Suppose that 𝒜 is a C^*-algebra. (a) Show that if 𝒜 has a unit, it is unique (call it I ); furth

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Suppose that 𝒜 is a C^*-algebra. (a) Show that if 𝒜 has a...

Suppose that $\mathscr{A}$ is a $\mathrm{C}^{*}$-algebra. (a) Show that if $\mathscr{A}$ has a unit, it is unique (call it $I$ ); furthermore $I^{*}=I$ and $\|I\|=1$ (provided $\|A\| \neq 0$ for some $A \in \mathscr{A}$ ). (b) Suppose $\mathscr{A}$ is unital. Show that if $A$ is invertible, so is $A^{*}$, with $\left(A^{*}\right)^{-1}=\left(A^{-1}\right)^{*}$. (c) Every $A \in \mathscr{A}$ can be written as $A=X+i Y$ where $X$ and $Y$ are self-adjoint. (d) If $\mathscr{A}$ is unital and $U$ is unitary (meaning $U U^{*}=U^{*} U=I$ ), then $\|U\|=1$.

Suppose that $\mathscr{A}$ is a $\mathrm{C}^{*}$-algebra.
(a) Show that if $\mathscr{A}$ has a unit, it is unique (call it $I$ ); furthermore $I^{*}=I$ and $\|I\|=1$ (provided $\|A\| \neq 0$ for some $A \in \mathscr{A}$ ).
(b) Suppose $\mathscr{A}$ is unital. Show that if $A$ is invertible, so is $A^{*}$, with $\left(A^{*}\right)^{-1}=\left(A^{-1}\right)^{*}$.
(c) Every $A \in \mathscr{A}$ can be written as $A=X+i Y$ where $X$ and $Y$ are self-adjoint.
(d) If $\mathscr{A}$ is unital and $U$ is unitary (meaning $U U^{*}=U^{*} U=I$ ), then $\|U\|=1$.

Key Concepts

Unitary Element

A unitary element in a C*-algebra is one that satisfies the condition that its product with its adjoint (in either order) equals the unit element. Unitary elements generalize the concept of rotations in finite-dimensional spaces and are characterized by having a norm equal to one. They are central to many structural aspects of C*-algebras, including symmetry and group representations.

Self-Adjoint Decomposition

Every element in a C*-algebra can be uniquely expressed as a sum of a self-adjoint element and an imaginary multiple of another self-adjoint element. This decomposition parallels expressing a complex number in terms of its real and imaginary components, enabling analysis and simplification of questions involving spectral theory and operator dynamics.

Invertible Element

An element of a C*-algebra is invertible if there exists another element (its inverse) that multiplies with it to yield the unit element. A crucial property in C*-algebras is that the adjoint of an invertible element is also invertible, and the inverse of the adjoint is the adjoint of the inverse. This relationship is instrumental in the analysis of operator algebras and their spectrum.

Unit Element

In a unital C*-algebra, the unit (or identity) element is the unique element such that multiplying any element by this unit returns the original element. This concept is fundamental since it lays the groundwork for defining invertibility. In any C*-algebra, the unit, if it exists, is self-adjoint (equal to its own adjoint) and typically has norm one, ensuring consistency within the algebraic structure.

C*-algebra

A C*-algebra is a Banach algebra equipped with an involution (a *-operation) that satisfies the C*-identity, meaning that the norm of any element squared is equal to the norm of its product with its adjoint. This structure allows for a rich theory replete with spectral, geometric, and algebraic properties, particularly important in functional analysis and quantum mechanics.

Adjoint Operation

The adjoint operation in a C*-algebra is an involution that assigns to each element a new element, generalizing the idea of a conjugate transpose. It satisfies properties like (AB)* = B*A* and (A*)* = A for any elements A and B. This operation is essential for defining self-adjoint and unitary elements, which play key roles in the structure and representation of the algebra.

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