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Analogs of Cuntz algebras on Lp spaces

Banff International Research Station for Mathematical Innovation and Discovery

Graph algebras have been intensively studied in the C* and purely algebraic contexts. We propose another context: Banach algebras of operators on Lp spaces. We report on results on the analogs in this context of Cuntz algebras and UHF algebras. (The analogs of UHF algebras are needed for the proofs of some of the results on the analogs of Cuntz algebras.) For p ∈ [1, ∞) and d ∈ {2, 3, . . .}, we identify a “good” analog Odp of the Cuntz algebra Od which acts on spaces of the form Lp(X,μ). We prove uniqueness and simplicity of Odp. Unlike the C* case, these two results seem to be independent, and our proofs of the two results are entirely unrelated. We further prove that Odp is purely infinite and amenable as a Banach algebra, and we compute its topological K-theory, getting the expected answer. We prove that for d1,d2 ∈ {2,3,...} and for p1 ≠ p2, there is no nonzero continuous homomorphism from Op1 to Op2. We leave a number of problems open. In particular, we do pd1d2pp p not know whether the L spatial tensor product O2 ⊗p O2 is isomorphic to O2 for p ̸= 2. We prove that the “good” Lp analogs of UHF algebras are unique in a suitable sense, simple, amenable, have a unique continuous trace, and have the expected K-theory. We show that for each p and each supernatural number, there are uncountably many mutually nonisomorphic Lp UHF algebras with the same supernatural number and having all the properties given above except uniqueness and amenability. This talk will be primarily a survey, but will explain some of the key ideas.

Mathematics; Functional analysis; Associative rings and algebras; Operator theory/algebras

video/mp4

Author affiliation: University of Oregon

2014-08-07

Attribution-NonCommercial-NoDerivs 2.5 Canada

http://hdl.handle.net/2429/49469

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/2.5/ca/