on A (resp. B). Suppose that A is an abstract Segal algebra in B. We find conditions on A and B which imply that M1(A) is topologically algebra isomorphic to M1(B). As a special case we obtain the result of [8] which states that if A is an A*-algebra that is a*-ideal in its B*-algebra completion B and A2 is dense in A then M1(A) is topologically algebra isomorphic to M1(B). We make an application of our main result to right complemented Banach algebras.

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36

Illoussamen, El Hossein, and Volker Runde. "Topologically simple Banach algebras with derivation." Bulletin of the Australian Mathematical Society 60, no. 1 (August 1999): 153–61. http://dx.doi.org/10.1017/s0004972700033414.

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It is not known if a commutative, topologically simple, radical Banach algebra exists. If, however, every derivation on such an algebra is continuous, this yields the automatic continuity of all derivations on commutative, semiprime Banach algebras. Utilising techniques used by Thomas in his proof of the Singer-Wermer conjecture, we show that, if A is a commutative, topologically simple Banach algebra with a non-zero derivation on it, then a quotient of a certain localisation of A has a power series structure. A pivotal role is played by what we call ample sets of denominators.

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37

Oshobi, E. O., and J. S. Pym. "Banach algebras whose duals consist of multipliers." Mathematical Proceedings of the Cambridge Philosophical Society 102, no. 3 (November 1987): 481–505. http://dx.doi.org/10.1017/s0305004100067542.

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A few years ago, the authors considered briefly Banach algebras whose duals could be identified ‘naturally’ with their multiplier algebras [17]. In this context, naturalness can be interpreted as meaning that, for each element b of the algebra B and each pair of elements u, v of the dual B′,where 〈, 〉 denotes the dual pairing and the products are of elements of B′ regarded as left or right multipliers on B. In the present paper we return to the same circle of ideas but begin with a more general situation. We assume only that the algebra B is injectively embedded in its algebra of left, and in its algebra of right, multipliers and that its dual B′ can be injectively embedded in the algebra M(B) of double multipliers on B (definition below) in such a way that the above relation holds. From these assumptions we shall prove that there is a normed algebra A such that M(B) is the dual of A and is the algebra of continuous left multipliers on A (or, equally, right multipliers).

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38

Nicolau, Artur, and Daniel Suárez. "Approximation by invertible functions of $H^{\infty}$." MATHEMATICA SCANDINAVICA 99, no. 2 (December 1, 2006): 287. http://dx.doi.org/10.7146/math.scand.a-15013.

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We provide an analytic proof that if $H^\infty$ is the algebra of bounded analytic functions on the unit disk, $A$ is a Banach algebra and $f: H^\infty \rightarrow A$ is a Banach algebras morphism with dense image, then $f((H^\infty)^{-1})$ is dense in $A^{-1}$.

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39

Seddighin, Morteza. "Supersets for the spectrum of elements in extended Banach algebras." International Journal of Mathematics and Mathematical Sciences 12, no. 4 (1989): 823–24. http://dx.doi.org/10.1155/s016117128900102x.

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If A is a Banach Algebra with or without an identity, A can be always extended to a Banach algebraA¯with identity, whereA¯is simply the direct sum of A and C, the algebra of complex numbers. In this note we find supersets for the spectrum of elements ofA¯.

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40

Ebadian, A., and A. Jabbari. "ГИПЕРТАУБЕРОВЫ АЛГЕБРЫ, ОПРЕДЕЛЕННЫЕ ГОМОМОРФИЗМОМ БАНАХОВОЙ АЛГЕБРЫ." Вестник КРАУНЦ. Физико-математические науки, no. 1 (May 4, 2019): 18–28. http://dx.doi.org/10.26117/2079-6641-2019-26-1-18-28.

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Let A and B be Banach algebras and T: B→A be a continuous homomorphism. We consider left multipliers from A×TB into its the first dual i.e., A*×B* and we show that A×TB is a hyper-Tauberian algebra if and only if A and B are hyper-Tauberian algebras. Пусть A и B – банаховы алгебры, а T: B→A – непрерывный гомоморфизм. Мы рассматриваем левые мультипликаторы из A×TB в его первое двойственное, т.е. A*×B*, и показываем, что A×TB является гипертауберовой алгеброй тогда и только тогда, когда A и B являются гипертауберовыми алгебрами.

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41

Dashti, Mahshid, and Sima Soltani Renani. "The retraction of certain banach right modules associated to a character." Mathematica Slovaca 69, no. 4 (August 27, 2019): 891–900. http://dx.doi.org/10.1515/ms-2017-0260.

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Abstract Let 𝓐 be a Banach algebra and let 𝓜 be a unital Banach algebra. For a homomorphism Φ from 𝓐 into 𝓜, we consider 𝓜 as a Banach right 𝓐-module and investigate when 𝓜 is a retract of 𝓐 with respect to Φ. We also give characterizations of admitting vector-valued invariant Φ-means in terms of projectivity and injectivity. Finally, we apply these results to abstract Segal algebras.

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42

Ganiev, I. G., and O. I. Egamberdiev. "The Arens Algebras of Vector-Valued Functions." Journal of Function Spaces 2014 (2014): 1–4. http://dx.doi.org/10.1155/2014/248925.

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43

Vasylyshyn, Taras, and Kostiantyn Zhyhallo. "Entire Symmetric Functions on the Space of Essentially Bounded Integrable Functions on the Union of Lebesgue-Rohlin Spaces." Axioms 11, no. 9 (September 7, 2022): 460. http://dx.doi.org/10.3390/axioms11090460.

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The class of measure spaces which can be represented as unions of Lebesgue-Rohlin spaces with continuous measures contains a lot of important examples, such as Rn for any n∈N with the Lebesgue measure. In this work we consider symmetric functions on Banach spaces of all complex-valued integrable essentially bounded functions on such unions. We construct countable algebraic bases of algebras of continuous symmetric polynomials on these Banach spaces. The completions of such algebras of polynomials are Fréchet algebras of all complex-valued entire symmetric functions of bounded type on the abovementioned Banach spaces. We show that each such Fréchet algebra is isomorphic to the Fréchet algebra of all complex-valued entire symmetric functions of bounded type on the complex Banach space of all complex-valued essentially bounded functions on [0,1].

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44

Das, Dipankar, Nilakshi Goswami, and Vishnu Narayan Mishra. "Some results on the projective cone normed tensor product spaces over banach algebras." Boletim da Sociedade Paranaense de Matemática 38, no. 1 (February 19, 2018): 197. http://dx.doi.org/10.5269/bspm.v38i1.36450.

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For two real Banach algebras $\mathbb{A}_1$ and $\mathbb{A}_2$, let $K_p$ be the projective cone in $\mathbb{A}_1\otimes_\gamma \mathbb{A}_2$. Using this we define a cone norm on the algebraic tensor product of two vector spaces over the Banach algebra $\mathbb{A}_1\otimes_\gamma \mathbb{A}_2$ and discuss some properties. We derive some fixed point theorems in this projective cone normed tensor product space over Banach algebra with a suitable example. For two self mappings $S$ and $T$ on a cone Banach space over Banach algebra, the stability of the iteration scheme $x_{2n+1}=Sx_{2n}$, $x_{2n+2}=Tx_{2n+1},\;n=0,1,2,...$ converging to the common fixed point of $S$ and $T$ is also discussed here.

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Zarei, G., A. Pourabbas, M. Rostami, and A. Sahami. "Approximate Biprojectivity of ℓ 1 -Munn Banach Algebras." Journal of Mathematics 2022 (November 14, 2022): 1–9. http://dx.doi.org/10.1155/2022/2112167.

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In the present paper, we study the approximate biprojectivity and weak approximate biprojectivity of ℓ 1 -Munn Banach algebras when the related sandwich matrix is regular over Inv A . In fact, we show that a ℓ 1 -Munn Banach algebra with the regular sandwich matrix over Inv A is approximately biprojective (weak approximately biprojective) if and only if A is approximately biprojective (weak approximately biprojective), respectively. We also study approximate biprojectivity of upper triangular Banach algebra when the associated sandwich matrix with elements in Inv A is invertible. Finally, we apply our results to Rees semigroup algebras.

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46

Galkin, Oleg E., and Svetlana Y. Galkina. "On the invertibility of solutions of first order linear homogeneous differential equations in Banach algebras." Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva 21, no. 4 (December 30, 2019): 430–42. http://dx.doi.org/10.15507/2079-6900.21.201904.430-442.

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This work is devoted to the study of some properties of linear homogeneous differential equations of the first order in Banach algebras. It is found (for some types of Banach algebras), at what right-hand side of such an equation, from the invertibility of the initial condition it follows the invertibility of its solution at any given time. Associative Banach algebras over the field of real or complex numbers are considered. The right parts of the studied equations have the form [F(t)](x(t)), where {F(t)} is a family of bounded operators on the algebra, continuous with respect to t∈R. The problem is to find all continuous families of bounded operators on algebra, preserving the invertibility of elements from it, for a given Banach algebra. In the proposed article, this problem is solved for only three cases. In the first case, the algebra consists of all square matrices of a given order. For this algebra, it is shown that all continuous families of operators, preserving the invertibility of elements from the algebra at zero must be of the form [F(t)](y)=a(t)⋅y+y⋅b(t), where the families {a(t)} and {b(t)} are also continuous. In the second case, the algebra consists of all continuous functions on the segment. For this case, it is shown that all families of operators, preserving the invertibility of elements from the algebra at any time must be of the form [F(t)](y)=a(t)⋅y, where the family {a(t)} is also continuous. The third case concerns those Banach algebras in which all nonzero elements are invertible. For example, the algebra of complex numbers and the algebra of quaternions have this property. In this case, any continuous families of bounded operators preserves the invertibility of the elements from the algebra at any time. The proposed study is in contact with the research of the foundations of quantum mechanics. The dynamics of quantum observables is described by the Heisenberg equation. The obtained results are an indirect argument in favor of the fact, that the known form of the Heisenberg equation is the only correct one.

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47

Dixon, P. G. "Topologically nilpotent Banach algebras and factorisation." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 119, no. 3-4 (1991): 329–41. http://dx.doi.org/10.1017/s0308210500014876.

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SynopsisA Banach algebra A is said to be topologically nilpotent if sup {‖x1x2…xn‖1/n: xi ∈ A, ‖xi‖ ≦ 1 (1 ≦ i ≦ n)} tends to zero as n → ∞. A Banach algebra A is uniformly topologically nil if sup {‖xn‖ 1/n: x ∈ A, ‖x‖ ≦ 1} tends to zero as n → ∞. These notions are equivalent for commutative algebras and a topological version of the Nagata-Higman Theorem gives a partial result for the non-commutative case. Topologically nilpotent algebras have a strong non-factorisation property and this yields theorems of the type “factorisation implies the existence of arbitrarily slowly decreasing powers”. Extensions of topologically nilpotent algebras by topologically nilpotent algebras are topologically nilpotent.

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48

Momeni, M., T. Yazdanpanah, and M. R. Mardanbeigi. "-Approximately Contractible Banach Algebras." Abstract and Applied Analysis 2012 (2012): 1–20. http://dx.doi.org/10.1155/2012/653140.

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We investigate -approximate contractibility and -approximate amenability of Banach algebras, which are extensions of usual notions of contractibility and amenability, respectively, where is a dense range or an idempotent bounded endomorphism of the corresponding Banach algebra.

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49

Ettefagh, Mina. "Biprojectivity and biflatness of generalized module extension Banach algebras." Filomat 32, no. 17 (2018): 5895–905. http://dx.doi.org/10.2298/fil1817895e.

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We investigate biprojectivity and biflatness of generalized module extension Banach algebra A Z B, in which A and B are Banach algebras and B is an algebraic Banach A-bimodule, with multiplication: (a, b)?(a',b') = (aa', ab' + ba' + bb')

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50

Srivastav, Anand. "Extreme Points of Positive Functionals and Spectral States on Real Banach Algebras." Canadian Journal of Mathematics 44, no. 4 (August 1, 1992): 856–66. http://dx.doi.org/10.4153/cjm-1992-051-9.

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AbstractExtreme points of positive functionals and spectral states on real commutative Banach algebras are investigated and characterized as multiplicative functionals extending the well-known results from complex to real Banach algebras. As an application a new and short proof of the existence of the Shilov boundary of a real commutative Banach algebra with nonempty maximal ideal space is given.

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