Automorphism Of The C ∗ C^* C ∗ -algebra C ( T ) C(\mathbb{T}) C ( T )

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Introduction to C*-algebras and Automorphisms

In functional analysis, C-algebras* hold a pivotal role as they provide an abstract framework for studying operators on Hilbert spaces. These algebras, equipped with a norm and an involution operation, capture essential properties of operators, making them invaluable in both pure mathematics and applications to quantum mechanics. To fully appreciate the automorphisms of the specific C*-algebra C(T), we must first understand the foundational aspects of C*-algebras and automorphisms in a broader context.

A C-algebra* is essentially a Banach algebra A over the complex numbers, equipped with an involution x → x**, which is an anti-linear map satisfying (xy) = yx* and (x*)* = x for all x, y in A. Furthermore, it satisfies the crucial C* identity: ||xx*|| = ||x||² for all x in A. This identity links the algebraic structure (multiplication and involution) with the analytic structure (norm), making C*-algebras particularly rigid and well-behaved. The quintessential example of a C*-algebra is the algebra B(H) of all bounded linear operators on a Hilbert space H, with the involution being the adjoint operation. Subalgebras of B(H) that are closed in the norm topology and closed under the adjoint operation are also C*-algebras.

An automorphism of a C*-algebra A is an isomorphism from A onto itself. In simpler terms, it is a bijective map α: A → A that preserves the algebraic structure (addition, multiplication, and involution) and the scalar multiplication. Specifically, for all x, y in A and complex numbers λ, an automorphism α must satisfy:

  1. α(x + y) = α(x) + α(y)
  2. α(xy) = α(x)α(y)
  3. α(λx) = λ**α(x)
  4. α(x*) = α(x)*

Additionally, since we are dealing with Banach algebras, we require α to be continuous. A fundamental result in functional analysis ensures that any algebraic isomorphism between C*-algebras is automatically continuous, making this a natural requirement.

Automorphisms are crucial for understanding the symmetries and structural properties of C*-algebras. They allow us to transform the algebra while preserving its fundamental characteristics. The set of all automorphisms of a C*-algebra A forms a group, denoted by Aut(A), under the operation of composition. Studying this group provides deep insights into the structure of A.

In the context of C*-algebras, automorphisms can arise from various sources. For instance, if A is the algebra B(H) of bounded operators on a Hilbert space H, then any unitary operator U in B(H) gives rise to an automorphism α by conjugation: α(T) = UTU* for all T in B(H). Such automorphisms are called inner automorphisms. However, not all automorphisms are inner, and the outer automorphisms (those that are not inner) often reveal more subtle aspects of the algebra's structure.

The C*-algebra C(T) and its Significance

To deeply understand the automorphisms of the C*-algebra C(T), it is essential to first explore the nature and significance of C(T) itself. The algebra C(T), representing continuous functions on the unit circle, serves as a fundamental example in the theory of C*-algebras, bridging topological concepts with algebraic structures.

The C*-algebra C(T) is formally defined as the set of all continuous complex-valued functions on the unit circle T in the complex plane. The unit circle T can be represented as the set z ∈ C |z| = 1. The algebraic operations in C(T) are pointwise addition and multiplication of functions, and scalar multiplication is also defined pointwise. The involution operation f → f** is given by complex conjugation, i.e., f**(z) = f(z) for all z in T, where the overline denotes the complex conjugate. The norm in C(T) is the supremum norm, defined as ||f|| = sup|f(z)| z ∈ T for f ∈ C(T).

The C*-identity, ||ff***|| = ||f||², holds in C(T) due to the properties of continuous functions and complex conjugation. Indeed, (ff***)(z) = f(z)f(z), and taking the supremum over T yields the desired identity. This makes C(T) a quintessential example of a commutative C*-algebra.

Significance of C(T) in C*-algebra Theory

C(T) holds a special place in the landscape of C*-algebras due to several reasons:

  1. Commutativity: C(T) is a commutative C*-algebra, which means that for any functions f, g in C(T), fg = gf. Commutative C*-algebras are significantly simpler to analyze than their non-commutative counterparts, and C(T) serves as a prototypical example for understanding the structure of commutative C*-algebras.

  2. Gelfand Representation: The Gelfand representation theorem provides a profound connection between commutative C*-algebras and topological spaces. Specifically, it states that every commutative C*-algebra is isometrically -isomorphic to the algebra of continuous functions on its Gelfand spectrum, which is the space of maximal ideals equipped with the weak topology. In the case of C(T), the Gelfand spectrum is homeomorphic to the unit circle T itself, making C(T) a natural realization of this abstract theory.

  3. Universal Properties: C(T) possesses several universal properties that make it a fundamental building block in C*-algebra theory. For example, it is the universal C*-algebra generated by a unitary element. This means that any C*-algebra containing a unitary element can be mapped homomorphically into C(T), providing a powerful tool for studying unitary operators and their associated algebras.

  4. Applications in Harmonic Analysis: C(T) is intimately connected with harmonic analysis on the circle. The Fourier series of a function in C(T) provides a decomposition into basic oscillatory components, and the study of these Fourier series has deep connections with the representation theory of the circle group. The algebraic structure of C(T) provides a natural framework for studying these harmonic properties.

  5. Topological Interpretation: The unit circle T is a fundamental topological space, and the algebra C(T) captures many of its topological properties. For example, the homotopy groups of C(T) are closely related to the homotopy groups of T, and the K-theory of C(T) encodes topological information about vector bundles over T. This interplay between topology and algebra makes C(T) a rich and rewarding object of study.

Defining the Automorphism α on C(T)

Now, let's delve into the specific automorphism α on the C*-algebra C(T), which is the central focus of our discussion. Understanding this automorphism requires a precise definition and a clear grasp of its action on functions within C(T). This automorphism is defined through a rotation on the unit circle, offering a geometric interpretation that is both intuitive and mathematically rigorous.

We define the automorphism α: C(T) → C(T) by its action on a function f in C(T). For a fixed real number θ, the automorphism α is defined as:

α(f)(z) = f(e^(-2πiθ)z)

for all z in the unit circle T. In this definition, e^(-2πiθ) represents a rotation in the complex plane by an angle of -2πθ radians. Thus, α(f)(z) evaluates the function f at a point on the unit circle that has been rotated by this angle. This geometric interpretation is crucial for understanding the properties and behavior of α.

Verification of α as an Automorphism

To ensure that α is indeed an automorphism, we need to verify that it satisfies all the necessary properties, including being a *-isomorphism and bijective. Let's break down the verification step by step:

  1. Linearity: For any functions f, g in C(T) and complex number λ, we need to show that α(f + λg) = α(f) + λ**α(g). By definition,

    α(f + λg)(z) = (f + λg)(e^(-2πiθ)z) = f(e^(-2πiθ)z) + λg(e^(-2πiθ)z) = α(f)(z) + λ**α(g)(z)

    This holds for all z in T, so α(f + λg) = α(f) + λ**α(g), confirming linearity.

  2. Multiplicativity: We need to show that α(fg) = α(f)α(g). By definition,

    α(fg)(z) = (fg)(e^(-2πiθ)z) = f(e(-2πiθ)z)g(e(-2πiθ)z) = α(f)(z)α(g)(z)

    This holds for all z in T, so α(fg) = α(f)α(g), confirming multiplicativity.

  3. Preservation of Involution: We need to show that α(f*) = α(f)*. By definition,

    α(f*)(z) = f*(e^(-2πiθ)z) = f(e^(-2πiθ)z) = α(f)(z)

    This holds for all z in T, so α(f*) = α(f)*, confirming the preservation of involution.

  4. Bijectivity: To show that α is bijective, we need to demonstrate that it is both injective (one-to-one) and surjective (onto).

    • Injectivity: Suppose α(f) = 0, which means α(f)(z) = f(e^(-2πiθ)z) = 0 for all z in T. Since the map z → e^(-2πiθ)z is a bijection on T, it follows that f(z) = 0 for all z in T, so f = 0. Thus, α is injective.
    • Surjectivity: For any g in C(T), we need to find an f in C(T) such that α(f) = g. Define f(z) = g(e^(2πiθ)z). Then, α(f)(z) = f(e^(-2πiθ)z) = g(e(2πiθ)e(-2πiθ)z) = g(z) for all z in T. Thus, α(f) = g, and α is surjective.

Since α is linear, multiplicative, preserves the involution, and is bijective, it is indeed a *-isomorphism. Moreover, since it maps C(T) onto itself, it is an automorphism.

Geometric Interpretation

The geometric interpretation of α provides valuable intuition. The map α effectively rotates the argument of the function f by an angle of -2πθ. This rotation preserves the continuity of the function and its algebraic properties, making it a natural automorphism to consider. The parameter θ controls the amount of rotation, and different values of θ will yield different automorphisms. Understanding this geometric action is key to exploring the broader properties of automorphisms on C(T).

Analyzing Properties and Implications of the Automorphism

Now that we have established the definition and validity of the automorphism α on C(T), we can delve into analyzing its properties and implications. Understanding these aspects provides deeper insights into the structure of C(T) and the nature of its automorphisms. Key properties include its continuity, its action on specific functions, and its relationship to the group of automorphisms of C(T).

Continuity of α

Since α is an automorphism of a C*-algebra, it is automatically continuous. However, it is instructive to directly verify its continuity using the supremum norm. Recall that the norm in C(T) is defined as ||f|| = sup|f(z)| z ∈ T. To show that α is continuous, we need to demonstrate that it is bounded, i.e., there exists a constant M > 0 such that ||α(f)|| ≤ M||f|| for all f in C(T).

Consider the norm of α(f):

||α(f)|| = sup|α(f)(z)| z ∈ T = sup|f(e^(-2πiθ)z)| z ∈ T

Since |f(e^(-2πiθ)z)| is simply evaluating the absolute value of f at a rotated point on the unit circle, and the supremum is taken over all points on the unit circle, we have:

sup|f(e^(-2πiθ)z)| z ∈ T = sup|f(w)| w ∈ T = ||f||

Thus, ||α(f)|| = ||f||, which means that α is an isometry (a norm-preserving map). This implies that α is continuous with a bound M = 1. The fact that α is an isometry is a strong property, indicating that it preserves distances in C(T).

Action on Specific Functions

To further understand α, it is helpful to examine its action on specific functions within C(T). A particularly important class of functions to consider are the monomials, z^n, where n is an integer. These functions play a fundamental role in the Fourier analysis on the circle.

Let f_n(z) = z^n for z in T and n ∈ Z (the set of integers). Then,

α(f_n)(z) = f_n(e^(-2πiθ)z) = (e(-2πiθ)z)n = e(-2πinθ)zn = e^(-2πinθ)f_n(z)

This shows that α maps the monomial z^n to a scalar multiple of itself, specifically e(-2πinθ)zn. This scalar factor represents a phase shift. The monomials z^n form an orthonormal basis for the Hilbert space L²(T), and their linear span is dense in C(T) by the Stone-Weierstrass theorem. Therefore, understanding the action of α on these monomials is crucial for understanding its action on the entire algebra C(T).

Furthermore, this calculation reveals that the monomials z^n are eigenvectors for the operator α, with eigenvalues e^(-2πinθ). This spectral information provides a deeper understanding of the automorphism's structure.

Relationship to the Automorphism Group of C(T)

The set of all automorphisms of C(T) forms a group under the operation of composition, denoted by Aut(C(T)). The automorphisms of the form α we have defined constitute a subgroup of Aut(C(T)). To see this, consider two automorphisms α_θ₁ and α_θ₂ defined by rotations θ₁ and θ₂, respectively:

α_θ₁(f)(z) = f(e^(-2πiθ₁)z) α_θ₂(f)(z) = f(e^(-2πiθ₂)z)

The composition of these automorphisms is:

(α_θ₁ ◦ α_θ₂)(f)(z) = α_θ₁(α_θ₂(f))(z) = α_θ₂(f)(e^(-2πiθ₁)z) = f(e(-2πiθ₂)(e(-2πiθ₁)z)) = f(e^(-2πi(θ₁ + θ₂))z)

This shows that the composition α_θ₁ ◦ α_θ₂ is an automorphism corresponding to the rotation θ₁ + θ₂. Thus, the set of automorphisms α_θ θ ∈ R forms a group under composition, which is isomorphic to the additive group of real numbers modulo 1 (i.e., the circle group R/ Z).

This subgroup represents a continuous family of automorphisms, parameterized by the rotation angle θ. However, it does not exhaust all possible automorphisms of C(T). There are other types of automorphisms, such as those induced by reflections or more general homeomorphisms of the unit circle. The full automorphism group Aut(C(T)) is much larger and more complex, but the subgroup of rotations provides a fundamental starting point for its analysis.

Implications and Further Explorations

The analysis of the automorphism α and its properties has several important implications and opens avenues for further explorations:

  1. Dynamical Systems: The automorphism α can be viewed as a dynamical system on C(T), where iterates of α describe the evolution of functions under repeated rotations. This perspective connects C*-algebra theory with the study of dynamical systems, providing a powerful framework for analyzing long-term behavior and invariant structures.

  2. Invariant Subalgebras: Investigating the subalgebras of C(T) that are invariant under α provides insights into the structure of C(T) and the properties of α. For example, the fixed-point subalgebra (the set of functions f such that α(f) = f) corresponds to functions that are invariant under the rotation, which can reveal symmetries and patterns in C(T).

  3. Crossed Products: The crossed product C*-algebra C(T) ⋊ Z (where Z represents the integers) associated with the action of α is a fundamental construction in C*-algebra theory. This crossed product captures the interplay between C(T) and the automorphism α, and its structure reflects the dynamical properties of the system. Studying crossed products provides a powerful tool for analyzing non-commutative C*-algebras.

  4. Generalizations: The automorphism α is a specific example of a more general class of automorphisms induced by homeomorphisms of the unit circle. Exploring automorphisms induced by other types of transformations can lead to a deeper understanding of the automorphism group Aut(C(T)) and its structure.

Conclusion

The study of automorphisms of C*-algebras, particularly the C*-algebra C(T), provides a rich landscape for exploring connections between functional analysis, topology, and algebra. The automorphism α defined by α(f)(z) = f(e^(-2πiθ)z) serves as a fundamental example, illustrating how rotations on the unit circle induce automorphisms on C(T). Through detailed analysis, we have verified that α is indeed an automorphism, examined its continuity, elucidated its action on monomials, and placed it within the context of the automorphism group of C(T).

The automorphism α encapsulates the essence of how transformations on the underlying space (the unit circle T) translate into transformations on the algebra of functions defined on that space. This interplay between geometry and algebra is a recurring theme in the study of C*-algebras, and C(T) provides a concrete and accessible setting for exploring these ideas.

Furthermore, the analysis of α opens doors to more advanced topics, such as dynamical systems, invariant subalgebras, crossed products, and generalizations to other automorphisms. These avenues of exploration underscore the significance of studying automorphisms in the broader context of C*-algebra theory.

In summary, the automorphism α on C(T) is not just a mathematical construct; it is a gateway to understanding the profound connections between algebraic structures and geometric transformations. By studying such examples, we gain deeper insights into the world of C*-algebras and their applications in various fields of mathematics and physics.