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Professor Herbert Kamowitz andCohomology of Banach Algebras
Zinaida Lykova
Newcastle University
Banach algebras and its applications, 15th July 2019
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Herbert M. Kamowitz, 1932 – 2018
Figure: Photo courtesy of Levine Chapels
He was a professor of mathematics at the University ofMassachusetts, Boston.
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PhD thesis, 1960
Kamowitz wrote his PhD thesis “Cohomology Groups ofCommutative Banach Algebra” under the supervision of Prof.John Wermer in Brown University in 1960.
The main results from his thesis were published inTransactions of the American Mathematical Society, 102(1962), No. 2, 352-372.
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Professor John Wermer
Figure: from Yiddish Book Center
Prof. Wermer is a mathematician specializing in complexanalysis. He received his Ph.D. from Harvard University in1951.
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“Cohomology groups of commutative Banach
algebras”
Kamowitz wrote: “The aim of this paper is to extend thecohomology theory of Hochschild to commutative Banachalgebras and to investigate some consequences that mayderived from this extension.”
G. Hochschild, On the cohomology groups of an associativealgebra, Ann. of Math. 46 (1945), 58–67.
The cohomology groups of a Banach algebra with coefficientsin a Banach bimodule are defined in a similar way toHochschild cohomology of an abstract algebra, except thatone takes the topology into account, and so all cochains andcoboundary operators are continuous.
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Main results - I
Kamowitz used continuous Hochschild cohomology groupsH2(A,X ) of commutative Banach algebras A with coefficientsin Banach A-bimodules X as a tool for studying the splittingof extensions of commutative Banach algebras.
Theorem 1
Let A be a commutative Banach algebra, let B be a (notnecessarily commutative) Banach algebra with the radical Rsatisfying B/R = A. Suppose R2 = 0 and R is finitedimensional. If H2(A,X ) = 0 for all finite dimensionalBanach A-bimodules X , then there exists a closed subalgebraA′ of B with B = A′ ⊕ R . If H1(A,X ) = 0 for all finitedimensional Banach A-bimodules X , and ifB = A1 ⊕ R = A2 ⊕ R , then A1 and A2 are quasi-innerautomorphic, that is, there exists an automorphism σ of A1
onto A2 and r ∈ R such that σ(a) = a− ar + ra for all a ∈ A1.
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Main results - II
Theorem 2
Let Ω be a compact Hausdorff space, let C (Ω) be a Banachalgebra of complex-valued continuous functions on Ω, and letX be Banach C (Ω)-bimodule. If X is finite dimensional, thenH1(C (Ω),X ) = 0 and H2(C (Ω),X ) = 0.
He wrote:“However it is an open question whetherH2(C (Ω),X ) = 0 for all Banach C (Ω)-bimodules”.
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Topological homology, 1966-1972
A. Guichardet, Homology and cohomology of Banachalgebras, C. R. Acad. Sci. Paris Ser. A 262 (1966), 38 - 41.(France)
In 1970-1972 several very influential papers appeared inEngland, Moscow and USA on continuous cohomology ofBanach and topological algebras.
Richard V. Kadison and John R. Ringrose, Cohomologyof operator algebras: I. Type I von Neumann algebras, ActaMath., Volume 126 (1971), 227– 243.
Barry Johnson, Cohomology in Banach algebras, Memoirs ofthe American Mathematical Society, 127 (1972), Providence,R.I.: American Mathematical Society.
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Topological homology, 1972
Alexander Ya. Helemskii, Global dimension of a Banachfunction algebra is different from one, Funktsional. Anal. iPrilozhen., 6 (1972), Issue 2, 95–96.
Joseph L. Taylor, Homology and cohomology for topologicalalgebras, Advances in Maths 9 (1972), 137-182 andA general framework for a multi-operator functional calculus,Advances in Maths 9 (1972), 183-252.
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Some other books and papers on topological
homology
Alexander Ya. Helemskii, The homology of Banach andtopological algebras, Mathematics and its Applications (SovietSeries) 1986; 41(1989), Dordrecht: Kluwer AcademicPublishers Group.
Cyclic cohomology was introduced by A. Connes in 1981.His original motivation came from operator theory. AlainConnes. Noncommutative differential geometry. Inst. HautesEtudes Sci. Publ. Math., 62 (1985) 257–360.
William Bade, Garth Dales and Zinaida Lykova,Algebraic and strong splittings of extensions of Banachalgebras, Memoirs of the American Mathematical Society, 137(1999), 113pp.
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Fereidoun Ghahramani about Herbert Kamowitz,
June 2018
Herb will be fondly remembered; he was a kind man and it wasalways pleasure for me to co-participate with him inconferences.
Mathematically, I owe it to him and Stephan Scheinberg forthe significant ground-work that they laid in their 1969 paperon Volterra algebra that finally resulted in the resolution of thequestion on connectedness of the automorphism group of theVolterra algebra. I believe he was also one of the pioneers inthe cohomology of Banach algebras that later became animportant component of research in Banach algebras theory.Herbert Kamowitz and Stephen Scheinberg, Derivationsand Automorphisms of L1(0, 1), Transactions of the AmericanMathematical Society Vol. 135 (1969), 415-427.
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Herbert Kamowitz and Joel and Uta Feinstein
Figure: Photo courtesy of Margaret JohnsonZinaida Lykova Herbert Kamowitz and Cohomology of Banach algebras
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Wieslaw Zelazko, Herbert Kamowitz, George
Willis
Figure: Photo courtesy of Margaret Johnson
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Thank you
Zinaida Lykova Herbert Kamowitz and Cohomology of Banach algebras
