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1

PHILLIPS, N. CHRISTOPHER. "K-THEORY FOR FRÉCHET ALGEBRAS". International Journal of Mathematics 02, nr 01 (luty 1991): 77–129. http://dx.doi.org/10.1142/s0129167x91000077.

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We define K-theory for Fréchet algebras (assumed to be locally multiplicatively convex) so as to simultaneously generalize K-theory for σ-C*-algebras and K-theory for Banach algebras. The main results on K-theory of σ-C*-algebras, which are analogs of standard theorems on representable K-theory of spaces, carry over to the more general case. Our theory also gives the expected results in two other cases. If the invertible elements of a Fréchet algebra are an open set, as is the case for dense subalgebras of C*-algebras closed under holomorphic functional calculus, then our theory agrees with the result of applying the Banach algebra definition. For commutative unital Fréchet algebras, our K-theory is the same as the representable K-theory of the maximal ideal space with its compactly generated topology.

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2

Martinez-Moreno, J., i A. Rodriguez-Palacios. "Imbedding elements whose numerical range has a vertex at zero in holomorphic semigroups". Proceedings of the Edinburgh Mathematical Society 28, nr 1 (luty 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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3

Gourdeau, Frédéric. "Amenability of Banach algebras". Mathematical Proceedings of the Cambridge Philosophical Society 105, nr 2 (marzec 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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4

Hadwin, Don, i Mehmet Orhon. "A noncommutative theory of Bade functionals". Glasgow Mathematical Journal 33, nr 1 (styczeń 1991): 73–81. http://dx.doi.org/10.1017/s0017089500008053.

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Since the pioneering work of W. G. Bade [3, 4] a great deal of work has been done on bounded Boolean algebras of projections on a Banach space ([11, XVII.3.XVIII.3], [21, V.3], [16], [6], [12], [13], [14], ]17], [18], [23], [24]). Via the Stone representation space of the Boolean algebra, the theory can be studied through Banach modules over C(K), where K is a compact Hausdorff space. One of the key concepts in the theory is the notion of Bade functionals. If X is a Banach C(K)-module and x ε X, then a Bade functional of x with respect to C(K) is a continuous linear functional α on X such that, for each a in C(K) with a ≥ 0, we have(i) α (ax) ≥0,(ii) if α (ax) = 0, then ax = 0.

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5

Kubota, Yosuke. "Notes on twisted equivariant K-theory for C*-algebras". International Journal of Mathematics 27, nr 06 (czerwiec 2016): 1650058. http://dx.doi.org/10.1142/s0129167x16500580.

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In this paper, we study a generalization of twisted (groupoid) equivariant K-theory in the sense of Freed–Moore for [Formula: see text]-graded [Formula: see text]-algebras. It is defined by using Fredholm operators on Hilbert modules with twisted representations. We compare it with another description using odd symmetries, which is a generalization of van Daele’s K-theory for [Formula: see text]-graded Banach algebras. In particular, we obtain a simple presentation of the twisted equivariant K-group when the [Formula: see text]-algebra is trivially graded. It is applied for the bulk-edge correspondence of topological insulators with CT-type symmetries.

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6

Feeman, Timothy G. "The Bourgain algebra of a nest algebra". Proceedings of the Edinburgh Mathematical Society 40, nr 1 (luty 1997): 151–66. http://dx.doi.org/10.1017/s0013091500023518.

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In analogy with a construction from function theory, we herein define right, left, and two-sided Bourgain algebras associated with an operator algebra A. These algebras are defined initially in Banach space terms, using the weak-* topology on A, and our main result is to give a completely algebraic characterization of them in the case where A is a nest algebra. Specifically, if A = alg N is a nest algebra, we show that each of the Bourgain algebras defined has the form A + K ∩ B, where B is the nest algebra corresponding to a certain subnest of N. We also characterize algebraically the second-order (and higher) Bourgain algebras of a nest algebra, showing for instance that the second-order two-sided Bourgain algebra coincides with the two-sided Bourgain algebra itself in this case.

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7

Ludkovsky, S., i B. Diarra. "Spectral integration and spectral theory for non-Archimedean Banach spaces". International Journal of Mathematics and Mathematical Sciences 31, nr 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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8

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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9

Pfaffenberger, W. E., i J. Phillips. "Commutative Gelfand Theory for Real Banach Algebras: Representations as Sections of Bundles". Canadian Journal of Mathematics 44, nr 2 (1.04.1992): 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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10

Rupp, R., i A. Sasane. "Reducibility in Aℝ(K), Cℝ(K), and A(K)". Canadian Journal of Mathematics 62, nr 3 (1.06.2010): 646–67. http://dx.doi.org/10.4153/cjm-2010-025-9.

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AbstractLet K denote a compact real symmetric subset of ℂ and let Aℝ(K) denote the real Banach algebra of all real symmetric continuous functions on K that are analytic in the interior K◦ of K, endowed with the supremum norm. We characterize all unimodular pairs ( f , g) in Aℝ(K)2 which are reducible. In addition, for an arbitrary compact K in ℂ, we give a new proof (not relying on Banach algebra theory or elementary stable rank techniques) of the fact that the Bass stable rank of A(K) is 1. Finally, we also characterize all compact real symmetric sets K such that Aℝ(K), respectively Cℝ(K), has Bass stable rank 1.

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11

Magyar, Zoltán, i Zoltán Sebestyén. "On the Definition of C*-Algebras II". Canadian Journal of Mathematics 37, nr 4 (1.08.1985): 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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12

Schochet, Claude L. "Banach algebras, Samelson products, and the Wang differential". Journal of Topology and Analysis 06, nr 02 (9.04.2014): 281–303. http://dx.doi.org/10.1142/s1793525314500113.

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Assume that given a principal G bundle ζ : P → Sk (with k ≥ 2) and a Banach algebra B upon which G acts continuously. Let [Formula: see text] denote the associated bundle and let [Formula: see text] denote the associated Banach algebra of sections. Then π* GL Aζ⊗B is determined by a mostly degenerate spectral sequence and by a Wang differential [Formula: see text] We show that if B is a C*-algebra then the differential is given explicitly in terms of an enhanced Samelson product with the clutching map of the principal bundle. Analogous results hold after localization and in the setting of topological K-theory. We illustrate our technique with a close analysis of the invariants associated to the C*-algebra of sections of the bundle [Formula: see text] constructed from the Hopf bundle ζ : S7 → S4 and by the conjugation action of S3 on M2 = M2(ℂ). We compare and contrast the information obtained from the homotopy groups π*( U ◦Aζ⊗M2), the rational homotopy groups π*( U ◦Aζ⊗M2) ⊗ ℚ and the topological K-theory groups K*(Aζ⊗M2), where U ◦B is the connected component of the unitary group of the C*-algebra B.

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13

Abel, Mati. "Dense subalgebras in noncommutative Jordan topological algebras". Acta et Commentationes Universitatis Tartuensis de Mathematica 1 (31.12.1996): 65–70. http://dx.doi.org/10.12697/acutm.1996.01.07.

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Wilansky conjectured in [12] that normed dense Q-algebras are full subalgebras of Banach algebras. Beddaa and Oudadess proved in [2] that Wilansky’s conjecture was true. They showed that k-normed Q-algebras are full subalgebras of k-Banach algebras for each k∈(0,1]. Moreover, J. Pérez, L. Rico and A. Rodríguez showed in [8], Theorem 4, that this was also true in the case of noncommutative Jordan-Banach algebras. In the present paper this problem has been studied in a more general case. It is proved that all dense Q-subalgebras of topological algebras and of noncommutative Jordan topological algebras with continuous multiplication are full subalgebras. Some equivalent conditions that a dense subalgebra would be a Q-algebra (in subspace topology) in Q-algebras and in nonassociative Jordan Q-algebras with continuous multiplication are given.

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14

Chung, Yeong Chyuan. "Quantitative K-theory for Banach algebras". Journal of Functional Analysis 274, nr 1 (styczeń 2018): 278–340. http://dx.doi.org/10.1016/j.jfa.2017.08.016.

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15

Heinrich, S., C. Ward Henson i L. C. Moore. "A note on elementary equivalence of C(K) spaces". Journal of Symbolic Logic 52, nr 2 (czerwiec 1987): 368–73. http://dx.doi.org/10.2307/2274386.

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In this paper we give a closer analysis of the elementary properties of the Banach spaces C(K), where K is a totally disconnected, compact Hausdorff space, in terms of the Boolean algebra B(K) of clopen subsets of K. In particular we sharpen a result in [4] by showing that if B(K1) and B(K2) satisfy the same sentences with ≤ n alternations of quantifiers, then the same is true of C(K1) and C(K2). As a consequence we show that for each n there exist C(K) spaces which are elementarily equivalent for sentences with ≤ n quantifier alternations, but which are not elementary equivalent in the full sense. Thus the elementary properties of Banach spaces cannot be determined by looking at sentences with a bounded number of quantifier alternations.The notion of elementary equivalence for Banach spaces which is studied here was introduced by the second author [4] and is expressed using the language of positive bounded formulas in a first-order language for Banach spaces. As was shown in [4], two Banach spaces are elementarily equivalent in this sense if and only if they have isometrically isomorphic Banach space ultrapowers (or, equivalently, isometrically isomorphic nonstandard hulls.)We consider Banach spaces over the field of real numbers. If X is such a space, Bx will denote the closed unit ball of X, Bx = {x ϵ X∣ ∣∣x∣∣ ≤ 1}. Given a compact Hausdorff space K, we let C(K) denote the Banach space of all continuous real-valued functions on K, under the supremum norm. We will especially be concerned with such spaces when K is a totally disconnected compact Hausdorff space. In that case B(K) will denote the Boolean algebra of all clopen subsets of K. We adopt the standard notation from model theory and Banach space theory.

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16

Laustsen, Niels Jakob. "K-Theory for the Banach Algebra of Operators on James's Quasi-reflexive Banach Spaces". K-Theory 23, nr 2 (czerwiec 2001): 115–27. http://dx.doi.org/10.1023/a:1017573608843.

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17

Paravicini, Walther. "Induction for Banach Algebras, Groupoids and KKban". Journal of K-theory 4, nr 3 (23.10.2009): 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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18

Paravicini, Walther. "Morita equivalences and KK-theory for Banach algebras". Journal of the Institute of Mathematics of Jussieu 8, nr 3 (15.12.2008): 565–93. http://dx.doi.org/10.1017/s1474748008000340.

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AbstractVincent Lafforgue's bivariant K-theory for Banach algebras is invariant in the second variable under a rather general notion of Morita equivalence. In particular, the ordinary topological K-theory for Banach algebras is invariant under Morita equivalences.

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19

Crabb, M. J., i C. M. McGregor. "Interpolation and inequalities for functions of exponential type: the Arens irregularity of an extremal algebra". Glasgow Mathematical Journal 35, nr 3 (wrzesień 1993): 325–26. http://dx.doi.org/10.1017/s0017089500009897.

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For any compact convex set K ⊂ ℂ there is a unital Banach algebra Ea(K) generated by an element h in which every polynomial in h attains its maximum norm over all Banach algebras subject to the numerical range V(h) being contained in K, [1]. In the case of K a line segment in ℝ, we show here that Ea(K) does not have Arens regular multiplication. We also show that ideas about Ea(K) give simple proofs of, and extend, two inequalities of C. Frappier [4].

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20

Farjoun, Emmanuel Dror, i Claude L. Schochet. "Spaces of sections of Banach algebra bundles". Journal of K-Theory 10, nr 2 (4.04.2012): 279–98. http://dx.doi.org/10.1017/is012002001jkt183.

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AbstractSuppose thatBis a G-Banach algebra over= ℝ or ℂXis a finite dimensional compact metric space, ζ :P → Xis a standard principalG-bundle, andAζ= Γ(X,P×GB) is the associated algebra of sections. We produce a spectral sequence which converges to π*(GLoAζ) withA related spectral sequence converging toK*+1(Aζ) (the real or complex topologicalK-theory) allows us to conclude that ifBis Bott-stable, (i.e., if π*(GLoB) →K*+1(B) is an isomorphism for all * > 0) then so isAζ.

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21

Yost, David. "Strictly Convex Banach Algebras". Axioms 10, nr 3 (11.09.2021): 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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22

Ozawa, Masanao. "Boolean valued interpretation of Banach space theory and module structures of von Neumann algebras". Nagoya Mathematical Journal 117 (marzec 1990): 1–36. http://dx.doi.org/10.1017/s0027763000001793.

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Recently, systematic applications of the Scott-Solovay Boolean valued set theory were done by several authors; Takeuti [25, 26, 27, 28, 29, 30], Nishimura [13, 14] Jech [8] and Ozawa [15, 16, 17, 18, 19, 20] in analysis and Smith [23], Eda [2, 3] in algebra. This approach seems to be providing us with a new and powerful machinery in analysis and algebra. In the present paper, we shall study Banach space objects in the Scott-Solovay Boolean valued universe and provide some useful transfer principles from theorems of Banach spaces to theorems of Banach modules over commutative AW*-algebras. The obtained machinery will be applied to resolve some problems concerning the module structures of von Neumann algebras.

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23

Wang, Xin, i Peng Cao. "Perturbation Theory for Quasinilpotents in Banach Algebras". Mathematics 8, nr 7 (15.07.2020): 1163. http://dx.doi.org/10.3390/math8071163.

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In this paper, we prove the following result by perturbation technique. If q is a quasinilpotent element of a Banach algebra and spectrum of p + q for any other quasinilpotent p contains at most n values then q n = 0 . Applications to C* algebras are given.

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24

Sahami, Amir, Mehdi Rostami, Seyedeh Fatemeh Shariati i Salman Babayi. "On Some Homological Properties of Hypergroup Algebras with Relation to Their Character Spaces". Journal of Mathematics 2022 (29.01.2022): 1–5. http://dx.doi.org/10.1155/2022/4939971.

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In this paper, we study the notion of approximate biprojectivity and left φ -biprojectivity of some Banach algebras, where φ is a character. Indeed, we show that approximate biprojectivity of the hypergroup algebra L 1 K implies that K is compact. Moreover, we investigate left φ -biprojectivity of certain hypergroup algebras, namely, abstract Segal algebras. As a main result, we conclude that (with some mild conditions) the abstract Segal algebra B is left φ -biprojective if and only if K is compact, where K is a hypergroup. We also study the approximate biflatness and left φ -biflatness of hypergroup algebras in terms of amenability of their related hypergroups.

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25

Van Daele, Alfons. "K-theory for graded Banach algebras. II". Pacific Journal of Mathematics 134, nr 2 (1.10.1988): 377–92. http://dx.doi.org/10.2140/pjm.1988.134.377.

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26

DAELE, A. VAN. "K-THEORY FOR GRADED BANACH ALGEBRAS I". Quarterly Journal of Mathematics 39, nr 2 (1988): 185–99. http://dx.doi.org/10.1093/qmath/39.2.185.

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27

Batkunde, Harmanus. "ALJABAR-C* KOMUTATIF". BAREKENG: Jurnal Ilmu Matematika dan Terapan 7, nr 1 (1.03.2013): 31–35. http://dx.doi.org/10.30598/barekengvol7iss1pp31-35.

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These notes in this paper will discuss about C*-algebras commutative and its properties. The theory of algebra-*, Banach-* algebra, C*-algebras and *-homomorphism are included. We also give some examples of commutative C*-algebras. We shall prove and discuss some important properties of commutative C*-algebras and *-homomorphism.

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28

Medghalchi, A. R. "Cohomology on hypergroup algebras". Studia Scientiarum Mathematicarum Hungarica 39, nr 3-4 (1.11.2002): 297–307. http://dx.doi.org/10.1556/sscmath.39.2002.3-4.4.

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There are concepts which are related to or can be formulated by homological techniques, such as derivations, multipliers and lifting problems. Moreover, a Banach algebra A is said to be amenable if H1(A,X*)=0 for every A-dual module X*. Another concept related to the theory is the concept of amenability in the sense of Johnson. A topological group G is said to be amenable if there is an invariant mean on L 8(G). Johnson has shown that a topological group is amenable if and only if the group algebra L1(G) is amenable. The aim of this research is to define the cohomology on a hypergroup algebra L(K) and extend the results of L1(G) over to L(K). At first stage it is viewed that Johnson's theorem is not valid so more. If A is a Banach algebra and h is a multiplicative linear functional on A, then (A,h) is called left amenable if for any Banach two-sided A-module X with ax=h(a)x(a? A, x? X),H1(A,X*)=0. We prove that (L(K),h) is left amenable if and only if K is left amenable. Where, the latter means that there is a left invariant mean m on C(K), i. e., m(lf)=m(f)x, where lxf(µ)=f(dx*µ). In this case we briefly say that L(K) is left amenable. Johnson also showed that L1(G) is amenable if and only if the augmentation ideal I={f? L1(G)|∫Gf=0}0 has abounded right approximate identity. We extend this result to hypergroups.

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29

Johnson, B. E. "Symmetric amenability and the nonexistence of Lie and Jordan derivations". Mathematical Proceedings of the Cambridge Philosophical Society 120, nr 3 (październik 1996): 455–73. http://dx.doi.org/10.1017/s0305004100075010.

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A. M. Sinclair has proved that if is a semisimple Banach algebra then every continuous Jordan derivation from into is a derivation ([12, theorem 3·3]; ‘Jordan derivation’ is denned in Section 6 below). If is a Banach -bimodule one can consider Jordan derivations from into and ask whether Sinclair's theorem is still true. More recent work in this area appears in [1]. Simple examples show that it cannot hold for all modules and all semisimple algebras. However, for more restricted classes of algebras, including C*-algebras one does get a positive result and we develop two approaches. The first depends on symmetric amenability, a development of the theory of amenable Banach algebras which we present here for the first time in Sections 2, 3 and 4. A Banach algebra is symmetrically amenable if it has an approximate diagonal consisting of symmetric tensors. Most, but not all, amenable Banach algebras are symmetrically amenable and one can prove results for symmetric amenability similar to those in [8] for amenability. However, unlike amenability, symmetric amenability does not seem to have a concise homological characterisation. One of our results [Theorem 6·2] is that if is symmetrically amenable then every continuous Jordan derivation into an -bimodule is a derivation. Special techniques enable this result to be extended to other algebras, for example all C*-algebras. This approach to Jordan derivations appears in Section 6.

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30

Blecher, David P. "One-sided ideals and approximate identities in operator algebras". Journal of the Australian Mathematical Society 76, nr 3 (czerwiec 2004): 425–48. http://dx.doi.org/10.1017/s1446788700009964.

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AbstractA left ideal on any C*-algebra is an example of an operator algebra with a right contractive approximate indentiy (r.c.a.i.). Indeed, left ideal in C*-algebras may be charcterized as the class of such operator algebras, which happen also to be triple systems. Conversely, we show here and in a sequel to this paper, that operator algebras with r.c.a.i. shoulod be studied in terms of a certain let ideal of a C*-algebra. We study left ideals from the perspective of ‘Hamana theory’ and using the multiplier algebras of an operator space studied elsewhere by the author. More generally, we develop some general theory for operator algebras which have a 1-sided identity or approzimate indentity, including a Banach-Stone theorem for these algebras, and an analysis of the ‘multiplier operator algebra’.

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31

Aparicio, C., F. Ocaña, R. Payá i A. Rodríguez. "A non-smooth extension of Frechet differentiability of the norm with applications to numerical ranges". Glasgow Mathematical Journal 28, nr 2 (lipiec 1986): 121–37. http://dx.doi.org/10.1017/s0017089500006443.

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The following result in the theory of numerical ranges in Banach algebras is well known (see [3, Theorem 12.2]). The numerical range of an element F in the bidual of a unital Banach algebra A is the closure of the set of values at F of the w*-continuous states of . As a consequence of the results in this paper the following

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32

Christopher Phillips, N., i Maria Grazia Viola. "Classification of spatial Lp AF algebras". International Journal of Mathematics 31, nr 13 (grudzień 2020): 2050088. http://dx.doi.org/10.1142/s0129167x20500883.

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We define spatial [Formula: see text] AF algebras for [Formula: see text], and prove the following analog of the Elliott AF algebra classification theorem. If [Formula: see text] and [Formula: see text] are spatial [Formula: see text] AF algebras, then the following are equivalent: [Formula: see text] and [Formula: see text] have isomorphic scaled preordered [Formula: see text]-groups. [Formula: see text] as rings. [Formula: see text] (not necessarily isometrically) as Banach algebras. [Formula: see text] is isometrically isomorphic to [Formula: see text] as Banach algebras. [Formula: see text] is completely isometrically isomorphic to [Formula: see text] as matricial [Formula: see text] operator algebras. As background, we develop the theory of matricial [Formula: see text] operator algebras, and show that there is a unique way to make a spatial [Formula: see text] AF algebra into a matricial [Formula: see text] operator algebra. We also show that any countable scaled Riesz group can be realized as the scaled preordered [Formula: see text]-group of a spatial [Formula: see text] AF algebra.

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33

Plaschinsky, P. "ONE FUNCTIONAL OPERATOR INVERSION FORMULA". Mathematical Modelling and Analysis 6, nr 1 (30.06.2001): 138–46. http://dx.doi.org/10.3846/13926292.2001.9637153.

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Some results about inversion formula of functional operator with generalized dilation are given. By means of commutative Banach algebra theory the explicit form of inversion operator is expressed. Some commutative Banach algebras with countable generator systems are constructed, their maximal ideal spaces are investigated.

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34

Chen, Huanyin, i Tugce Calci. "ps-drazin inverses in banach algebras". Filomat 33, nr 7 (2019): 2125–33. http://dx.doi.org/10.2298/fil1907125c.

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An element a in a Banach algebra A has ps-Drazin inverse if there exists p2 = p ? comm2(a) such that (a - p)k ? J(A) for some k ? N. Let A be a Banach algebra, and let a,b ? A have ps-Drazin inverses. If a2b = aba and b2a = bab, we prove that 1. ab ? A has ps-Drazin inverse. 2. a + b ? A has ps-Drazin inverse if and only if 1 + adb ? A has ps-Drazin inverse. As applications, we present various conditions under which a 2 x 2 matrix over a Banach algebra has ps-Drazin inverse.

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35

Javanshiri, Hossein, i Mehdi Nemati. "Amalgamated duplication of the Banach algebra 𝔄 along a 𝔄-bimodule 𝒜". Journal of Algebra and Its Applications 17, nr 09 (23.08.2018): 1850169. http://dx.doi.org/10.1142/s0219498818501694.

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Let [Formula: see text] and [Formula: see text] be Banach algebras such that [Formula: see text] is a Banach [Formula: see text]-bimodule with compatible actions. We define the product [Formula: see text], which is a strongly splitting Banach algebra extension of [Formula: see text] by [Formula: see text]. After characterization of the multiplier algebra, topological center, (maximal) ideals and spectrum of [Formula: see text], we restrict our investigation to the study of semisimplicity, regularity, Arens regularity of [Formula: see text] in relation to that of the algebras [Formula: see text], [Formula: see text] and the action of [Formula: see text] on [Formula: see text]. We also compute the first cohomology group [Formula: see text] for all [Formula: see text] as well as the first-order cyclic cohomology group [Formula: see text], where [Formula: see text] is the [Formula: see text]th dual space of [Formula: see text] when [Formula: see text] and [Formula: see text] itself when [Formula: see text]. These results are not only of interest in their own right, but also they pave the way for obtaining some new results for Lau products and module extensions of Banach algebras as well as triangular Banach algebra. Finally, special attention is devoted to the cyclic and [Formula: see text]-weak amenability of [Formula: see text]. In this context, several open questions arise.

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36

Vasylyshyn, Taras, i Kostiantyn Zhyhallo. "Entire Symmetric Functions on the Space of Essentially Bounded Integrable Functions on the Union of Lebesgue-Rohlin Spaces". Axioms 11, nr 9 (7.09.2022): 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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37

KARLOVICH, YU I. "AN ALGEBRA OF PSEUDODIFFERENTIAL OPERATORS WITH SLOWLY OSCILLATING SYMBOLS". Proceedings of the London Mathematical Society 92, nr 3 (18.04.2006): 713–61. http://dx.doi.org/10.1017/s0024611505015674.

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Let $V(\mathbb{R})$ denote the Banach algebra of absolutely continuous functions of bounded total variation on $\mathbb{R}$, and let $\mathcal{B}_p$ be the Banach algebra of bounded linear operators acting on the Lebesgue space $L^p(\mathbb{R})$ for $1 < p < \infty$. We study the Banach algebra $\mathfrak{A}\subset\mathcal{B}_p$ generated by the pseudodifferential operators of zero order with slowly oscillating $V(\mathbb{R})$-valued symbols on $\mathbb{R}$. Boundedness and compactness conditions for pseudodifferential operators with symbols in $L^\infty (\mathbb{R}, V(\mathbb{R}))$ are obtained. A symbol calculus for the non-closed algebra of pseudodifferential operators with slowly oscillating $V(\mathbb{R})$-valued symbols is constructed on the basis of an appropriate approximation of symbols by infinitely differentiable ones and by use of the techniques of oscillatory integrals. As a result, the quotient Banach algebra $\mathfrak{A}^\pi = {\mathfrak A} / \mathcal{K}$, where $\mathcal{K}$ is the ideal of compact operators in $\mathcal{B}_p$, is commutative and involutive. An isomorphism between the quotient Banach algebra $\mathfrak{A}^\pi$ of pseudodifferential operators and the Banach algebra $\widehat{\mathfrak{A}}$ of their Fredholm symbols is established. A Fredholm criterion and an index formula for the pseudodifferential operators $A \in \mathfrak{A}$ are obtained in terms of their Fredholm symbols.

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38

Kheawborisut, Araya, Siriluk Paokanta, Jedsada Senasukh i Choonkil Park. "Ulam stability of hom-ders in fuzzy Banach algebras". AIMS Mathematics 7, nr 9 (2022): 16556–68. http://dx.doi.org/10.3934/math.2022907.

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<abstract><p>This paper aims to investigate a new type of derivations in a fuzzy Banach algebra. Moreover, by using the fixed point method, we obtain some stability results of the hom-der in fuzzy Banach algebras associated with the functional equation</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ f(x+{\textbf{k}}y) = f(x)+{\textbf{k}}f(y) $\end{document} </tex-math></disp-formula></p> <p>where $ {\textbf{k}} $ is a fixed positive integer greater than $ 1 $.</p></abstract>

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39

Bridges, Douglas, Robin Havea i Peter Schuster. "Ideals in constructive Banach algebra theory". Journal of Complexity 22, nr 6 (grudzień 2006): 729–37. http://dx.doi.org/10.1016/j.jco.2006.03.004.

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40

Corach, Gustavo, i Angel R. Larotonda. "Unimodular matrices in Banach algebra theory". Proceedings of the American Mathematical Society 96, nr 3 (1.03.1986): 473. http://dx.doi.org/10.1090/s0002-9939-1986-0822443-7.

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41

Dupré, Maurice J., James F. Glazebrook i Emma Previato. "Differential Algebras with Banach-Algebra Coefficients I: from C*-Algebras to the K-Theory of the Spectral Curve". Complex Analysis and Operator Theory 7, nr 4 (17.02.2012): 739–63. http://dx.doi.org/10.1007/s11785-012-0221-2.

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42

Zhang, Yong. "Approximate complementation and its applications in studying ideals of Banach algebras". MATHEMATICA SCANDINAVICA 92, nr 2 (1.06.2003): 301. http://dx.doi.org/10.7146/math.scand.a-14407.

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We show that a subspace of a Banach space having the approximation property inherits this property if and only if it is approximately complemented in the space. For an amenable Banach algebra a closed left, right or two-sided ideal admits a bounded right, left or two-sided approximate identity if and only if it is bounded approximately complemented in the algebra. If an amenable Banach algebra has a symmetric diagonal, then a closed left (right) ideal $J$ has a right (resp. left) approximate identity $(p_{\alpha})$ such that, for every compact subset $K$ of $J$, the net $(a\cdot p_{\alpha})$ (resp. $(p_{\alpha}\cdot a)$) converges to $a$ uniformly for $a \in K$ if and only if $J$ is approximately complemented in the algebra.

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43

Feeman, Timothy G. "Nest Algebras of Operators and the Dunford-Pettis Property". Canadian Mathematical Bulletin 34, nr 2 (1.06.1991): 208–14. http://dx.doi.org/10.4153/cmb-1991-033-7.

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AbstractA Banach space X is said to have the Dunford-Pettis Property if every weakly compact linear operator T: X —> Y, where Y is any Banach space, is completely continuous (that is, T maps weakly convergent sequences to strongly convergent ones). In this paper, we prove that if A is a nest algebra of operators on a separable, infinite dimensional Hilbert space, then A fails to have the Dunford-Pettis Property. We also investigate a certain algebra associated to A, analogous to a construction used by Bourgain and others in connection with the Dunford-Pettis Property for function algebras. We show that this algebra must lie between A and the quasi-triangular algebra A + K and we give examples to show that either extreme or something in between is possible. Finally, we consider the algebra of analytic Toeplitz operators and give a result for the corresponding associated algebra which is analogous to a result of Cima, Jansen, and Yale for H∞.

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44

Stokke, Ross. "Fourier Spaces and Completely Isometric Representations of Arens Product Algebras". Canadian Journal of Mathematics 71, nr 03 (7.01.2019): 717–47. http://dx.doi.org/10.4153/cjm-2018-023-5.

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AbstractMotivated by the definition of a semigroup compactication of a locally compact group and a large collection of examples, we introduce the notion of an (operator) homogeneous left dual Banach algebra (HLDBA) over a (completely contractive) Banach algebra $A$ . We prove a Gelfand-type representation theorem showing that every HLDBA over A has a concrete realization as an (operator) homogeneous left Arens product algebra: the dual of a subspace of $A^{\ast }$ with a compatible (matrix) norm and a type of left Arens product $\Box$ . Examples include all left Arens product algebras over $A$ , but also, when $A$ is the group algebra of a locally compact group, the dual of its Fourier algebra. Beginning with any (completely) contractive (operator) $A$ -module action $Q$ on a space $X$ , we introduce the (operator) Fourier space $({\mathcal{F}}_{Q}(A^{\ast }),\Vert \cdot \Vert _{Q})$ and prove that $({\mathcal{F}}_{Q}(A^{\ast })^{\ast },\Box )$ is the unique (operator) HLDBA over $A$ for which there is a weak $^{\ast }$ -continuous completely isometric representation as completely bounded operators on $X^{\ast }$ extending the dual module representation. Applying our theory to several examples of (completely contractive) Banach algebras $A$ and module operations, we provide new characterizations of familiar HLDBAs over A and we recover, and often extend, some (completely) isometric representation theorems concerning these HLDBAs.

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45

Perov, Anatoly I., i Irina D. Kostrub. "On differential equations in Banach algebras". Russian Universities Reports. Mathematics, nr 132 (2020): 410–21. http://dx.doi.org/10.20310/2686-9667-2020-25-132-410-421.

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We consider higher-order linear differential equations with constant coefficients in Banach algebras (this is a direct generalization of higher-order matrix differential equations). The presentation is based on higher algebra, differential equations and functional analysis. The results obtained can be used in the study of matrix equations, in the theory of small oscillations in physics, and in the theory of perturbations in quantum mechanics. The presentation is based on the original research of the authors.

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46

Rennison, J. F. "The quasi-centre of a Banach algebra". Mathematical Proceedings of the Cambridge Philosophical Society 103, nr 2 (marzec 1988): 333–37. http://dx.doi.org/10.1017/s0305004100064914.

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An element a of a Banach algebra A over ࠶ will be called quasi-central if, for some K ≥ 1,The set Q(A) of all quasi-central elements of A will be called the quasi-centre of A and the set of elements a which satisfy (1) for a particular value of K will be denoted by Q(K, A). The motivation for these definitions is the result of Le Page ([1], proposition 3) that Q(1, A) coincides with the centre Z(A) of A. The reader is referred to [2] and [3] for a study of the properties of quasi-central elements.

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47

Chen, Huanyin, i Abdolyousefi Sheibani. "Cline’s formula and Jacobson’s lemma for g-Drazin inverse". Filomat 35, nr 15 (2021): 5083–91. http://dx.doi.org/10.2298/fil2115083c.

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We present new conditions under which Cline?s formula and Jacobson?s lemma for g-Drazin inverse hold. Let A be a Banach algebra, and let a,b ? A satisfying akbkak = ak+1 for some k ? N. We prove that a has g-Drazin inverse if and only if bkak has g-Drazin inverse. In this case, (bkak)d = bk(ad)2ak and ad = ak[(bkak)d]k+1. Further, we study Jacobson?s lemma for g-Drazin inverse in a Banach algebra under the preceding condition. The common spectral property of bounded linear operators on a Banach space is thereby obtained.

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48

Iwata, Yoritaka. "Theory of B(X)-Module: Algebraic Module Structure of Generally Unbounded Infinitesimal Generators". Advances in Mathematical Physics 2020 (29.11.2020): 1–27. http://dx.doi.org/10.1155/2020/3989572.

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The concept of logarithmic representation of infinitesimal generators is introduced, and it is applied to clarify the algebraic structure of bounded and unbounded infinitesimal generators. In particular, by means of the logarithmic representation, the bounded components can be extracted from generally unbounded infinitesimal generators. In conclusion, the concept of module over a Banach algebra is proposed as the generalization of the Banach algebra. As an application to mathematical physics, the rigorous formulation of a rotation group, which consists of unbounded operators being written by differential operators, is provided using the module over a Banach algebra.

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49

Laustsen, Niels Jakob. "K -Theory for Algebras of Operators on Banach Spaces". Journal of the London Mathematical Society 59, nr 2 (kwiecień 1999): 715–28. http://dx.doi.org/10.1112/s0024610799007206.

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50

HU, CHUANPU. "A GENERALIZATION OF K-THEORY FOR COMPLEX BANACH ALGEBRAS". Quarterly Journal of Mathematics 39, nr 3 (1988): 349–59. http://dx.doi.org/10.1093/qmath/39.3.349.

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