## AW$$^*$$-algebras which are enveloping $$C^*$$-algebras of JC-algebras.(English)Zbl 1271.46054

Summary: The authors consider enveloping $$C^*$$-algebras of AJW-algebras. Conditions are given for when the enveloping $$C^*$$-algebra of an AJW-algebra is an AW$$^*$$-algebra, and corresponding theorems are proved. In particular, we prove that if $$\mathcal{A}$$ is a real AW$$^*$$-algebra, $$\mathcal{A}_{sa}$$ is the JC-algebra of all self-adjoint elements of $$\mathcal{A}$$, $$\mathcal{A}+i\mathcal{A}$$ is an AW$$^*$$-algebra and $$\mathcal{A}\cap i\mathcal{A} = \{0\}$$, then the enveloping $$C^*$$-algebra $$C^*(\mathcal{A}_{sa})$$ of the JC-algebra $$\mathcal{A}_{sa}$$ is an AW$$^*$$-algebra. Moreover, if $$\mathcal{A}+i\mathcal{A}$$ does not have nonzero direct summands of type I$$_{2}$$, then $$C^*(\mathcal{A}_{sa})$$ coincides with the algebra $$\mathcal{A}+i\mathcal{A}$$, i.e., $$C^*(\mathcal{A}_{sa})= \mathcal{A}+i\mathcal{A}$$.

### MSC:

 46L70 Nonassociative selfadjoint operator algebras 17C65 Jordan structures on Banach spaces and algebras 47L30 Abstract operator algebras on Hilbert spaces 46L35 Classifications of $$C^*$$-algebras

### Keywords:

JC-algebra; real AW$$^*$$-algebra; AW$$^*$$-algebra; AJW-algebra
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### References:

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