$$K$$ -Groups of a $$C^{*}$$ -Algebra Generated by a Single Operator
Tóm tắt
In this paper, we compute
$$K$$
-groups
$$\{K_{n}(C^{*}(x))\}_{n=0}^{\infty }$$
of the
$$C^{*}$$
-subalgebra
$$C^{*}(x)$$
of
$$B(H),$$
generated by a single operator
$$x,$$
where
$$H$$
is a separable infinite dimensional Hilbert space, and
$$B(H)$$
is the operator algebra consisting of all (bounded linear) operators on
$$H.$$
These computations not only provide nice examples in
$$K$$
-theory, but also characterize-and-classify projections in a
$$C^{*}$$
-algebra generated by a single operator. The main result of this paper shows that: the
$$K$$
-groups of
$$C^{*}(x)$$
are completely characterized by those of
$$C^{*}(q),$$
where
$$q$$
is the positive-operator part of
$$x$$
in the polar decomposition of
$$x.$$
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