Definition:C*-Algebra
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Definition
Let $A$ be a Banach algebra over $\C$.
Let $A$ have a conjugate-linear anti-automorphic involution $*$ satisfying the following axioms:
This article, or a section of it, needs explaining. In particular: Do we need to leave the above missing links conjugate-linear and anti-automorphic? I think the former is $(C^*2)$+$(C^*4)$, while the latter is $(C^*3)$. "Think" is inadequate. Find a source or leave it alone. I mean it looks like that this definition is incomplete because of these two missing links but in fact "conjugate-linear anti-automorphic involution" exactly means $(C^*1)$-$(C^*4)$. Only $(C^*5)$ is the really additional condition. So, the clarity of the statement might be improved. I wrote 'think' because I do not know the definition of anti-automorphic.
I am only saying that the current exposition is suboptimal. After 'satisfying the following axioms:', $(C^*1)$+$(C^*4)$ are just rephrasing the aforementioned "conjugate-linear anti-automorphic involution", then $(C^*5)$ is suddenly an additional thing. It currently looks like that all $(C^*1)$+$(C^*5)$ are additional things and the above unlinked and unexplained assumptions conjugate-linear anti-automorphic make this definition incomplete. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code. |
\((\text C^* 1)\) | $:$ | \(\ds \forall x \in A:\) | \(\ds x^{**} \) | \(\ds = \) | \(\ds x \) | ||||
\((\text C^* 2)\) | $:$ | \(\ds \forall x \in A:\) | \(\ds x^* + y^* \) | \(\ds = \) | \(\ds \paren {x + y}^* \) | ||||
\((\text C^* 3)\) | $:$ | \(\ds \forall x, y \in A:\) | \(\ds x^* y^* \) | \(\ds = \) | \(\ds \paren {y x}^* \) | ||||
\((\text C^* 4)\) | $:$ | \(\ds \forall x \in A, c \in \C:\) | \(\ds \paren {c x}^* \) | \(\ds = \) | \(\ds \overline c \paren x^* \) | ||||
\((\text C^* 5)\) | $:$ | \(\ds \forall x \in A:\) | \(\ds \norm {x x^*} \) | \(\ds = \) | \(\ds \norm x^2 \) |
Then $A$ is referred to as a $\text C^*$-algebra.
Also known as
An older term for $C^*$-algebra is $B^*$-algebra.
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): $\text B^*$-algebra
- 2017: Manfred Einsiedler and Thomas Ward: Functional Analysis, Spectral Theory, and Applications Section $11.2$: $C^*$-algebras: Definition $11.16$