Algebraic K-theory of Normed Algebras

Inassaridze, Hvedri

doi:10.1023/a:1007885125771

Cited by 8 publications

(14 citation statements)

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“…This conjecture was confirmed by Higson for Karoubi-Villamayor algebraic K-functors [Hi] and by Suslin and Wodzicki for Quillen algebraic K-functors [SW]. Further it was proved by the first author [In1] that algebraic and topological K-functors are isomorphic for polynomial extensions of stable C * -algebras.…”

mentioning

confidence: 78%

“…This assertion is proved similarly to the particular case of polynomial extensions of C * -algebras (See [In1], Lemma 16) and is omited. 2 Corollary 6.…”

Section: Proposition 1 (I) a Is A Generalized Operator Algebramentioning

confidence: 93%

“…In effect, they are split exact, since tensoring by K, then taking the monoid algebra preserve split exactness and use Corollary 6 for the case K * . The stability follows from the Morita invariance property and the equality Theorem 11 is proved in [In1], Theorem 18, Remark 19 and Proposition 21, for particular cases of the monoid M when A is a stable C * -algebra.…”

Section: Proofmentioning

confidence: 97%

“…(1) By Corollary 6 the functor K n (−⊗K) : Pr C * → Ab, n ≥ 1, is split exact and it is standard using Morita invariance property (see [SW], [In1]) to show the stability of the functor K n (−⊗K), n ≥ 1. By Theorem 8 these functors are homotopy functors and it remains to apply Theorem A.…”

Section: Proofmentioning

confidence: 99%

“…Theorems A and B have been proved in detail for normed R-algebras in [In1], Theorems 10 and 22. The proof for topological R-algebras is completely similar and will be omited.…”

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confidence: 99%

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K-Theory of Stable Generalized Operator Algebras

Inassaridze¹,

Kandelaki²

2002

K-Theory

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Abstract. It is proved that algebraic and topological K-functors are isomorphic on the category of stable generalized operator algebras which are K i -regular for all i > 0.In 1979 Karoubi conjectured the isomorphism of algebraic and topological Kfunctors on the category of stable C * -algebras [Ka]. This conjecture was confirmed by Higson for Karoubi-Villamayor algebraic K-functors [Hi] and by Suslin and Wodzicki for Quillen algebraic K-functors [SW]. Further it was proved by the first author [In1] that algebraic and topological K-functors are isomorphic for polynomial extensions of stable C * -algebras.Our aim is to confirm Karoubi's conjecture for a wider class of complex topological algebras. Namely it is proved the equality of algebraic and topological K-functors on the category of stable generalized operator algebras (Theorems 9 and 11).Generalized operator algebras are projective limits of C * -algebras in the category of complex involutive algebras equipped with Hausdorff topology such that all algebraic operations are continuous. In our paper generalized operator algebras will be called shortly pro-C * -algebras. Such algebras have been introduced and studied by Arveson [Ar] for countable inverse systems of C * -algebras and by Weidner [We] for arbitrary inverse systems of C * -algebras.We begin by the formulation of sufficient conditions establishing the relationship between algebraic and topological K-functors on the category of real (or complex) topological algebras.Let A be a real topological algebra with unit. K n (A) denotes the Quillen algebraic K-functor for n ≥ 0 and the negative algebraic K-functor for n < 0, κ n (A), n > 0, denotes the Karoubi-Villamayor algebraic K-functor.

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confidence: 78%

“…This assertion is proved similarly to the particular case of polynomial extensions of C * -algebras (See [In1], Lemma 16) and is omited. 2 Corollary 6.…”

Section: Proposition 1 (I) a Is A Generalized Operator Algebramentioning

confidence: 93%

Section: Proofmentioning

confidence: 97%

Section: Proofmentioning

confidence: 99%

“…Theorems A and B have been proved in detail for normed R-algebras in [In1], Theorems 10 and 22. The proof for topological R-algebras is completely similar and will be omited.…”

mentioning

confidence: 99%

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K-Theory of Stable Generalized Operator Algebras

Inassaridze¹,

Kandelaki²

2002

K-Theory

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Comparison Between Algebraic and Topological K-Theory for Banach Algebras and C*-Algebras

Rosenberg¹

2005

Handbook of K-Theory

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For a Banach algebra, one can define two kinds of K-theory: topological Ktheory, which satisfies Bott periodicity, and algebraic K-theory, which usually does not. It was discovered, starting in the early 80's, that the "comparison map" from algebraic to topological K-theory is a surprisingly rich object. About the same time, it was also found that the algebraic (as opposed to topological) K-theory of operator algebras does have some direct applications in operator theory. This article will summarize what is known about these applications and the comparison map.

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To Professor Hvedri Inassaridze’s 80th birthday

Janelidze

2013

J. Homotopy Relat. Struct.

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This letter is about my former supervisor, who will see it, and I hope will forgive me for all its imperfections. It also says a few words about a remarkable event in the history of the Georgian science-the creation of the first and so far unique successful and internationally recognized Georgian school of abstract algebra. This creation was based on Inassaridze's mathematical talent, originality of thoughts, and mathematical achievements-especially in homological and homotopical algebra, algebraic topology, and algebraic K-theory. I only indicate some of these achievements, and the best way to learn about them is to read his papers and books of course; they are all listed at the end of the letter. Hvedri Inassaridze was born in Paris on 6 December, 1932. As his webpage http:// www.rmi.ge/~hvedri/ says (but I am slightly changing the wording), he studied at Lycée Claude Bernard, Paris, until 1947, and then completed his school education in Tbilisi in 1950. In the same year he became a student of Tbilisi State University, but one year later his family was sent for the so-called permanent exile to Kazakhstan. By the way, I was also there for a similar reason, but we could not have known each other since Inassaridze family was rehabilitated in 1954, when I was two years old, not to mention that Kazakhstan is very large … Hvedri continued his study at Tbilisi State University, and graduated in 1958. A PhD ("Candidate of Science") degree in those years, and especially if your supervisor was George Chogoshvili, would mean that you are an established mathematician with several substantial publications. However, Hvedri's PhD dissertation was exceptionally good even at that level-so good that it was transformed into DSc dissertation. This involved a complicated bureaucratic procedure, and as a result, he became a Candidate of Science in 1966 and a Doctor of Science in 1970. For the readers not familiar with the Eastern European academic system, I should add: the difference between

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Algebraic K-theory of Normed Algebras

Cited by 8 publications

References 4 publications

K-Theory of Stable Generalized Operator Algebras

K-Theory of Stable Generalized Operator Algebras

Comparison Between Algebraic and Topological K-Theory for Banach Algebras and C*-Algebras

To Professor Hvedri Inassaridze’s 80th birthday

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