Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On Defining AW*-algebras and Rickart C*-algebras   Let A be a C*-algebra. It is shown that A is an AW*-algebra if, and only if,
each maximal abelian self--adjoint subalgebra of A is monotone complete. An
analogous result is proved for Rickart C*-algebras; a C*-algebra is a Rickart
C*-algebra if, and only if, it is unital and each maximal abelian self--adjoint
subalgebra of A is monotone {\sigma}-complete.

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The categories of this paper are: alg-geom
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The categories of this paper are: gr-qc
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The categories of this paper are: physics
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The categories of this paper are: acc-phys
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The categories of this paper are: bayes-an
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The categories of this paper are: funct-an
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The categories of this paper are: cond-mat
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The categories of this paper are: cs
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The categories of this paper are: math-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: chao-dyn
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The categories of this paper are: adap-org
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The categories of this paper are: q-fin
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The categories of this paper are: stat
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The categories of this paper are: nucl-th
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The categories of this paper are: ao-sci