PlanetPhysics/Cstar Algebra

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C*- and von Neumann algebras: Quantum operator algebra in quantum theories

Introduction

C*-algebra has evolved as a key [[../PreciseIdea/|concept]] in Quantum Operator Algebra after the introduction of the von Neumann algebra for the mathematical foundation of [[../QuantumParadox/|quantum mechanics]]. The von Neumann algebra [[../TrivialGroupoid/|classification]] is simpler and studied in greater depth than that of general C*-algebra classification theory.

The importance of C*-algebras for understanding the geometry of [[../QuantumSpinNetworkFunctor2/|quantum state spaces]] (Alfsen and Schultz, 2003 [1]) cannot be overestimated. The theory of C*-algebras has numerous applications in the theory of [[../CategoricalGroupRepresentation/|representations]] of [[../TrivialGroupoid/|groups]] and symmetric algebras, the theory of [[../ContinuousGroupoidHomomorphism/|dynamical systems]], statistical physics and [[../SpaceTimeQuantizationInQuantumGravityTheories/|quantum field theory]], and also in the theory of [[../QuantumOperatorAlgebra4/|operators]] on a [[../NormInducedByInnerProduct/|Hilbert space]].

Moreover, the introduction of [[../AbelianCategory3/|non-commutative]] C*-algebras in [[../NoncommutativeGeometry4/|noncommutative geometry]] has already played important roles in expanding the Hilbert space perspective of Quantum Mechanics developed by von Neumann. Furthermore, [[../ExtendedQuantumSymmetries/|extended quantum symmetries]] are currently being approached in terms of groupoid C*- [[../AssociatedGroupoidAlgebraRepresentations/|convolution]] algebra and their representations; the latter also enter into the construction of [[../QuantumCompactGroupoids/|compact quantum groupoids]] as developed in the Bibliography cited, and also briefly outlined here in the second [[../IsomorphicObjectsUnderAnIsomorphism/|section]]. The fundamental connections that exist between [[../Cod/|categories]] of C*-algebras and those of von Neumann and other [[../Groupoid/|quantum operator algebras]], such as JB- or JBL- algebras are yet to be completed and are the subject of in depth studies [1].

Basic definitions

A C*-algebra is simultaneously a *--algebra and a [[../NormInducedByInnerProduct/|Banach space]] -with additional conditions- as defined next.

Let us consider first the definition of an involution on a complex algebra 𝔄.

An involution on a complex algebra 𝔄 is a real--linear map T↦T* such that for all

S,T∈𝔄 and Failed to parse (unknown function "\bC"): {\displaystyle \lambda \in \bC} , we have T**=T,(ST)*=T*S*,(λT)*=λΒ―T*.

A *-algebra is said to be a [[../OrthomodularLatticeTheory/|complex associative algebra]] together with an [[../Cod/|operation]] of involution *~.

C*-algebra

A C*-algebra is simultaneously a *-algebra and a Banach space 𝔄, satisfying for all S,T∈𝔄~ the following conditions:

Failed to parse (syntax error): {\displaystyle \Vert S \circ T \Vert &\leq \Vert S \Vert ~ \Vert T \Vert~, \\ \Vert T^* T \Vert^2 & = \Vert T\Vert^2 ~. }

One can easily verify that ‖A*‖=‖A‖~.

By the above axioms a C*--algebra is a special case of a Banach algebra where the latter requires the above C*-norm property, but not the involution (*) property.

Given Banach spaces E,F the space β„’(E,F) of (bounded) [[../Commutator/|linear operators]] from E to F forms a Banach space, where for E=F, the space β„’(E)=β„’(E,E) is a Banach algebra with respect to the [[../NormInducedByInnerProduct/|norm]] \bigbreak ‖T‖:=sup⁡{‖Tu‖:u∈E,‖u‖=1}. \bigbreak In quantum field theory one may start with a Hilbert space H, and consider the Banach algebra of bounded linear operators β„’(H) which given to be closed under the usual [[../CoIntersections/|algebraic]] operations and taking adjoints, forms a *--algebra of bounded operators, where the adjoint operation [[../Bijective/|functions]] as the involution, and for T∈β„’(H) we have~:

‖T‖:=sup⁡{(Tu,Tu):u∈H,(u,u)=1}, and </math> \Vert Tu \Vert^2 = (Tu, Tu) = (u, T^*Tu) \leq \Vert T^* T \Vert~ \Vert u \Vert^2~.Failed to parse (unknown function "\htmladdnormallink"): {\displaystyle By a ''\htmladdnormallink{morphism'' {http://planetphysics.us/encyclopedia/TrivialGroupoid.html} between C*-algebras} <math>\mathfrak A,\mathfrak B} we mean a linear map </math>\phi : \mathfrak A \lra \mathfrak B,suchthatforallS, T \in \mathfrak AFailed to parse (unknown function "\bigbreak"): {\displaystyle , the following hold~: \bigbreak <math>\phi(ST) = \phi(S) \phi(T)~,~ \phi(T^*) = \phi(T)^*~, } \bigbreak where a [[../Bijective/|bijective]] morphism is said to be an [[../IsomorphicObjectsUnderAnIsomorphism/|isomorphism]] (in which case it is then an isometry). A fundamental [[../Bijective/|relation]] is that any norm-closed *-algebra π’œ in β„’(H) is a C*-algebra, and conversely, any C*-algebra is isomorphic to a norm--closed *-algebra in β„’(H) for some Hilbert space H~. One can thus also define the category Failed to parse (syntax error): {\displaystyle \mathcal{C ^*} of C*-algebras and morphisms between C*-algebras}.

For a C*-algebra 𝔄, we say that T∈𝔄 is self--adjoint if </math>T = T^*.Accordingly,theself−−adjointpart\mathfrak A^{sa}of\mathfrak Aisareal[[../NormInducedByInnerProduct/|vectorspace]]sincewecandecompose<math>T∈𝔄sa as ~:

T=T′+T'′:=12(T+T*)+ι(−ι2)(T−T*).

A commutative C*--algebra is one for which the associative multiplication is commutative. Given a [[../OrthomodularLatticeTheory/|commutative C*--algebra]] 𝔄, we have 𝔄≅C(Y), the algebra of continuous functions on a compact Hausdorff space Y.

The classification of {C*-algebras} is far more complex than that of von Neumann algebras that provide the fundamental algebraic content of quantum state and [[../QuantumSpinNetworkFunctor2/|operator]] spaces in quantum theories.

All Sources

[1] [2] Cite error: The opening <ref> tag is malformed or has a bad name Cite error: The opening <ref> tag is malformed or has a bad name [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

References

  1. ↑ 1.0 1.1 1.2 E. M. Alfsen and F. W. Schultz: Geometry of State Spaces of Operator Algebras , Birkh\"auser, Boston--Basel--Berlin (2003).
  2. ↑ I. Baianu : Categories, Functors and Automata Theory: A Novel Approach to Quantum Automata through Algebraic--Topological Quantum Computations., Proceed. 4th Intl. Congress LMPS , (August-Sept. 1971).
  3. ↑ J. Dixmier, " C*-algebras" , North-Holland (1977) (Translated from French)
  4. ↑ S. Sakai, "C*-algebras and W* -algebras" , Springer (1971)
  5. ↑ D. Ruelle, "Statistical mechanics: rigorous results." , Benjamin (1974) '
  6. ↑ R.G. Douglas, "Banach algebra techniques in operator theory" , Acad. Press (1972)
  7. ↑ I. C. Baianu, J. F. Glazebrook and R. Brown.: A Non--Abelian, Categorical Ontology of Spacetimes and Quantum Gravity., Axiomathes 17 ,(3-4): 353-408(2007).
  8. ↑ M. R. Buneci.: Groupoid Representations , Ed. Mirton: Timishoara (2003).
  9. ↑ M. Chaician and A. Demichev: Introduction to Quantum Groups , World Scientific (1996).
  10. ↑ W. Drechsler and P. A. Tuckey: On quantum and parallel transport in a Hilbert bundle over spacetime., Classical and Quantum Gravity , 13 :611-632 (1996). doi: 10.1088/0264--9381/13/4/004
  11. ↑ V. G. Drinfel'd: Quantum groups, In Proc. Intl. Congress of Mathematicians, Berkeley 1986 , (ed. A. Gleason), Berkeley, 798-820 (1987).
  12. ↑ G. J. Ellis: Higher dimensional crossed modules of algebras, J. of Pure Appl. Algebra 52 (1988), 277-282.
  13. ↑ P.. I. Etingof and A. N. Varchenko, Solutions of the Quantum Dynamical Yang-Baxter Equation and Dynamical Quantum Groups, Comm.Math.Phys. , 196 : 591-640 (1998).
  14. ↑ P. I. Etingof and A. N. Varchenko: Exchange dynamical quantum groups, Commun. Math. Phys. 205 (1): 19-52 (1999)
  15. ↑ P. I. Etingof and O. Schiffmann: Lectures on the dynamical Yang--Baxter equations, in Quantum Groups and Lie Theory (Durham, 1999) , pp. 89-129, Cambridge University Press, Cambridge, 2001.
  16. ↑ B. Fauser: A treatise on quantum Clifford Algebras . Konstanz, Habilitationsschrift. (arXiv.math.QA/0202059). (2002).
  17. ↑ B. Fauser: Grade Free product Formulae from Grassman--Hopf Gebras. Ch. 18 in R. Ablamowicz, Ed., Clifford Algebras: Applications to Mathematics, Physics and Engineering , Birkh\"{a}user: Boston, Basel and Berlin, (2004).
  18. ↑ J. M. G. Fell.: The Dual Spaces of C*--Algebras., Transactions of the American Mathematical Society , 94 : 365--403 (1960).
  19. ↑ F.M. Fernandez and E. A. Castro.: (Lie) Algebraic Methods in Quantum Chemistry and Physics. , Boca Raton: CRC Press, Inc (1996).
  20. ↑ A.~Fr{\"o}hlich: Non--Abelian Homological Algebra. {I}. {D}erived functors and satellites, Proc. London Math. Soc. , 11 (3): 239--252 (1961).
  21. ↑ R. Gilmore: Lie Groups, Lie Algebras and Some of Their Applications. , Dover Publs., Inc.: Mineola and New York, 2005.
  22. ↑ P. Hahn: Haar measure for measure groupoids, Trans. Amer. Math. Soc . 242 : 1--33(1978).
  23. ↑ P. Hahn: The regular representations of measure groupoids., Trans. Amer. Math. Soc . 242 :34--72(1978).

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