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C*-algebra (Definition)

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

Introduction

C*-algebra has evolved as a key concept in Quantum Operator Algebra after the introduction of the von Neumann algebra for the mathematical foundation of quantum mechanics. The von Neumann algebra 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 quantum state spaces (Alfsen and Schultz, 2003 [1]) cannot be overestimated. The theory of C*-algebras has numerous applications in the theory of representations of groups and symmetric algebras, the theory of dynamical systems, statistical physics and quantum field theory, and also in the theory of operators on a Hilbert space.

Moreover, the introduction of non-commutative C*-algebras in noncommutative geometry has already played important roles in expanding the Hilbert space perspective of Quantum Mechanics developed by von Neumann. Furthermore, extended quantum symmetries are currently being approached in terms of groupoid C*- convolution algebra and their representations; the latter also enter into the construction of compact quantum groupoids as developed in the Bibliography cited, and also briefly outlined here in the second section. The fundamental connections that exist between categories of $C^*$-algebras and those of von Neumann and other 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 Banach space -with additional conditions- as defined next.

Let us consider first the definition of an involution on a complex algebra $\mathfrak{A}$.

Definition 0.1   An involution on a complex algebra $\mathfrak{A}$ is a real–linear map $T \mapsto T^*$ such that for all

$S, T \in \mathfrak{A}$ and $\lambda \in \mathbb{C}$, we have $T^{**} = T~,~ (ST)^* = T^* S^*~,~ (\lambda T)^* = \bar{\lambda} T^*~. $

A *-algebra is said to be a complex associative algebra together with an operation of involution $*$ .

C*-algebra

Definition 0.2   A C*-algebra is simultaneously a *-algebra and a Banach space $\mathfrak{A}$, satisfying for all $S, T \in \mathfrak{A}$  the following conditions:

$\begin{aligned}\Vert S \circ T \Vert &\leq \Vert S \Vert ~ \Vert T \Vert~, \\ \Vert T^* T \Vert^2 & = \Vert T\Vert^2 ~. \end{aligned}$

One can easily verify that $\Vert A^* \Vert = \Vert A \Vert$ .

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 $\mathcal L(E, F)$ of (bounded) linear operators from $E$ to $F$ forms a Banach space, where for $E=F$, the space $\mathcal L(E) = \mathcal L(E, E)$ is a Banach algebra with respect to the norm $\Vert T \Vert := \sup\{ \Vert Tu \Vert : u \in E~,~ \Vert u \Vert= 1 \}~. $ In quantum field theory one may start with a Hilbert space $H$, and consider the Banach algebra of bounded linear operators $\mathcal L(H)$ which given to be closed under the usual algebraic operations and taking adjoints, forms a $*$–algebra of bounded operators, where the adjoint operation functions as the involution, and for $T \in \mathcal L(H)$ we have :

$\Vert T \Vert := \sup\{ ( Tu , Tu): u \in H~,~ (u,u) = 1 \}~,$ and $\Vert Tu \Vert^2 = (Tu, Tu) = (u, T^*Tu) \leq \Vert T^* T \Vert~ \Vert u \Vert^2~.$

By a morphism between C*-algebras $\mathfrak{A},\mathfrak{B}$ we mean a linear map $\phi : \mathfrak{A} {\longrightarrow}\mathfrak{B}$, such that for all $S, T \in \mathfrak{A}$, the following hold : $\phi(ST) = \phi(S) \phi(T)~,~ \phi(T^*) = \phi(T)^*~, $ where a bijective morphism is said to be an isomorphism (in which case it is then an isometry). A fundamental relation is that any norm-closed $*$-algebra $\mathcal A$ in $\mathcal L(H)$ is a C*-algebra, and conversely, any C*-algebra is isomorphic to a norm–closed $*$-algebra in $\mathcal L(H)$ for some Hilbert space $H$ . One can thus also define the category $\mathcal{C}^*$ of C*-algebras and morphisms between C*-algebras.

For a C*-algebra $\mathfrak{A}$, we say that $T \in \mathfrak{A}$ is self–adjoint if $T = T^*$ . Accordingly, the self–adjoint part $\mathfrak{A}^{sa}$ of $\mathfrak{A}$ is a real vector space since we can decompose $T \in \mathfrak{A}^{sa}$ as :

$T = T' + T^{''} := \frac{1}{2} (T + T^*) + \iota (\frac{-\iota}{2})(T - T^*)~.$

A commutative C*–algebra is one for which the associative multiplication is commutative. Given a commutative C*–algebra $\mathfrak{A}$, we have $\mathfrak{A} \cong C(Y)$, 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 operator spaces in quantum theories.

Bibliography

1
E. M. Alfsen and F. W. Schultz: Geometry of State Spaces of Operator Algebras, Birkhäuser, 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).
4
I.M. Gel'fand, M.A. [M.A. Naimark] Neumark, “On the imbedding of normed rings in the rings of operators in Hilbert space” Mat. Sb. , 12 (54) : 2 (1943) pp. 197–213
4
M.A. Naimark, “Normed rings” , Reidel (1984) (Translated from Russian)
5
J. Dixmier, “ $C^*$-algebras” , North-Holland (1977) (Translated from French)
6
S. Sakai, “$C^*$-algebras and $W^*$ -algebras” , Springer (1971)
7
D. Ruelle, “Statistical mechanics: rigorous results.” , Benjamin (1974) '
8
R.G. Douglas, “Banach algebra techniques in operator theory” , Acad. Press (1972)
9
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).
10
M. R. Buneci.: Groupoid Representations, Ed. Mirton: Timishoara (2003).
11
M. Chaician and A. Demichev: Introduction to Quantum Groups, World Scientific (1996).
12
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
13
V. G. Drinfel'd: Quantum groups, In Proc. Intl. Congress of Mathematicians, Berkeley 1986, (ed. A. Gleason), Berkeley, 798-820 (1987).
14
G. J. Ellis: Higher dimensional crossed modules of algebras, J. of Pure Appl. Algebra 52 (1988), 277-282.
15
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).
16
P. I. Etingof and A. N. Varchenko: Exchange dynamical quantum groups, Commun. Math. Phys. 205 (1): 19-52 (1999)
17
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.
18
B. Fauser: A treatise on quantum Clifford Algebras. Konstanz, Habilitationsschrift. (arXiv.math.QA/0202059). (2002).
19
B. Fauser: Grade Free product Formulae from Grassman–Hopf Gebras. Ch. 18 in R. Ablamowicz, Ed., Clifford Algebras: Applications to Mathematics, Physics and Engineering, Birkhäuser: Boston, Basel and Berlin, (2004).
20
J. M. G. Fell.: The Dual Spaces of C*–Algebras., Transactions of the American Mathematical Society, 94: 365–403 (1960).
21
F.M. Fernandez and E. A. Castro.: (Lie) Algebraic Methods in Quantum Chemistry and Physics., Boca Raton: CRC Press, Inc (1996).
22
A. Fröhlich: Non–Abelian Homological Algebra. I. Derived functors and satellites, Proc. London Math. Soc., 11(3): 239–252 (1961).
23
R. Gilmore: Lie Groups, Lie Algebras and Some of Their Applications., Dover Publs., Inc.: Mineola and New York, 2005.
24
P. Hahn: Haar measure for measure groupoids, Trans. Amer. Math. Soc. 242: 1–33(1978).
25
P. Hahn: The regular representations of measure groupoids., Trans. Amer. Math. Soc. 242:34–72(1978).



"C*-algebra" is owned by bci1.

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See Also: general dynamic systems, dynamical system, groupoid C*-dynamical system

Also defines:  quantum operator algebra, groupoid, $W^*$-algebra, von Neumann algebra
Keywords:  C*-algebra, quantum operator algebra, groupoid, E. M. Alfsen and F. W. Schultz, I.M. Gel'fand, M.A. [M.A. Naimark] Neumark, J. Dixmier, von Neumann algebra, Haar system, Hopf algebra

Cross-references: operator, commutative C*--algebra, vector space, relation, isomorphism, bijective, morphism, functions, algebraic, norm, linear operators, operation, complex associative algebra, Banach space, quantum operator algebras, categories, section, compact quantum groupoids, convolution, extended quantum symmetries, noncommutative geometry, non-commutative, Hilbert space, operators, quantum field theory, dynamical systems, groups, representations, quantum state spaces, classification, quantum mechanics, concept
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This is version 14 of C*-algebra, born on 2009-01-10, modified 2009-01-29.
Object id is 369, canonical name is CstarAlgebra.
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Physics Classification: 03. (Quantum mechanics, field theories, and special relativity )
 03.65.Fd (Algebraic methods )
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