Category:C*-Algebras
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This category contains results about $\text C^*$-algebras.
Definitions specific to this category can be found in Definitions/C*-Algebras.
Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a Banach $\ast$-algebra over $\C$ such that:
| \((\text C^* 5)\) | $:$ | \(\ds \forall x \in A:\) | \(\ds \norm {x x^*} \) | \(\ds = \) | \(\ds \norm x^2 \) |
Then $\struct {A, \ast, \norm {\, \cdot \,} }$ is referred to as a $\text C^*$-algebra.
That is, a $\text C^*$-algebra is a Banach algebra $\struct {A, \norm {\, \cdot \,} }$ equipped with an involution $\ast : A \to A$ satisfying:
| \((\text C^* 1)\) | $:$ | \(\ds \forall x \in A:\) | \(\ds x^{**} \) | \(\ds = \) | \(\ds x \) | ||||
| \((\text C^* 2)\) | $:$ | \(\ds \forall x \in A:\) | \(\ds x^* + y^* \) | \(\ds = \) | \(\ds \paren {x + y}^* \) | ||||
| \((\text C^* 3)\) | $:$ | \(\ds \forall x, y \in A:\) | \(\ds x^* y^* \) | \(\ds = \) | \(\ds \paren {y x}^* \) | ||||
| \((\text C^* 4)\) | $:$ | \(\ds \forall x \in A, c \in \C:\) | \(\ds \paren {c x}^* \) | \(\ds = \) | \(\ds \overline c \paren x^* \) | ||||
| \((\text C^* 5)\) | $:$ | \(\ds \forall x \in A:\) | \(\ds \norm {x x^*} \) | \(\ds = \) | \(\ds \norm x^2 \) |
We call $(\text C^\ast 5)$ the $\text C^*$ identity.
Subcategories
This category has the following 12 subcategories, out of 12 total.
Pages in category "C*-Algebras"
The following 47 pages are in this category, out of 47 total.
*
B
C
- C* Identity implies Involution is Isometry
- C*-Algebra embeds into Multiplier Algebra as C*-Subalgebra
- C*-Algebra Generated by Commutative Self-Adjoint Set is Commutative
- Character on C*-Algebra is *-Algebra Homomorphism
- Character on Unital C*-Algebra has Modulus One at Unitary Elements
- Character on Unital C*-Algebra is Real at Hermitian Elements
- Closed *-Subalgebra of C*-Algebra is C*-Algebra
- Closed Ideal of C*-Algebra is Self-Adjoint
- Complex Numbers form Unital C*-Algebra
- Convergent Series of Positive Elements of C*-Algebra is Positive
E
I
N
- Non-Zero C*-Algebra contains Non-Zero Hermitian Element
- Norm of Element of C*-Algebra as Supremum over Closed Unit Ball
- Norm of Positive Linear Functional on Unital C*-Algebra
- Norm on C*-Algebra is Unique
- Normal Element of C*-Algebra is Hermitian iff Spectrum is Real
- Normal Element of C*-Algebra is Projection iff Spectrum contains only Zero and One
- Normal Element of Unital C*-Algebra is Unitary iff Spectrum is Subset of Unit Circle
P
- Positive Linear Functional on C*-Algebra induces Semi-Inner Product
- Positive Linear Functional on C*-Algebra is Bounded
- Positive Linear Functional on C*-Algebra is Increasing on Hermitian Elements
- Positive Linear Functional on C*-Algebra is Real on Hermitian Elements
- Positive Linear Functional on C*-Algebra preserves Star
S
- Space of Bounded Linear Operators on Hilbert Space is Unital C*-Algebra
- Space of Continuous Functions Vanishing at Infinity is C*-Algebra
- Spectral Radius of Normal Element of C*-Algebra Equal to Norm
- Spectrum of Commutative C*-Algebra is Non-Empty
- Spectrum of Element of Unital C*-Subalgebra of Unital C*-Algebra
- Spectrum of Hermitian Element in Unital C*-Algebra is Real
- Spectrum of Unitary Element in Unital C*-Algebra is Subset of Unit Circle
- Sufficient Condition for C* Identity