Commutators, commutativity and dimension in the socle of a Banach algebra: A generalized Wedderburn–Artin and Shoda's theorem

Journal of Mathematical Analysis and Applications

2016

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Let A be a complex semisimple Banach algebra with identity, and denote by σ ′ (x) and ρ(x) the nonzero spectrum and spectral radius of an element x ∈ A, respectively. We explore the relationship between elements a, b ∈ A that satisfy one of the following conditions:The latter problem was identified by Brešar andŠpenko in [7]. In particular, we use these conditions to spectrally characterize prime Banach algebras amongst the class of Banach algebras with nonzero socles, as well as to obtain spectral characterizations of socles which are minimal two-sided ideals.Problem 1. Suppose that a, b ∈ A satisfy σ(ax) = σ(bx) for all x ∈ A. Does this imply a = b?What is the relation between a and b?The first problem has been settled by G. Braatvedt and R. Brits in [5]:2010 Mathematics Subject Classification. 46H05, 46H10, 46H15, 47B10.

“…Other characterizations of such socles were recently obtained by the authors and G. Braatvedt (cf. [12, Theorem 3.8, Theorem 3.9] and [11,Theorem 4.4]).…”

Section: Introductionmentioning

confidence: 99%

Uniqueness under spectral variation in the socle of a Banach algebra

Journal of Mathematical Analysis and Applications

2016

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“…together with their properties will also be taken from [2]. Any further notation in the current paper follows [2,5,6], and, in particular, we shall denote the trace on the socle by Tr(·). Central to our discussion will be the following: Let p be a projection (idempotent) of A with rank (p) ≤ 1 (recall that rank one projections correspond to minimal projections).…”

Section: Introductionmentioning

confidence: 99%

“…Central to our discussion will be the following: Let p be a projection (idempotent) of A with rank (p) ≤ 1 (recall that rank one projections correspond to minimal projections). As in [5] we denote by J p the Theorem 1.5. [5, Theorem 3.3] Let {b, a 1 , .…”

Section: Introductionmentioning

confidence: 99%

“…The papers [5] and [6] also contain a number of characteristic properties of Shodacomplete Banach algebras, some of which will be useful for the current paper; we give a brief summary: Theorem 1.6 (Shoda-completeness). [5,6] Let A be a semisimple Banach algebra with Soc(A) = {0}. Then A is Shoda-complete if and only if any one the following conditions hold:…”

Section: Introductionmentioning

confidence: 99%

“…The socle, first introduced by J. Diuedonné over 70 years ago, is a well-known object in Banach algebras and in more general algebraic structures (rings etc.). As in the papers [5] and [6] we shall follow the relatively recent "spectral" point of view of the socle developed by Aupetit and Mouton in [2]; the associated notions of rank, trace, determinant etc. together with their properties will also be taken from [2].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Shoda-completion of a Banach algebra

Linear and Multilinear Algebra

2017

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In stark contrast to the case of finite rank operators on a Banach space, the socle of a general, complex, semisimple, and unital Banach algebra A may exhibit the 'pathological' property that not all traceless elements of the socle of A can be expressed as the commutator of two elements belonging to the socle. The aim of this paper is to show how one may develop an extension of A which removes the aforementioned problem. A naive way of achieving this is to simply embed A in the algebra of bounded linear operators on A, i.e. the natural embedding of A into L(A). But this extension is so large that it may not preserve the socle of A in the extended algebra L(A). Our proposed extension, which we shall call the Shoda-completion of A, is natural in the sense that it is small enough for the socle of A to retain the status of socle elements in the extension.2010 Mathematics Subject Classification. 15A60, 46H05, 46H10, 46H15, 47B10.

Identities, approximate identities and topological divisors of zero in Banach algebras

Journal of Mathematical Analysis and Applications

Melanie

2017

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In [3] S. J. Bhatt and H. V. Dedania exposed certain classes of Banach algebras in which every element is a topological divisor of zero. We identify a new (large) class of Banach algebras with the aforementioned property, namely, the class of non-unital Banach algebras which admits either an approximate identity or approximate units. This also leads to improvements of results by R. J. Loy and J. Wichmann, respectively. If we observe that every single example that appears in [3] belongs to the class identified in the current paper, and, moreover, that many of them are classical examples of Banach algebras with this property, then it is tempting to conjecture that the classes exposed in [3] must be contained in the class that we have identified here. However, we show somewhat elusive counterexamples. Furthermore, we investigate the role completeness plays in the results and show, by giving a suitable example, that the assumptions are not superfluous. The ideas considered here also yields a pleasing characterization: The socle of a semisimple Banach algebra is infinite-dimensional if and only if every socle-element is a topological divisor of zero in the socle.2010 Mathematics Subject Classification. 43H15, 46H05, 46H10.