Nuclear C*-algebra
In mathematics, a nuclear C*-algebra is a C*-algebra A such that the injective and projective C*-cross norms on A⊗B are the same for every C*-algebra B. This property was first studied by Takesaki (1964) under the name "Property T", which is not related to Kazhdan's property T.
Characterizations
Nuclearity admits the following equivalent characterizations:
- The identity map, as a completely positive map, approximately factors through matrix algebras. By this equivalence, nuclearity can be considered a noncommutative analogue of the existence of partitions of unity.
- The enveloping von Neumann algebra is injective.
- It is amenable as a Banach algebra.
- It is isomorphic to a C*-subalgebra B of the Cuntz algebra with the property that there exists a conditional expectation from to B. This condition is only equivalent to the others for separable C*-algebras.
gollark: Except it's addition.
gollark: <@563866872702042132> I already did the calling-is-multiplication thing.
gollark: This occurs in the final phase of potatoBIOS initialization:```luaif potatOS.registry.get "potatOS.immutable_global_scope" then setmetatable(_G, { __newindex = function(_, x) error(("cannot set _G[%q] - _G is immutable"):format(tostring(x)), 0) end })endif meta then _G.meta = meta.new() endif _G.textutilsprompt then textutils.prompt = _G.textutilsprompt endif process then process.spawn(keyboard_shortcuts, "kbsd") if http.websocket then process.spawn(skynet.listen, "skynetd") process.spawn(potatoNET, "systemd-potatod") end local autorun = potatOS.registry.get "potatOS.autorun" if type(autorun) == "string" then autorun = load(autorun) end if type(autorun) == "function" then process.spawn(autorun, "autorun") end if potatOS.registry.get "potatOS.extended_monitoring" then process.spawn(excessive_monitoring, "extended_monitoring") end if run then process.spawn(run_shell, "ushell") endelse if run then print "Warning: no process manager available. This should probably not happen - please consider reinstalling or updating. Fallback mode enabled." local ok, err = pcall(run_shell) if err then printError(err) end os.shutdown() endendwhile true do coroutine.yield() end```
gollark: (this applies after all the legacy `os.loadAPI` stuff loads anyway)
gollark: Too bad.
See also
References
- Connes, Alain (1976), "Classification of injective factors.", Annals of Mathematics, Second Series, 104 (1): 73–115, doi:10.2307/1971057, ISSN 0003-486X, JSTOR 1971057, MR 0454659
- Effros, Edward G.; Ruan, Zhong-Jin (2000), Operator spaces, London Mathematical Society Monographs. New Series, 23, The Clarendon Press Oxford University Press, ISBN 978-0-19-853482-2, MR 1793753
- Lance, E. Christopher (1982), "Tensor products and nuclear C*-algebras", Operator algebras and applications, Part I (Kingston, Ont., 1980), Proc. Sympos. Pure Math., 38, Providence, R.I.: Amer. Math. Soc., pp. 379–399, MR 0679721
- Pisier, Gilles (2003), Introduction to operator space theory, London Mathematical Society Lecture Note Series, 294, Cambridge University Press, ISBN 978-0-521-81165-1, MR 2006539
- Rørdam, M. (2002), "Classification of nuclear simple C*-algebras", Classification of nuclear C*-algebras. Entropy in operator algebras, Encyclopaedia Math. Sci., 126, Berlin, New York: Springer-Verlag, pp. 1–145, MR 1878882
- Takesaki, Masamichi (1964), "On the cross-norm of the direct product of C*-algebras", The Tohoku Mathematical Journal, Second Series, 16: 111–122, doi:10.2748/tmj/1178243737, ISSN 0040-8735, MR 0165384
- Takesaki, Masamichi (2003), "Nuclear C*-algebras", Theory of operator algebras. III, Encyclopaedia of Mathematical Sciences, 127, Berlin, New York: Springer-Verlag, pp. 153–204, ISBN 978-3-540-42913-5, MR 1943007
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