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Scientific Bulletin. Physical and Mathematical Research

Abstract

In the present paper the infinite Peirce decomposition of the algebra 𝐾(𝐻) of com-pact operators on an infinite dimensional separable Gilbert space 𝐻 is constructed, using the norm of the algebra 𝐾(𝐻) and a maximal family of mutually or-thogonal minimal projections, i.e., self-adjoint,idempotent elements. The infinite Peirce decompo-sition on the norm of a πΆβˆ—-algebra is also con-structed in 2012 by the first author. But, it turns, the condition, applied then, is not sufficient for the infi-nite Peirce decomposition on the norm constructed in 2012 to be an algebra. Therefore, in the present paper, the infinite Peirce decomposition on the norm we defined by an analog of the criterion of funda-mentality of a sequence and we proved that the infi-nite Peirce decomposition on the norm of the algebra 𝐾(𝐻) is a πΆβˆ—-algebra and a number of its properties. First, the idea of the infinite Peirce decomposition on the norm was realized in 2008 by the firs author.

Also, the present paper is devoted to the de-scription of local derivations on algebras compact operators. The history of local derivations and local automorphisms begins with the Gleason-Kahane-Ε»elazko theorem proved in 1967-1968. This theo-rem is a fundamental contribution in the theory of Banach algebras. This theorem asserts that every unital linear functional 𝐹 on a complex unital Ba-nach algebra 𝐴, such that 𝐹(π‘Ž) belongs to the spec-trum 𝜎(π‘Ž) of π‘Ž, for every π‘Žβˆˆπ΄, is multiplicative. In modern terminology this is equivalent to the fol-lowing condition: every unital linear local homo-morphism from a unital complex Banach algebra 𝐴 into 𝐂 is multiplicative. We recall that a linear map 𝑇 from a Banach algebra 𝐴 into a Banach algebra 𝐡 is said to be a local homomorphism if for every π‘Ž in 𝐴 there exists a homomorphism ο†π‘Žπ΄β†’π΅, depend-ing on π‘Ž, such that 𝑇(π‘Ž)=ο†π‘Ž(π‘Ž).

Later, in 1990, R. Kadison introduces the con-cept of local derivation and proves that each contin-uous local derivation from a von Neumann algebra into its dual Banach bemodule is a derivation. B. Jonson in 2001 extends the above result by proving that every local derivation from a C*-algebra into its Banach bimodule is a derivation. In particular, John-son gives an automatic continuity result by proving that local derivations of a C*-algebra 𝐴 into a Ba-nach 𝐴-bimodule 𝑋 are continuous even if not as-sumed a priori to be so. Based on these results, many authors have studied local derivations on operator algebras.

In the present paper, automorphisms and local automorphisms of compact operator algebras are studied. Recall that a bijective linear mapping  on an algebra 𝐴, satisfying, for each pair π‘₯, 𝑦 of ele-ments in 𝐴, (π‘₯𝑦)=(π‘₯)(𝑦), is called an auto-morphism. A linear mapping βˆ‡ on the algebra 𝐴 is called a local automorphism if, for every element π‘₯ in the algebra 𝐴, there exists an automorphism :𝐴→𝐴 such that βˆ‡(π‘₯)=(π‘₯).

In the present paper, every continuous bijective local automorphism on the infinite Peirce decompo-sition Ξ£π‘π‘–π‘œπ‘–π‘—π΅(𝐻)𝑝𝑗 on the norm of the algebra 𝐾(𝐻) of compact operators on an infinite dimen-sional separable Gilbert space 𝐻, is, weather an au-tomorphism, or antiautomorphism.

First Page

70

Last Page

75

DOI

517.98

References

[1] Arzikulov F.N. Infinite order and norm decompositions of C*-algebras. Int Journal of Math Analysis Vol. 2. No 5, 255-262. (2008)

[2] Arzikulov F.N. Infinite norm decompositions of C*-algebras. Operator Theory: Advances and Applications Springer Ba-sel AG. Vol 220, 11-21. (2012)

[3] Gleason A. A characterization of maximal ideals. J. Analyse Math. 19 (1967), 171-172.

[4] Johnson B. Local derivations on C*-algebras are derivations. Trans. Amer. Math. Soc. 353 (2001), 313-325.

[5] Kadison R. Local derivations. J. Algebra 130 (1990), 494-509.

[6] Kahane J., Ε»elazko W. A characterization of maximal ideals in commutative Banach algebras. Studia Math. 29 (1968), 339-343.

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